Economics of the Market

• Lebed is the first minor ever prosecuted for stock-market fraud.

• Lebed tools were
  • an America Online (AOL) internet connection,
  • an E*trade account, and
  • four email accounts in Yahoo Finance Message Boards.
• SEC accused him of making his money through a pump and dump strategy.

• Lebed forfeited \($285000\) in profit and interest he had made on \(11\) trades.

• He has never admitted any wrongdoing.
• He kept close to \($500000\) in profit.

People who trade stocks, trade based on what they feel will move and they can trade for profit. Nobody makes investment decisions based on reading financial filings. Whether a company is making millions or losing millions, it has no impact on the price of the stock. Whether it is analysts, brokers, advisors, Internet traders, or the companies, everybody is manipulating the market. If it wasn't for everybody manipulating the market, there wouldn't be a stock market at all…

(Jonathan Lebed, statement to his lawyer, Lewis, 2001)

• Economics studies the interactions of economic entities; namely firms, households, governments, and other organizations.

• How do these agents interact?
• A market is a coordination mechanism conveying information via prices among economic entities to organize production and allocate output.

• Physical markets are markets that manifest in specific geographical locations. Non physical markets are markets that are not physical. (e.g., a supermarket vs. a labor market.)
• Retail markets are final goods and services markets sold for consumption. Business markets are final or intermediate goods and services markets sold from one firm to another. (e.g., a fish market vs. a tractor market.)
• Financial markets are markets in which financial instruments and contracts are traded. Non financial markets are markets in which commodities and services are traded. (e.g., a stock exchange vs. a grocery store.)
• Authorized markets are markets where the trade of commodities occurs via channels authorized by the manufacturers. Unauthorized markets (or gray markets) are markets with channels that the manufacturer does not recognize. (e.g., US version video games market in the US vs. Japanese version video games in the US)
• Legal markets are markets in which commodities and services are traded according to the governing set of rules. Illegal markets (or black markets) are markets in which transactions have some aspect of non compliance with the governing rules. (e.g., car market vs. illegal drug market)

• How inclusively a market is defined entails a level of arbitrariness.
• An electric automobile sold in Frankfurt belongs in
  • Frankfurt's automobile market
  • Hessen's automobile market
  • Germany's automobile market
  • Germany's electric automobile market
  • European automobile market
• The definition of what a market includes is crucial in legal cases concerning competition law.

• Coordination mechanisms based on prices do not always exist.
• Some commodities and services have special characteristics that
  • either prevent efficient coordination based on prices,
  • or even lead to a complete production shutdown!

• A market failure is a situation in which the price-based coordination of economic agents is impossible or inefficient.

• Looking within the firm (Economics of the Firm):
  • A firm is treated as an administrative entity in which a team of managers is needed to coordinate and monitor its activities and the division of labor.
  • How do managers and workers interact?
• Looking across firms (This course):
  • A firm becomes a pool of know-how, physical, and financial capital.
  • It uses its resources to produce output.
  • For-profit firms are the primary instruments in capitalist economies for producing and distributing commodities and services.

• Firms may compete by choosing prices.
• They may follow price discrimination strategies.
• They may compete by choosing quantities.
• They may differentiate their products to stand out.

1. Consider a market with demand \(d(p) = 100 - 4 p\) and supply \(s(p) = 40 + 2p\).

   1. Calculate the demanded and supplied quantities for \(p=2\). Does the market clear? Does it have a shortage or a surplus?
   2. Calculate the market clearing price.
   1. For \(p=2\), we have \(d(2) = 92\) and \(s(2) = 44\). The market does not clear because \(d(2)>s(2)\). It has a shortage equal to \(d(2) - s(2) = 48\)
   2. The market clearing price is obtained by setting \(d(p) = s(p)\) and solving for \(p\). This gives \(p=10\).

2. Consider a market with demand \(d(p) = 250/p\) and supply \(s(p) = 10p\).

   1. Calculate the market clearing price (assume that the price cannot be negative).
   2. Calculate the price elasticity of demand.
   3. Calculate the price elasticity of supply.
   1. By \(d(p) = s(p)\), we get \(p=5\). The assumption of the exercise excludes the negative root.
   2. The elasticity of demand is given by \[e_{d}(p) = d'(p) \frac{p}{d(p)} = -\frac{250}{p^{2}}\frac{p^{2}}{250} = -1.\] Demand elasticity is constant for all prices.
   3. The elasticity of supply is given by \[e_{s}(p) = s'(p) \frac{p}{s(p)} = 10\frac{p}{10 p} = 1.\] Supply elasticity is also constant for all prices.

3. Consider a market with demand \(d(p) = 22 - 3p\) and supply \(s(p) = 10 + p\).

   1. Calculate the market clearing price.
   2. Calculate the price elasticity of demand at the market clearing price.
   3. Calculate the price elasticity of supply at the market clearing price.
   1. We calculate \(p=3\).
   2. The elasticity of demand at \(p=3\) is \[e_{d}(3) = \left. -3 \frac{p}{22 - 3p} \right|_{p=3} = -\frac{9}{13}.\]
   3. The elasticity of supply at \(p=3\) is \[e_{s}(3) = \left. \frac{p}{10 + p} \right|_{p=3} = \frac{3}{13}.\]

1. Consider a market with demand \(d(p) = \alpha - \beta p\) and supply \(s(p) = \gamma + \delta p\), where all Greek letters are positive parameters.

   1. Calculate the market clearing price.
   2. Calculate the price elasticity of demand at the market clearing price.
   3. Calculate the price elasticity of supply at the market clearing price.
   1. The market clearing price is given by \(d(p) = s(p)\), or equivalently \[\alpha - \beta p = \gamma + \delta p.\] We can solve for prices to get \[p_{m} = \frac{\alpha - \gamma}{\delta + \beta}.\]
   2. The elasticity of demand at \(p_{m}\) is \[e_{d}(p_{m}) = -\beta \frac{1}{\frac{\alpha}{p_{m}} - \beta} = -\beta\frac{\alpha-\gamma}{\alpha\delta + \beta\gamma}.\]
   3. The elasticity of supply at \(p_{m}\) is \[e_{s}(p_{m}) = \delta \frac{1}{\frac{\gamma}{p_{m}} + \delta} = \delta\frac{\alpha-\gamma}{\alpha\delta + \beta\gamma}.\]

• Textile production has two main sub-processes:
  • Spinning
  • Weaving

• Weaving productivity increased \(\rightarrow\)
• Cloth production increased \(\rightarrow\)
• Yarn demand increased \(\rightarrow\)
• Spinsters could not keep up! \(\rightarrow\)
• The new technology changed the allocation of capital and labor in textile production!
• (And Spinning jenny was invented a bit later)

• A production function is monotonic if output increases whenever the input factors increase.
• A production technology exhibits the free disposal property if inputs can be discarded at no cost.

• If \(f(L) = 5 \sqrt{L}\),
• then \(\mathrm{MP}(L) = f'(L) = \frac{5}{2} \frac{1}{\sqrt{L}}\)

• Suppose that \(f(K, L) = 10 K^{\alpha}L^{\beta}\).
• Then
  • \(\alpha+\beta > 1\),
  • decreasing returns to scale if \(\alpha+\beta < 1\), and
  • constant returns to scale if \(\alpha+\beta = 1\).

• Suppose that \(f(K, L) = K^{0.4}L^{0.6}\).
• Then \(f\) has
  • \(\mathrm{MP}(K;L) = \frac{\partial f (K,L)}{\partial K} = 0.4 K^{-0.6}L^{0.6}\),
  • \(\mathrm{MP}(L;K) = \frac{\partial f (K,L)}{\partial L} = 0.6 K^{0.4}L^{-0.4}\), and
  • \(\mathrm{MRTS}(K,L) = \frac{\mathrm{MP}(K;L)}{\mathrm{MP}(L;K)} = \frac{2}{3}\frac{L}{K}\).

Suppose that some production technology induces a production set \(\mathcal{Q}\). For single output production technologies, the production function is defined as the maximum amount of output that is feasible for a given combination of input factors. Thus, with one factor \[f(x) = \max \left\{ q \ge 0 \colon \left(x, q \right) \in \mathcal{Q} \right\},\] and with two factors \[f(x_{1}, x_{2}) = \max \left\{ q \ge 0 \colon \left(x_{1}, x_{2}, q \right) \in \mathcal{Q} \right\}.\]

Suppose that the original production function has two factors, namely \(f_{1}(x_{1}, x_{2})\). Due to some physical or legal constraint, the firm cannot change \(x_{2}\), which is kept constant equal to \(\hat{x_{2}}\). This can happen if the time frame of interest is too short to allow the firm to adjust \(x_{2}\). The resulting production is \[ f_{2}(x_{1}) := f_{1}(x_{1}, \hat{x_{2}}). \]

The marginal product is the rate of change of the production output with respect to an input factor. For single-dimensional factor spaces, by the definition of the derivative, we have \[f'(x_{0}) = \lim_{x \to x_{0}} \frac{f(x) - f(x_{0})}{x-x_{0}}.\] For multidimensional factor spaces, by the definition of the partial derivative, we have \[\frac{\partial f}{\partial x_{0}}(x_{0}, x_{1}) = f_{x_{i}} = \lim_{x \to x_{0}} \frac{f(x, x_{1}) - f(x_{0}, x_{1})}{x-x_{0}}.\]

The marginal product of factor \(x_{i}\) is diminishing when \(f_{x_{i}}\) is decreasing. If \(f_{x_{i}}\) is differentiable, then the marginal product of factor \(x_{i}\) is diminishing if and only if \(f_{x_{i}x_{i}} \le 0\). For a single factor production function, the last condition is true if and only if the function is concave.

For a fixed level of output, say \(q = f(x_{1}, x_{2})\), we can think of the implicit mapping \[g(x_{1}) = \left\{ x_{2} \in \mathbb{R} \colon \ q = f(x_{1}, x_{2}) \right\}.\] This mapping gives us the isoquant for \(q\). Under some invertibility and smoothness conditions that most commonly used production functions satisfy, the isoquant is differentiable. The opposite of its derivative typically denoted as \[-\frac{\mathrm{d} x_{2}}{\mathrm{d} x_{1}} = \mathrm{MRTS}(x_{1}, x_{2}) = \frac{\partial f(x_{1}, x_{2}) / \partial x_{1}}{\partial f(x_{1}, x_{2}) / \partial x_{2}},\] is the rate of change of \(x_{2}\) with respect to \(x_{1}\) given that production is kept constant at a level \(q\). This rate of change is called the marginal rate of technical substitution.

The elasticity of substitution is defined by \[e \left(x_{2}, x_{1}\right) = \left(\frac{\mathrm{d}\, \ln \mathrm{MRTS}\left(x_{1}, x_{2}\right)}{\mathrm{d}\, \ln \left(x_{2}/x_{1}\right) }\right)^{-1}.\] It constitutes a measure of the substitutability between input factors in a production function.

1. The following graph plots the isoquants of a production function for various levels of output. Determine the regions where the production function exhibits constant, increasing, and decreasing returns to scale.

   Initial q	Final q	Input scale	Final q / Initial q	Returns to scale
   3	9	2	3	Increasing
   9	18	1.5	2	Increasing
   18	24	1.3333333	1.3333333	Constant
   24	30	1.25	1.25	Constant
   30	36	1.2	1.2	Constant
   36	40	1.1666667	1.1111111	Decreasing
   40	42	1.1428571	1.05	Decreasing

2. Consider the family of production sets \[ \mathcal{Q}^{r} =\left\{\left(x, q \right) \in \mathbb{R}^2_{\ge 0} \colon 0 \le q \le x^{r} \right\} \quad\quad (0 < r < \infty). \]

   1. Plot in a single graph the production functions for \(0< r_{1} < 1 = r_{2} < r_{3}\).
   2. Sketch the production set for a fixed \(r < 1\).
   3. Sketch the production set for a fixed \(r > 1\).
   4. For which values of \(r\) the marginal product of the production function corresponding to \(\mathcal{Q}^{r}\) is diminishing / constant / increasing?
   1. The production function corresponding to \(\mathcal{Q}^{r}\) is \(f(x) = x^{r}\).
   2. For \(r < 1\), the production set is convex.
   3. For \(r > 1\), the production set is not convex.
   4. Given any \(r>0\), the marginal product of the production function is \(f'(x) = r x^{r - 1}\).
      • If \(r < 1\), then \(f''(x) = r (r - 1) x^{r - 2} \le 0\), which implies that the marginal product of \(f\) is decreasing.
      • If \(r = 1\), then \(f''(x) = r (r - 1) x^{r - 2} = 0\), which implies that the marginal product of \(f\) is constant.
      • If \(r > 1\), then \(f''(x) = r (r - 1) x^{r - 2} \ge 0\), which implies that the marginal product of \(f\) is increasing.

1. Consider the production function \(f(x_{1}, x_{2}) = x_{1}^{\alpha} x_{2}^{\beta}\) for parameters \(\alpha, \beta > 0\). For which values of \(\alpha\) and \(\beta\) does \(f\) exhibit decreasing / constant / increasing returns to scale?

   For every \(t>1\), we have

   \begin{align*} f(t x_{1}, t x_{2}) &= \left(t x_{1}\right)^{\alpha} \left(t x_{2}\right)^{\beta} \\ &= t^{\alpha + \beta} x_{1}^{\alpha} x_{2}^{\beta} \\ &= t^{\alpha + \beta} f(x_{1}, x_{2}). \end{align*}

   • If \(\alpha + \beta > 1\), then \(t^{\alpha + \beta} > t\) and \(f(t x_{1}, t x_{2}) > t f(x_{1}, x_{2})\), which means that \(f\) exhibits increasing returns to scale.
   • If \(\alpha + \beta < 1\), then \(t^{\alpha + \beta} < t\) and \(f(t x_{1}, t x_{2}) < t f(x_{1}, x_{2})\), which means that \(f\) exhibits decreasing returns to scale.
   • If \(\alpha + \beta = 1\), then \(t^{\alpha + \beta} = t\) and \(f(t x_{1}, t x_{2}) = t f(x_{1}, x_{2})\), which means that \(f\) exhibits constant returns to scale.

1. Show that the elasticity of the constant elasticity production function is constant and equal to \((1 - \rho)^{-1}\).

   The marginal product of the constant elasticity production function with respect to the first input factor is

   \begin{align*} \frac{\partial f\left(x_{1}, x_{2}\right)}{\partial x_{1}} &= A \frac{r}{\rho} \left(\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}\right)^{\frac{r}{\rho}-1} \alpha \rho x_{1}^{\rho-1} \\ &= \frac{f\left(x_{1}, x_{2}\right)}{\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}} \alpha r x_{1}^{\rho-1}. \end{align*}

   Analogously, the marginal product with respect to the second input factor is

   \begin{align*} \frac{\partial f\left(x_{1}, x_{2}\right)}{\partial x_{2}} &= \frac{f\left(x_{1}, x_{2}\right)}{\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}} \left(1 - \alpha\right) r x_{2}^{\rho-1}. \end{align*}

   One can then calculate the marginal rate of technical substitution by

   \begin{align*} \mathrm{MRTS}(x_{1}, x_{2}) &= \frac{\partial f\left(x_{1}, x_{2}\right) / \partial x_{1}}{\partial f\left(x_{1}, x_{2}\right) / \partial x_{2}} \\ &= \frac{\alpha}{1-\alpha} \left(\frac{x_{1}}{x_{2}}\right)^{\rho-1}, \end{align*}

   and, thus,

   \begin{align*} \ln \mathrm{MRTS}(x_{1}, x_{2}) &= \ln\frac{\alpha}{1-\alpha} + (\rho-1) \ln\frac{x_{1}}{x_{2}}. \end{align*}

   Then, we have

   \begin{align*} \frac{\mathrm{d}\, \ln \mathrm{MRTS}\left(x_{1}, x_{2}\right)}{\mathrm{d}\, \ln \left(x_{2}/x_{1}\right) } = 1 - \rho, \end{align*}

   which implies that

   \begin{align*} e \left(x_{2}, x_{1}\right) = \frac{1}{1 - \rho}. \end{align*}

2. Consider the constant elasticity of substitution function \[ f\left(x_{1}, x_{2}\right) = A \left(\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}\right)^{\frac{r}{\rho}}. \]

   1. Show that for \(\rho\to -\infty\) \(f\) converges to the fixed proportion production function \(f\left(x_{1}, x_{2}\right) \to A \min\left\{x_{1}, x_{2}\right\}^{r}\).
   2. Show that for \(\rho=1\) \(f\) reduces to the perfect substitutes production function \(f\left(x_{1}, x_{2}\right) = A \left(\alpha x_{1} + \left(1 - \alpha\right) x_{2}\right)^{r}\).
   3. Show that for \(\rho\to 0\) \(f\) converges to the Cobb-Douglas production function \(f\left(x_{1}, x_{2}\right) \to A x_{1}^{\alpha r} x_{2}^{\left(1 - \alpha\right)r}\).

   1. Suppose that \(x_{1} = \min\left\{x_{1}, x_{2}\right\}\). For every \(\rho<0\), we can rewrite the constant elasticity of substitution function as \[ f\left(x_{1}, x_{2}\right) = A x_{1}^{r}\left(\alpha + \left(1 - \alpha\right) \left(\frac{x_{2}}{x_{1}}\right)^{\rho}\right)^{\frac{r}{\rho}}. \] Since \(x_{2}/x_{1} > 1\), we have \(0 \le (x_{2}/x_{1})^{\rho} \le 1\) for all \(\rho<0\). Hence \[ A x_{1}^{r}\alpha^{\frac{r}{\rho}} \le f\left(x_{1}, x_{2}\right) \le A x_{1}^{r}, \] which implies \[ A x_{1}^{r} = \lim_{\rho \to -\infty} A x_{1}^{r}\alpha^{\frac{r}{\rho}} \le \lim_{\rho \to -\infty} f\left(x_{1}, x_{2}\right) \le \lim_{\rho \to -\infty} A x_{1}^{r} = A x_{1}^{r}. \]

      When \(x_{2} = \min\left\{x_{1}, x_{2}\right\}\), a similar argument shows that \[ \lim_{\rho \to -\infty} f\left(x_{1}, x_{2}\right) = A x_{2}^{r}. \]

      We can combine the two cases by writing \[ \lim_{\rho \to -\infty} f\left(x_{1}, x_{2}\right) = A \min\left\{x_{1}, x_{2}\right\}^{r}. \]

   2. The result is obtained by substituting \(\rho =1\).

   3. We can rewrite \[ f\left(x_{1}, x_{2}\right) = A \exp\left(r \frac{\ln\left(\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}\right)}{\rho}\right). \] By L'Hospital's

      \begin{align*} \lim_{\rho \to 0} \frac{\ln\left( \alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho} \right)}{\rho} &= \lim_{\rho \to 0} \frac{\alpha x_{1}^{\rho} \ln x_{1} + \left(1 - \alpha\right) x_{2}^{\rho} \ln x_{2}}{\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}} \\ &= \alpha \ln x_{1} + \left(1 - \alpha\right) \ln x_{2} \\ &= \ln x_{1}^{\alpha} + \ln x_{2}^{\left(1 - \alpha\right)}. \end{align*}

      By the continuous mapping theorem, one can then conclude

      \begin{align*} \lim_{\rho \to 0} f\left(x_{1}, x_{2}\right) &= A \exp\left(r \ln x_{1}^{\alpha} + r\ln x_{2}^{\left(1 - \alpha\right)} \right) \\ &= A x_{1}^{\alpha r} x_{2}^{\left(1 - \alpha\right)r}. \end{align*}

• One shareholder is the National Center for Public Policy Research (NCPPR) investor group.
  • A conservative think tank.
  • Does not own much stock in Apple (not in the top stockholders).
• Apple had many programs targeting environmental sustainability.
• NCPPR pushed a proposal about disclosing the costs of such sustainability programs.
  • Argued for more transparency in actions towards the "amorphous concept of environmental sustainability".
  • Rejected by Apple shareholders, with only \(3\%\) voting in favor.

• In practice, only chasing profits can lead to tunnel vision.
• Short-sighted strategies do not always have the desired effect in the long-run.
• Sometimes companies pursue other strategic goals (e.g., social and environmental responsibility).
• Antiwar campaigns, environmental programs, and free software updates might not be profitable.
• However, they help create a customer ecosystem and build customer loyalty.
• This, in turn, can lead to sustainable, greater profits in the long-run.
• Nonetheless, ignoring profits for too long can lead to insolvency!
• Where is the golden ratio? Finding it distinguishes good and bad managers!

How much from each factor would a profit maximizing firm like to use?

• We check the factor demand function.

• The factor demand is a function that gives the profit maximizing input factor quantity for given input and output prices.

For which price level is an input factor choice profit maximizing?

• We check the inverse factor demand function.

• The inverse factor demand is a function that gives the input price factor for which a factor choice maximizes profits.

• If we find combinations of inputs, outputs, and prices for which the resulting inequality does not hold, then the firm was not maximizing profits on at least one of the dates.
• If all combinations of inputs, outputs, and prices in our data satisfy the resulting inequality, then we cannot reject the possibility of profit maximizing behavior.

• If the prices of input factors remain constant (i.e., \(\Delta r = \Delta w = 0\)), then

  \begin{align*} \Delta p \Delta q \ge 0. \end{align*}
• Changes in output quantities should have the same sign as changes in output prices.

• If the output price and the price of an input factor remain constant (say \(\Delta p = \Delta w = 0\)), then

  \begin{align*} \Delta r \Delta K \le 0. \end{align*}
• Changes in the price of an input factor should have the opposite sign of changes in the corresponding demanded factor quantities.

The general profit function of a \(m\text{-output}\), \(n\text{-input}\) production process is given by \[\pi\left(q_{1},... , q_{m}, x_{1},..., x_{n}\right) = \underbrace{\sum_{i=1}^m p_{i}q_{i}}_{Revenue} - \underbrace{\sum_{i=1}^n w_{i}x_{i}}_{Cost}.\]

The isoprofit loci give the combinations of inputs and outputs resulting in a given level of profit \(\hat \pi\), i.e., \[\left\{ (q, x) \in \mathbb{R}^{m}\times\mathbb{R}^{n} \colon\ \hat \pi = \sum_{i=1}^m p_{i}q_{i} - \sum_{i=1}^n w_{i}x_{i} \right\}.\] For the single output, single input case (\(m=1=n\)), we obtain the isoprofit line in the output-input space by solving for \(q\). Thus, \[q = \frac{\hat \pi}{p} + \frac{w}{p}x .\]

If a factor is fixed, then there is a fixed cost component. E.g., say that \(x_{j}\) is fixed equal to \(\hat x_{j}\). Then \[\pi = \sum_{i=1}^n p_{i}q_{i} - \underbrace{\sum_{i\neq j} w_{i}x_{i}}_{Variable\ cost} - \underbrace{w_{j}\hat x_{j}}_{fixed\ cost}.\]

The firm maximizes its profit by choosing the level of the input factor

Because profits are increasing in \(q\), if the production function is monotonic (the usual case), the problem simplifies to \[\max_{x} \left\{ p f(x) - w x \right\}.\] For non-boundary solutions, the marginal rates of revenue and cost are equalized at the maximum (why?), namely \[p f'(x) = w.\]

The firm maximizes its profit by choosing the levels of the input factors

Because profit is increasing in \(q\), as long as the production function is monotonic (the usual case), the problem simplifies to \[\max_{x_{1}, x_{2}} \left\{ p f(x_{1}, x_{2}) - w_{1} x_{1} - w_{2} x_{2} \right\}.\] For non-boundary solutions, the marginal rates of revenue and cost are equalized for each factor at the maximum, namely

Combining the first order conditions, we see that the marginal rate of technical substitution is equal to relative prices at any interior maximum, i.e., \[\mathrm{MRTS}(x_{1}, x_{2}) = \frac{\partial f(x_{1}, x_{2}) / \partial x_{1}}{\partial f(x_{1}, x_{2}) / \partial x_{2}} = \frac{w_{1}}{w_{2}}.\]

Suppose that \((x_{1}, x_{2})\) maximizes profits under a constant returns to scales production technology. Then, for any \(t>1\),

If \(\pi\left(x_{1}, x_{2}\right)\) is positive, then \(t \pi\left(x_{1}, x_{2}\right) > \pi\left(x_{1}, x_{2}\right)\). Since \((x_{1}, x_{2})\) is profit maximizing, \(\pi\left(x_{1}, x_{2}\right)\) cannot be positive. Repeating the argument with \(0 < t < 1\), we can exclude the possibility of negative profits. Therefore, only zero maximizing profits are compatible with constant returns to scales.

1. Suppose that a firm produces output \(q\) using only labor \(L\) according to the production function \(q=\sqrt{L}\). The price of output is \(p=2\), and the price of labor is \(w=1\). Find the profit maximizing labor and the corresponding output. What is the optimal profit?

   Maximize \(\pi(L) = 2\sqrt{L} - L\) to find \(L=1\) and \(q=1\). Then, the optimal profit is \(\pi=1\).

2. Suppose that a firm produces output \(q\) using labor \(L\) and capital \(K\) according to the production function \(q=K^{1/4}L^{1/4}\). The price of output is \(p=2\), the wage is \(w=1\), and the interest rate is \(r=1\).

   1. What happens to output when capital and labor are doubled?
   2. Find the profit maximizing labor, capital, and output.
   3. Find the maximum profits.
   4. Is there a unique maximizing output if the production is given by \(q=K^{1/2}L^{1/2}\)? Why, or why not?
   1. The production function exhibits decreasing returns to scale.
   2. Maximize \(\pi(K, L) = 2K^{1/4}L^{1/4} - K - L\) to find \(L=K\). Substituting into the production function, we find that the maximum satisfies \(\pi(L) = 2L^{1/2} - 2L\). Not all of these points are maximizing allocations. Maximizing \(\pi(L) = 2L^{1/2} - 2L\) gives \(L=1/4\). Thus, \(K=1/4\) and \(q=1/2\).
   3. Substituting \(L=K=1/4\) into the profit function gives \(\pi=1/2\).
   4. If we attempt to follow the same process, we find \(\pi(L) = 2L - 2L = 0\). Thus, profits are zero irrespective of the chosen allocation. Every choice of output produced by \(K=L\) constitutes a maximizing allocation. This is because the production function exhibits constant returns to scales.

1. Consider a firm with a two dimensional production function \(f\).

   1. What happens to the firm's profits if \(f\) exhibits increasing returns to scale and the firm doubles its scale of operation while all prices remain fixed?
   2. What happens to the firm's profits if \(f\) exhibits decreasing returns to scale and the firm halves its scale of operation while all prices remain fixed?

   1. The profits of the firm more than double because the output more than doubles, while the cost doubles.

      \begin{align*} \pi\left(2 x_{1}, 2 x_{2}\right) &= p f(2 x_{1}, 2 x_{2}) - w_{1} 2 x_{1} - w_{2} 2 x_{2} \\ & > 2 \left( p f(x_{1}, x_{2}) - w_{1} x_{1} - w_{2} x_{2} \right) \\ & = 2 \pi\left(x_{1}, x_{2}\right). \end{align*}

   2. The profits of the firm less than halve because the output less than halves, while the cost halves.

      \begin{align*} \pi\left(x_{1}, x_{2}\right) &= \pi\left(2 \frac{1}{2} x_{1}, 2 \frac{1}{2} x_{2}\right) \\ &= p f\left(2 \frac{1}{2} x_{1}, 2 \frac{1}{2} x_{2}\right) - w_{1} 2 \frac{1}{2} x_{1} - w_{2} 2 \frac{1}{2} x_{2} \\ & < 2 \left( p f\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right) - w_{1} \frac{1}{2} x_{1} - w_{2} \frac{1}{2} x_{2} \right) \\ & = 2 \pi\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right). \end{align*}

      Hence,

      \begin{align*} \pi\left(\frac{1}{2} x_{1}, \frac{1}{2} x_{2}\right) > \frac{1}{2} \pi\left(x_{1}, x_{2}\right). \end{align*}

2. Consider a firm with a single input factor production technology \(f\) having diminishing marginal product. The output and input prices, \(p\) and \(w\), are fixed. The firm produces at a positive output level for which \(p f'(x) > w\) holds and makes positive profits. Is this a profit maximizing firm? If not, and the firm wishes to increase its profits, should it increase or decrease the amount of the input factor?

   Since the firm makes positive profits, it is better off than not producing at all, in which case its profits are non-positive. However, the firm is not producing the profit maximizing output because the variational condition of interior solutions, namely \(p f'(x) = w\), is not satisfied. Since \(p\) and \(w\) are fixed, \(f'(x)\) needs to decrease for the variational condition to hold. This implies that the input factor level should increase because the marginal product is decreasing.

3. Consider a firm with a production function \(f(x_{1}, x_{2}) = x_{1}^{1/2} x_{2}^{1/4}\). The output price is \(p\), and the input prices are \(w_{1}\) and \(w_{2}\). All prices are fixed.

   1. Show that, when maximizing profits, the marginal product of each factor is equal to the corresponding real factor price.
   2. Use your answer in the previous part to calculate the input factor demands.
   3. Suppose that \(p=4\), \(w_{1} = 2\), and \(w_{2} = 1\). What are the optimal demanded quantities of the input factors? What is the optimal supplied quantity? How much profit does the firm make?

   1. The maximization problem of the firm is

      \begin{align*} \max_{x_{1}, x_{2}} &\left\{ p x_{1}^{1/2} x_{2}^{1/4} - w_{1} x_{1} - w_{2} x_{2} \right\}, \end{align*}

      with first order conditions

      \begin{align*} \frac{w_{1}}{p} &= \frac{1}{2} x_{1}^{-1/2} x_{2}^{1/4} \\ \underbrace{\frac{w_{2}}{p}}_{Real\ factor\ price} &= \underbrace{\frac{1}{4} x_{1}^{1/2} x_{2}^{-3/4}}_{Marginal\ product} \end{align*}

   2. Multiplying the two necessary conditions results in

      \begin{align*} \frac{w_{1} w_{2}}{p^2} &= \frac{1}{8} x_{1}^{1/2-1/2} x_{2}^{1/4-3/4}, \end{align*}

      and solving for \(x_{2}\) gives

      \begin{align*} x_{2} = \frac{p^{4}}{64 w_{1}^{2} w_{2}^{2}}. \end{align*}

      To obtain the factor demand for \(x_{1}\), we substitute the demand for \(x_{2}\) in the first variational condition to get

      \begin{align*} \frac{w_{1}}{p} &= \frac{1}{2} x_{1}^{-1/2} \frac{p}{2 \sqrt{2} w_{1}^{1/2} w_{2}^{1/2}}. \end{align*}

      Then solving for \(x_{1}\) gives

      \begin{align*} x_{1} &= \frac{p^{4}}{32 w_{1}^{3} w_{2}}. \end{align*}
   3. Lastly, substituting the prices into the demands gives \(x_{1} = x_{2} = 1\). Substituting the optimal demanded quantities into the production function gives \(q = f(1,1) = 1\). Lastly, the firm's profit is calculated as \(\pi = 4 - 2 - 1 = 1\).

4. Suppose that at the beginning of a given month, we observe that a firm produces an output level \(q=20\) at a price level \(p=2\). After a month, we observe that the same firm produces \(q = 15\) at a price of \(p=4\). No other changes are observed in the prices of the market. Does the firm maximize its profit?

   Since all input prices remain constant, we have \(\Delta w_{i} = 0\) for all input factors between the two dates. This implies that the revealed profit maximization inequality reduces to \(\Delta p \Delta q \ge 0\). However, we observe \(\Delta p = 2\) and \(\Delta q = -5\). Thus, by the revealed profit maximization inequality, the firm did not maximize its profit on one of these two (or both) dates.

1. Consider a firm with a Cobb-Douglas production function \(f(x_{1}, x_{2}) = A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r}\) for \(0 < \alpha < 1\) and \(r \neq 1\). Output and input prices, namely \(p\), \(w_{1}\), and \(w_{2}\), are fixed.

   1. Write the profit maximization problem of the firm.
   2. Calculate the first order conditions of the profit maximization problem.
   3. Calculate the supply function.
   4. Calculate the input factor demands.
   5. Calculate the profit of the firm.
   6. What happens to supply, input factor demands, and profit for \(r=1\)?

   1. The maximization problem of the firm is

      \begin{align*} \max_{x_{1}, x_{2}} &\left\{ p A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r} - w_{1} x_{1} - w_{2} x_{2} \right\} \end{align*}

   2. The necessary conditions for interior solutions are

      \begin{align*} w_{1} &= p A \alpha r x_{1}^{\alpha r-1} x_{2}^{(1 - \alpha)r} = p \alpha r \frac{f(x_{1}, x_{2})}{x_{1}} \\ w_{2} &= p (1 - \alpha) r x_{1}^{\alpha r} x_{2}^{(1 - \alpha)r - 1} = p (1 - \alpha) r \frac{f(x_{1}, x_{2})}{x_{2}} \end{align*}

   3. To shorten the notation, let \(q = f(x_{1}, x_{2})\). With this notation, we can rewrite the first order conditions as

      \begin{align*} x_{1} &= p \alpha r \frac{q}{w_{1}} \\ x_{2} &= p (1 - \alpha) r \frac{q}{w_{2}}. \end{align*}

      Substituting in the production function gives

      \begin{align*} q &= f(x_{1}, x_{2}) \\ &= A \left(p \alpha r \frac{q}{w_{1}}\right)^{\alpha r} \left(p (1 - \alpha) r \frac{q}{w_{2}}\right)^{(1 - \alpha) r} \\ &= A (p r q)^{r} \left(\frac{\alpha}{w_{1}}\right)^{\alpha r} \left(\frac{1 - \alpha}{w_{2}}\right)^{(1 - \alpha) r}, \end{align*}

      which implies

      \begin{align*} q(w_{1}, w_{2}) &= A^{\frac{1}{1-r}} (p r)^{\frac{r}{1-r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1-r}} \left(\frac{1 - \alpha}{w_{2}}\right)^{\frac{(1 - \alpha) r}{1-r}}. \end{align*}

   4. We can obtain the input factor demands by replacing the supply function in the first order conditions. This gives

      \begin{align*} x_{1} &= p \alpha r \frac{A^{\frac{1}{1-r}} (p r)^{\frac{r}{1-r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1-r}} \left(\frac{1 - \alpha}{w_{2}}\right)^{\frac{(1 - \alpha) r}{1-r}}}{w_{1}} \\ &= (A p r)^{\frac{1}{1-r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{1 - (1 - \alpha)r}{1-r}} \left(\frac{1 - \alpha}{w_{2}}\right)^{\frac{(1 - \alpha) r}{1-r}}, \end{align*}

      and analogously

      \begin{align*} x_{2} &= (A p r)^{\frac{1}{1-r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1-r}} \left(\frac{1 - \alpha}{w_{2}}\right)^{\frac{1 - \alpha r }{1-r}}. \end{align*}

   5. The easiest way to calculate the profit of the firm is to use the first order conditions of part 2 and the supply function. We then have

      \begin{align*} \pi(p, w_{1}, w_{2}) &= p q - w_{1} p \alpha r \frac{q}{w_{1}} - w_{2} p (1 - \alpha) r \frac{q}{w_{2}} \\ &= p q\left(1 - \alpha r - (1 - \alpha) r \right) \\ &= p (1 - r) q \\ &= (p A)^{\frac{1}{1-r}} \left(1 - r \right) r^{\frac{r}{1-r}} \left(\frac{\alpha}{w_{1}}\right)^{\frac{\alpha r}{1-r}} \left(\frac{1 - \alpha}{w_{2}}\right)^{\frac{(1 - \alpha) r}{1-r}} . \end{align*}
   6. If \(r=1\), the production technology has constant returns to scale. Then, the firm has zero profit, and it is indifferent about its level of supply.

• The minimum wage increases the labor cost of production \(\rightarrow\)
• Firms have an incentive to substitute labor with other input factors (e.g., capital) while keeping production constant (based on the MRTS) \(\rightarrow\)
• This can decrease labor demand (conditional factor demand for labor) \(\rightarrow\)
• Lead to job losses and unemployment.

• The Ifo Institute of Munich was painting a horror scenario at the time (2014).
• It calculated that a minimum wage of \(8.50\) Euros would jeopardize \(0.9\) million jobs!
• Its predictions in 2010 were even worse, with \(1.2\) million jobs on the line!
• The reports of Ifo cannot be found online anymore.
• But you can follow the story in 20/03/2014: ifo Institut, (Krüsemann, 2018)

• The cost minimization channel is sensible!
• However, it is not the only channel affecting labor demand.
• Migration affects the number of available workers and changes the labor supply.
• Changes in demand for exports affect labor demand.
• What happens in the market is a combination of demand and supply.
• These effects may reinforce or cancel out each other.

• The total production expenditures for using capital \(K\) and labor \(L\) are calculated by

\[E(K, L) = r K + w L\]

• Expenditures are measured in monetary units (e.g., Euros, Dollars, etc.).

• They do not directly depend on production technology.
• They do not directly depend on the produced quantity.
• They are the same if we use \(K=2\) and \(L=1\) to produce \(q=1\) or \(q=2\).
• Not very helpful.
• How can we find the minimum cost of producing a given quantity?

• Suppose production is only based on labor, i.e.,

\[q = \sqrt{L}.\]

• We want to find the minimum cost of producing \(q_{0}=2\) when the wage is \(w=1\).
• This means that \(L=4\).
• So, the minimum cost is \(w L = 4\).

• Sometimes, production involves costs that cannot be adjusted (at least in the short-run).
• For example, the offices rented by a firm cannot change on short notice.
• Then, one component of the cost function is fixed, i.e.,

\[c(q) = \alpha + \beta q\]

• Fixed costs are business costs that do not depend on the level of production. They are the same for every amount of produced output.

• Fixed costs may either be recoverable or not.
  • Suppose that a firm owns the property where its offices are located. If production ceases at some point, it can liquidate the property and recover some of its costs.
  • Suppose that a firm rents the property where its offices are located. If production ceases at some point, it cannot recover the rents that it paid during its operation.
• Sunk costs are fixed business costs that cannot be recovered if production stops.
• Bygones are bygones: Sunk costs do not affect the future choices of (rational decision making) firms.

• In business practice, it is more useful to talk about the average production cost.
• The average cost is the mean production cost of a cost minimizing firm producing an output quantity under given input prices.

\[\bar{c}(q, w_1, w_2) = \frac{Total\ cost}{\#Produced\ units} = \frac{c(q)}{q}\]

A firm employing input factors \(\left(x_{1}, \dots, x_{n}\right)\) under prices \(\left(w_{1}, \dots, w_{n}\right)\) has expenditures \[E\left(x_{1}, \dots, x_{n}\right) = \sum_{j=1}^{n} x_{j} w_{j}\] If a factor is fixed, then there is a fixed component to expenditures. For instance if \(x_{1}\) is fixed to \(\hat x_{1}\), we have \[c\left(x_{2}, \dots, x_{n}\right) := c\left(\hat x_{1}, x_{2}, \dots, x_{n}\right) = \underbrace{\hat x_{1} w_{1}}_{fixed\ cost,\ say\ w_{0}} + \sum_{j=2}^{n} x_{j} w_{j}\]

The isocost loci give the combinations of inputs resulting in a given level of costs \(\hat c\), i.e., \[\left\{ x \in \mathbb{R}^{n} \colon\ \hat c = \sum_{j=1}^n w_{j}x_{j} \right\}. \] For the case of two inputs (\(n=2\)), we obtain the isocost line in the input plane by solving for \(x_{2}\). Thus, \[x_{2} = \frac{\hat c}{w_{2}} - x_{1} \frac{w_{1}}{w_{2}}\]

The problem statement is

If the production function is (strictly) monotonic (for instance, a root production function) and the factor price \(w\) is positive, then the solution to the problem is obtained by merely inverting the production function. Thus,

The problem is given by

For non-boundary solutions, a variations arguments shows that the ratio of marginal products is equal to the ratio of factor prices at interior, cost minimizing allocations. This means

If the production technology exhibits constant returns to scale, then \[ c(t q) = t c(q) \quad \quad (t>0).\] If the production technology exhibits increasing returns to scale, then \[ c(t q) < t c(q) \quad \quad (t>1).\] Lastly, for production technologies exhibiting decreasing returns to scale, we have \[ c(t q) > t c(q) \quad \quad (t>1).\]

1. Consider a firm with a production function \(f(x_{1}, x_{2}) = 4 x_{1}^{1/2} x_{2}^{1/2}\). The input prices are \(w_{1} = 40\) and \(w_{2} = 10\).

   1. What is the slope of the isocost lines with respect to the first input factor?
   2. What is the cost minimizing marginal rate of technical substitution between \(x_{1}\) and \(x_{2}\)?
   3. Find the cost minimizing conditional factor demanded quantities and the minimum production cost when the firm produces an output level \(q\).
   4. Calculate the cost minimizing conditional factor demanded quantities and the minimum production cost when \(q=40\).
   1. For a fixed level of cost \(\hat c\), we have \(\hat c = 40 x_{1} + 10 x_{2}\). Solving for \(x_{2}\) gives \[ x_{2} = \frac{1}{10}\hat c - 4 x_{1}, \] so the slope of the isocost with respect to \(x_{1}\) is equal to \(-4\).
   2. From the cost minimizing condition, one can calculate \[ \mathrm{MRTS}(x_{1}, x_{2}) = \frac{w_{1}}{w_{2}} = 4. \] The firm substitutes a unit of \(x_{1}\) with \(4\) units of \(x_{2}\) at the minimizing cost level of production.
   3. Using the cost minimizing condition, we obtain \[ 4 = \mathrm{MRTS}(x_{1}, x_{2}) = \frac{f_{x_{1}}(x_{1}, x_{2})}{f_{x_{2}}(x_{1}, x_{2})} = \frac{2 x_{1}^{-1/2} x_{2}^{1/2}}{2 x_{1}^{1/2} x_{2}^{-1/2}} = \frac{x_{2}}{x_{1}}. \] Solving for \(x_{2}\) and substituting into the production function gives \[ q = 4 x_{1}^{1/2} x_{2}^{1/2} = 4 x_{1}^{1/2} 2 x_{1}^{1/2} = 8 x_{1}. \] The last equation implies that \(x_{1} = q / 8\) and \(x_{2} = q / 2\). Hence, the cost function is \[ c(q) = w_{1} x_{1} + w_{2} x_{2} = 40 \frac{q}{8} + 10 \frac{q}{2} = 10 q. \]
   4. Substituting the given output level results in \(x_{1} = 5\) and \(x_{2} = 20\). Moreover, the minimum cost is \(400\).

2. Consider a firm with a production technology \(f\) having two factors that exhibit diminishing marginal rates of technical substitution. The input factor prices, \(w_{1}\) and \(w_{2}\), are fixed. The firm produces an output level \(q\) using input factors quantities such that \(\mathrm{MP}(x_{1}; x_{2}) / w_{1} > \mathrm{MP}(x_{2}; x_{1}) / w_{2}\) holds. Is this a cost minimizing firm? If not, how can the firm reduce its cost?

   The firm does not minimize its cost because the variational condition \(\mathrm{MRTS}(x_{1}, x_{2}) = w_{1} / w_{2}\) is not satisfied. Since \(w_{1}\) and \(w_{2}\) are fixed, \(\mathrm{MRTS}(x_{1}, x_{2})\) needs to decrease for the variational condition to hold. Since the marginal rate of technical substitution is decreasing, the firm can reduce its cost by increasing the input factor level of \(x_{1}\) and decreasing that of \(x_{2}\).

3. Suppose that at the beginning of a given month, we observe that a firm produces an output level \(q\) using input factors \(x_{1}=1\) and \(x_{2}=3\) with corresponding prices \(w_{1}=3\) and \(w_{2}=1\). After a month, we observe that the same firm produces the same level of output using \(x_{1}=5\) and \(x_{2}=1\) under prices \(w_{1}=5\) and \(w_{2}=3\). Does the firm minimize its cost?

   We calculate \(\Delta w_{1}=2\), \(\Delta w_{2}=2\), \(\Delta x_{1} =4\), and \(\Delta x_{2} =-2\). Therefore \[ \Delta w_{1} \Delta x_{1} + \Delta w_{2} \Delta x_{2} = 2 \cdot 4 + 2 \cdot (-2) = 4 > 0. \] Since the revealed cost minimization inequality is not satisfied, the firm does not minimize its cost at one of these two (or both) dates.

1. Prove that if a firm maximizes its profits, then it minimizes its costs.

   Suppose that a firm maximizes its profits under prices \(p\), \(w_{1}\), \(w_{2}\) when producing \(\hat q\) and demanding \(\hat x_{1}\) and \(\hat x_{2}\). Towards contradiction, suppose that \(\hat x_{1}\) and \(\hat x_{2}\) do not minimize the firm's cost for producing \(\hat q\) under the given prices. Then, there exist \(\tilde x_{1}\) and \(\tilde x_{2}\), different than \(\hat x_{1}\) and \(\hat x_{2}\), that produce \(\hat q\) with less cost. Then

   \begin{align*} w_{1} \hat x_{1} + w_{2} \hat x_{2} &> w_{1} \tilde x_{1} + w_{2} \tilde x_{2} \implies \\ \hat \pi = p \hat q - w_{1} \hat x_{1} - w_{2} \hat x_{2} &< p \hat q - w_{1} \tilde x_{1} - w_{2} \tilde x_{2}, \end{align*}

   which implies that \(\hat q\), \(\hat x_{1}\), and \(\hat x_{2}\) are not profit maximizing.

2. What are the differences between the conditional factor demand and the factor demand? How are they related?

   The conditional factor demand is obtained by the cost minimization problem, while the factor demand is obtained by the profit maximization problem. The conditional factor demand depends on the output quantity, say \(q\), and factor prices, say \(w_{i}\). Instead, the factor demand depends on the output price, say \(p\), and factor prices.

   Let \(x_{i}^{c}\) and \(x_{i}\) correspondingly denote the conditional factor and factor demands. By exercise 1, if a firm is maximizing its profits, then its minimizes it costs. Therefore, if \(q(p, w_{1}, w_{2})\) is the supply function obtained by the profit maximization problem, then the two demands are interrelated by

   \begin{align*} x_{i}^{c}\left(q(p, w_{1}, w_{2}), w_{1}, w_{2}\right) = x_{i}\left(p, w_{1}, w_{2}\right). \end{align*}

3. Consider a firm with a Cobb-Douglas production function \(f(x_{1}, x_{2}) = A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r}\) for \(0 < \alpha < 1\) and \(r \neq 1\). Input prices, namely \(w_{1}\), and \(w_{2}\), are fixed.

   1. Write the Lagrangian of the cost minimization problem of the firm.
   2. Calculate the first order conditions of the cost minimization problem.
   3. Calculate the conditional input factor demands.
   4. Calculate the cost function of the firm.

   1. The Lagrangian of the problem is obtained by subtracting from the objective of the firm the production constraint multiplied by the Lagrange multiplier \(\lambda\). Specifically,

      \begin{align*} \mathcal{L} = w_{1} x_{1} + w_{2} x_{2} - \lambda \left(A x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r} - q\right). \end{align*}

   2. Differentiating the Lagrangian of the problem gives the first order conditions

      \begin{align*} w_{1} &= \lambda A \alpha r x_{1}^{\alpha r - 1} x_{2}^{(1 - \alpha) r} = \lambda \alpha r \frac{q}{x_{1}}, \\ w_{2} &= \lambda A (1 - \alpha) r x_{1}^{\alpha r} x_{2}^{(1 - \alpha) r - 1} = \lambda (1 - \alpha) r \frac{q}{x_{2}}. \end{align*}

   3. Solving the first order conditions for \(x_{1}\) and \(x_{2}\), respectively, and substituting to the production function results in

      \begin{align*} q = A \left(\lambda \alpha r \frac{q}{w_{1}}\right)^{\alpha r} \left(\lambda (1 - \alpha) r \frac{q}{w_{2}}\right)^{(1 - \alpha) r} = A \lambda^{r} r^{r} \left(\frac{\alpha}{w_{1}}\right)^{\alpha r} \left(\frac{1 - \alpha}{w_{2}}\right)^{(1 - \alpha) r} q^{r}, \end{align*}

      from which we obtain

      \begin{align*} \lambda = A^{-\frac{1}{r}} r^{-1} \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} q^{\frac{1-r}{r}}, \end{align*}

      Substituting back to the first order conditions, we can eliminate \(\lambda\), and solve for \(x_{1}\), \(x_{2}\) to obtain the conditional demand functions

      \begin{align*} x_{1} &= \lambda \alpha r \frac{q}{w_{1}} = \left(\frac{w_{2}}{w_{1}}\frac{\alpha}{1 - \alpha}\right)^{1-\alpha} \left(\frac{q}{A}\right)^{\frac{1}{r}}, \\ x_{2} &= \lambda (1 - \alpha) r \frac{q}{w_{2}} = \left(\frac{w_{1}}{w_{2}}\frac{1 - \alpha}{\alpha}\right)^{\alpha} \left(\frac{q}{A}\right)^{\frac{1}{r}}. \end{align*}

   4. The easiest way to calculate the cost function of the firm is to use the first order conditions of part 2 and the objective function. We then have

      \begin{align*} c(q, w_{1}, w_{2}) &= w_{1} \lambda \alpha r \frac{q}{w_{1}} + w_{2} \lambda (1 - \alpha) r \frac{q}{w_{2}} \\ &= \lambda r q \\ &= \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} \left(\frac{q}{A}\right)^{\frac{1}{r}}. \end{align*}

• Many behavioral economists in the 1980s contested that humans always make rational choices, as mainstream economic theory suggests.
• Among the examples of irrational behavior was the Sunk Cost Fallacy.
• Once time, money, or effort is invested in an act or a choice, humans tend to maintain their choices.
• Economists started conducting experiments to examine if the last conjecture was true.

Assume that you have spent \($100\) on a ticket for a weekend ski trip to Michigan. Several weeks later you buy a \($50\) ticket for a weekend ski trip to Wisconsin. You think you will enjoy the Wisconsin ski trip more than the Michigan ski trip. As you are putting your just-purchased Wisconsin ski trip ticket in your wallet, you notice that the Michigan ski trip and the Wisconsin ski trip are for the same weekend! It’s too late to sell either ticket, and you cannot return either one. You must use one ticket and not the other. Which ski trip will you go on?

(Arkes & Blumer, 1985 Experiment 1)

• Offered discounted seasonal theater tickets to subjects and checked how often people attended. (Arkes & Blumer, 1985 Experiment 2)

• The first \(60\) people approaching the ticket window to buy a ticket were randomly split into three groups:
  • Those who paid the normal \($15\) price,
  • those who took \($2\) discount, and
  • those who took a \($7\) discount.
• Tracked the number of times each individual visited the theater.

• Mainstream economic theory suggests that rational agents should not base their decisions on sunk costs.

• Only future benefits and costs determine the optimality of decisions, as those in the past cannot change anymore.
• Acting "irrationally" at a personal level is one thing.
• As a manager, falling to suck cost fallacies can cost money and endanger your business!

Consider the cost minimization problem with a root production function

Its solution, namely the (total) cost function, is given by

Having the cost function, one can calculate the remaining cost types as in the following table

The cost minimization problem for a Cobb-Douglas production function is

and its solution is given by

The minimum average cost is attained at the output level for which the marginal cost equals the average cost. Namely \[\min_{q} \bar{c}(q) = c'(\mathrm{arg\,min}_{q} \bar{c}(q))\] If \(c\) is differentiable, we can show this result by setting the derivative of average cost equal to zero, i.e.,

Analogously, we can show that the minimum average variable cost is attained at the output level for which the marginal cost equals the average variable cost. \[\min_{q} \bar{\mu}(q) = c'(\mathrm{arg\,min}_{q} \bar{\mu}(q)).\]

Consider again the cost minimization problem

with resulting cost function

What happens if one production function, say \(x_{2}\), is fixed? It is common for many production settings that some factors cannot be flexibly adjusted in short periods. This results in a fixed cost component. Fixing the level of \(x_{2} = \hat{x}_{2}\), the minimization problem becomes

Its solution is given by

Using the \(\beta\), \(B\) notation, we observe that the production function essentially becomes a root function (\(f(x_{1}) = B x_{1}^{\beta}\)), and the resulting cost function is very similar to the cost function obtained by single variable, root production functions.

1. Consider the cost function \(c(q) = 16 + 4q^{2}\). Calculate and plot

   1. the average cost function,
   2. the marginal cost function,
   3. the average variable cost function.
   4. the average fixed cost function.
   5. What is the level of output that yields the minimum average production cost?

   Cost type	Expression
   Total cost	\(c(q) = 16 + 4 q^{2}\)
   Average cost	\(\bar{c}(q) = \frac{16}{q} + 4 q\)
   Average variable cost	\(\bar{\mu}(q) = 4 q\)
   Average fixed cost	\(\bar{\sigma}(q) =\frac{16}{q}\)
   Marginal cost	\(c'(q) = 8 q\)

   The minimum average cost is obtained by the first order condition \[ -\frac{16}{q^{2}} + 4 = 0, \] which implies that \(q = 2\).

2. Consider the cost function obtained by a Cobb-Douglas production function

   \begin{align*} c(q; w_{1}, w_{2}) &= \left(\frac{\alpha}{w_{1}}\right)^{-\alpha} \left(\frac{1 - \alpha}{w_{2}}\right)^{-(1 - \alpha)} \left(\frac{q}{A}\right)^{\frac{1}{r}}, \end{align*}

   and let \(w_{1}=w_{2}=\alpha = 1 / 2\), \(A=1\), and \(r>0\). Show that the average cost function is

   1. increasing for decreasing returns to scale (\(r<1\)),
   2. decreasing for increasing returns to scale (\(r>1\)), and
   3. constant for constant returns to scale (\(r=1\)).

   The average cost function for the given parameters values is given by

   \begin{align*} \bar{c}(q) &= q^{\frac{1 - r}{r}}, \end{align*}

   with derivative

   \begin{align*} \bar{c}'(q) &= \frac{1 - r}{r}q^{\frac{1 - 2r}{r}}. \end{align*}

   If \(r<1\), then \(\bar{c}' >0\) and \(\bar{c}\) is increasing. If \(r>1\), then \(\bar{c}' <0\) and \(\bar{c}\) is decreasing. Finally, if \(r=1\), then \(\bar{c}' =0\) and \(\bar{c}\) is constant.

• In the early 1980s, the common practice of operating system companies was to charge hardware manufacturers for each operating system copy installed in a computer.

• Microsoft offered an alternative plan.
  • Charge computer manufacturers based on (the past number of) built computers.
  • The manufacturer was paying a general licensing fee and then could install the operating system in all the computers it produced.
  • Microsoft was offering low-priced licensing contracts making their operating system (MS-DOS) very attractive to manufacturers.

• Effectively, manufacturers could purchase Microsoft's operating system at much lower prices than the operating systems of other software companies.
  • A manufacturer had to pay \($50\ -\ $100 \) for installing an alternative operating system on an additional machine.
  • It cost nothing to install MS-DOS on an additional machine once a licensing contract with Microsoft has been signed.
  • MS-DOS ended up being the default operating system

• Android is free. Anyone can install the operating system on her device after accepting the terms and conditions.
• For mobile device manufacturers, Android is a cheaper operating system alternative for their products.
• This leads to more competitive prices for consumers too.
  • Microsoft's operating systems lost their primacy in the overall operating system market in \(2017\).
  • Microsoft's operating systems are still dominant in less portable devices, such as desktop PCs and Laptops.
  • The rise of Android has also impacted Microsoft's pricing strategies for its desktop operating systems.
  • Licensed users were able to upgrade to the last two versions of Microsoft's operating system without paying for a new license.

• In the long-run, a firm operates if it makes non-negative profit.
• In the short-run, a firm might also keep producing while it makes losses.
• A firm shutdowns if the market price is below its average variable cost.
• Equivalently, a firm shutdowns if its revenue is not enough to cover, besides variable cost, at least some part of its fixed cost.

• A price taking firm maximizes its profit when it produces at the point where its marginal cost is equal to the market price.
• For quantities where its marginal cost is above the market price,
  • the firm is making losses for the last unit of output it produces, and
  • it can increase its profit by lowering production.
• For quantities where its marginal cost is below the market price,
  • the firm is still making a profit for the last unit of output it produces, and
  • it can increase its profit even further by increasing production.

• The firm's inverse supply function is a mapping that gives at which price would a profit maximizing firm produce a particular quantity.
• It can be obtained by the first order condition of the firm's profit maximization problem.

• Suppose that the firm's cost function is \(c(q) = 2 q^{2}\).
• Suppose that the firm is a price taker, and the market price is \(p=8\).
• The firm maximizes its profit

\[\max_{q} \left\{ 8 q - 2 q^{2} \right\}.\]

• It solves the above to get

\[8 - 4 q \overset{!}{=} 0 \implies q = 2.\]

• The firm makes profit

\[\pi = 16 - 8 = 8.\]

The firm is willing to supply at prices covering its average variable cost. The cost function incorporates the technological constraints faced by the firm. Using the cost function, the profit maximization problem can be rewritten as

The profit can be decomposed into two parts:

• revenue \(pq\) and
• cost \(c(q)\).

For non-boundary solutions, the marginal revenue and marginal cost are equalized at the maximum (why?).

The above condition is necessary but not sufficient. The condition gives a maximum if

Therefore, marginal cost should be increasing (locally) at the maximum.

The firm opts out if for every level of output \(q>0\)

The solution to the profit maximization problem determines the supply of the firm. Supply \(s(p)\) gives the quantity that a profit maximizing firm would like to produce and sell in the market. On some occasions, it is easier to work with the inverse supply mapping \(p(q)\), which gives the price for which the supplied quantity \(q\) is profit maximizing. The inverse supply can be directly obtained from the variational condition \(p=c'(q)\).

Consider the cubic cost function

The inverse supply of a profit maximizing firm is

The surplus of the producer is the cumulative difference between the marginal revenue and cost of every produced unit. In a single-price market (such as in the case of perfect competition), the revenue of each unit (marginal revenue) is equal to the market price. The cost per unit of production is the marginal cost. Then, we can calculate

If production exhibits constant returns to scale, then \(c(q) = c_{1} q\). Therefore, a profit maximizing firm has supply

Its inverse supply is horizontal, constantly equal to \(c_{1}\) for all positive levels of output.

1. Consider a price-taking, profit maximizing firm with cost function \(c(q) = 1000 + 10 q^{2}\).

   1. Calculate the supply function.
   2. Find the output level that minimizes the average cost.
   3. Calculate the profit function.
   4. What is the marginal effect of a price change on profit?
   5. What is the minimum price for which the firm makes non negative profit?
   1. The first order condition of the profit maximization problem is \(20 q = p\). Solving for \(q\) results in the supply function \(s(p) = p / 20\).

   2. The average cost is determined by

      \begin{align*} \bar{c}(q) = \frac{1000}{q} + 10 q . \end{align*}

      We can calculate the minimum average cost by solving the equation \(\bar c(q) = c'(q)\), which is equivalent to

      \begin{align*} \frac{1000}{q} + 10 q = 20 q. \end{align*}

      This results to \(q = 10\).

   3. The profit function is given by

      \begin{align*} \pi(p) &= p s(p) - c\left(s(p)\right) \\ &= \frac{p^{2}}{20} - 1000 - 10 \frac{p^{2}}{400} \\ &= - 1000 + \frac{p^{2}}{40} . \end{align*}
   4. The effect of a marginal price change on profit is \(\pi'(p) = p / 20\).
   5. The firm makes non negative profit for prices such that \(\pi(p) \ge 0\), which implies that \(p \ge 200\).

2. Suppose that a firm has a supply function given by \(s(p) = 100 + 4 p\). Calculate the inverse supply function.

   We solve for \(p\) to get

   \begin{align*} p(q) &= - 25 + \frac{q}{4}. \end{align*}

• The CAP reduced production risks.
  • There was less uncertainty about the lower end of price distributions.

• This incentivized suppliers to produce more as they could calculate their production's minimum profit.
• It led to oversupplying certain CAP-supported agricultural products.
• The main goal of the CAP policy was achieved.
• Prices of many agricultural products remained (relatively) high during this period.
• The profitability of the agricultural sector was maintained, and the sector did not shrink.

• Taming the market forces is a very challenging problem.

• On many occasions, policy interventions fail despite the best of intentions.
• Even if a market intervention succeeds in the desired policy direction, there is a serious probability of unexpected side effects.

An excess supply or market surplus is a market state where the supplied quantities exceed the demanded quantities.

An excess demand or market shortage is a market state where the demanded quantities exceed the supplied quantities.

• One major difficulty in estimating the supply (or demand) equation is simultaneity.
• In market estimations, it boils down to the market clearing condition. \[ \begin{matrix}\text{Demanded}\\ \text{Quantity}\end{matrix} = \begin{matrix}\text{Supplied}\\ \text{Quantity}\end{matrix} = \begin{matrix}\text{Traded}\\ \text{Quantity}\end{matrix}\]

• Prices affect demanded quantities \(\rightarrow\)
• Demanded quantities affect traded quantities \(\rightarrow\)
• Traded quantities affect supplied quantities \(\rightarrow\)
• Supplied quantities affect prices.
• In short, prices are endogenous.

• We know from econometrics that a fundamental requirement for linear regressions is that the independent variables (right-hand side) should be exogenous.
• If we try to estimate

\[\mathrm{Supply}_{it} = \beta_{0} + \beta{_1} \mathrm{Price}_{it} + \dots + u_{it}, \] we will not obtain the correct coefficient for \(\beta_{1}\) because the exogeneity condition is not satisfied.

• This can be a real problem if we want to calculate the elasticity of supply.
• What can we do?

• We should take into account the complete market system.
• One way to do it is via a methodology called full information maximum likelihood.
• Without many details, the process
  1. models the market as a system of simulteneous equations, and
  2. estimates the demand and supply equations together.
• Luckily there is statistical software that automates this process.

• Install and load the R package markets (https://cran.r-project.org/package=markets).

install.packages('markets')
library(markets)

• Read the description of the houses dataset (https://markets.pikappa.eu/reference/houses.html).
• Estimate and summarize the supply equation using a linear regression.

ls <- lm(HS ~ RM + TREND + W + L1RM + MA6DSF + MA3DHF + MONTH, fair_houses())
summary(ls)

• Estimate and summarize the market system in equilibrium.

eq <- equilibrium_model(
  HS | RM | ID | TREND ~
  RM + TREND + W + CSHS + L1RM + L2RM + MONTH |
  RM + TREND + W + L1RM + MA6DSF + MA3DHF + MONTH,
  fair_houses(),  correlated_shocks = FALSE)
summary(eq)

• Compare the two supply price coefficients. Are both of them statistically significant?

• Plot the data points, the average demand, and supply equations on the quantity-price plane from the estimates of the equilibrium model.

plot(eq)

• Add the regression line to the plot.

abline(ls)

• For which estimate is supply more elastic?

The rate of change of aggregate supply is the sum of the rates of change of the individual supplies of each firm in the market. This is a direct consequence of the linearity of differentiation.

The elasticity of aggregate supply is a weighted average of the elasticities of the individual firms.

The weights are given by the market shares \(\frac{s_{i}(p)}{s(p)}\).

1. Suppose that a market consists of two firms, one with supply \(s_{1}(p) = -10 + p\) and one with \(s_{2}(p) = -15 + p\). Calculate the market supply function. At which prices does the market supply function have kinks?

   Note that \(s_{1}\) is valid for \(p\ge 10\) because the resulting supplied quantities are negative for smaller prices. Similarly, \(s_{2}\) is valid for \(p\ge 15\). Therefore, the market supply is given by

   \begin{align*} s(p) &= s_{1}(p) \mathbb{1}_{[10, \infty)}(p) + s_{2}(p) \mathbb{1}_{[15, \infty)}(p) \\ &= \left\{\begin{aligned} &0 & p < 10 \\ &-10 + p & 10 \le p < 15 \\ &-25 + 2p & 15 \le p . \end{aligned}\right. \end{align*}

   Thus, the market supply function has two kinks, one at \(p=10\) and one at \(p=15\).

2. Consider a market with multiple price-taking firms, all of which have cost functions \(c(q) = q^{2} + 1\) for \(q \ge 0\). Suppose that entry is free and instantaneous in the market, the market demand function is \(d(p) = 52 - p\), and the market perpetually clears.

   1. Calculate the supply and profit functions of an individual firm.
   2. Assume that \(n\) is a natural number (positive integer). What is the market supply function if there are \(n\) identical firms in the market?
   3. What does market-clearing imply for the relationship between the number of firms and the market price?
   4. Use the last condition to write the individual firm's profit as a function of the number of firms in the market.
   5. Find the equilibrium number of firms.
   6. What will be the equilibrium supplied quantity in the market?
   7. Suppose that demand shifts to \(d(p) = 52.5 - p\). Calculate the equilibrium number of firms and the equilibrium supplied quantity. Are firms' profits zero in this case?

   1. The firm \(i\)'s profit maximization condition gives the inverse supply function

      \begin{align*} p = c'(q) = 2 q, \end{align*}

      from which we get the supply \(s_{i}(p) = p / 2\). One can then calculate the profit function of the firm as

      \begin{align*} \pi_{i}(p) = p \frac{p}{2} - \frac{p^{2}}{4} - 1 = \frac{p^{2}}{4} - 1. \end{align*}

   2. With \(n\) identical firms, the market supply is

      \begin{align*} s(p) = \sum_{i=1}^{n} s_{i}(p) = \frac{n}{2}p. \end{align*}

   3. Equating market demand and supply gives

      \begin{align*} \frac{n}{2}p &= s(p) = d(p) = 52 - p \iff \\ p &= \frac{104}{n + 2}. \end{align*}

   4. The last equation implies that the firm i's profit can be rewritten as

      \begin{align*} \pi_{i}(n) = \left(\frac{52}{n + 2}\right)^{2} - 1. \end{align*}

   5. As long as entry is free and instantaneous, firms enter the market when they can make positive profits. Therefore, the equilibrium number of firms is exactly such that if one more firm enters the market, profits turn negative. Hence, the equilibrium number of firms is given by the greatest natural number \(n\) for which \(\pi(n)\ge 0\). The non-negative profit condition is satisfied if

      \begin{align*} \frac{52}{n + 2} \ge 1, \end{align*}

      or, equivalently, if \(n\le 50\). Therefore, the equilibrium number of firms is \(n=50\).

   6. Since \(n=50\), the equilibrium price is \(p=2\). Substituting the equilibrium number of firms and price in the market supply equation results in \(s(2) = 50\).

   7. With the shifted market demand, the market clearing condition implies that

      \begin{align*} p &= \frac{105}{n + 2}, \end{align*}

      hence, profits as a function of \(n\) become

      \begin{align*} \pi_{i}(n) = \left(\frac{52.5}{n + 2}\right)^{2} - 1. \end{align*}

      This implies that \(n\le 50.5\) and, thus, the equilibrium number of firms remains unchanged equal to \(n=50\). One then can calculate the equilibrium price by

      \begin{align*} p &= \frac{105}{52} > 2. \end{align*}

      The supplied market quantity is

      \begin{align*} s &= \frac{50}{2} \frac{105}{52} > 50 \end{align*}

      The equilibrium price increases, the supplied quantity also increases, and individual firm profits turn positive, i.e.

      \begin{align*} \pi_{i}(n) = \left(\frac{52.5}{52}\right)^{2} - 1 > 0. \end{align*}

• The US vs. AT&T case (1982) is perceived as one of the most successful US antitrust cases.

• The US Department of Justice argued that AT&T's market power led to higher prices and fewer services.
• The AT&T monopoly accounted for \(80-85\%\) of telephone lines in 1982.
• After the breakup, prices fell, and more services were provided.

• The AT&T case does not seem to fit well the price taking behavior of firms in perfect competition.

• Was the influence of prices a special AT&T characteristic?
• What do price setting firms consider when they choose their prices?

• The monopolist wants to produce a quantity level for which its marginal cost equals its marginal revenue.
• If marginal cost is greater than marginal revenue, then the monopolist can increase its profit by reducing production.
• If marginal revenue is greater than marginal cost, then the monopolist can increase its profit by increasing production.
• Therefore, at the maximum, we should have \[70 - 6 q = 4 q\]

• The monopolist produces less than the competition quantity.
• The monopolistic price is greater than the competition price.

• The monopolist makes profit

  \begin{align*} \pi_{m} &= p_{m} q_{m} - c\left(q_{m}\right) \\ &= 343 - (98 + 2) \\ &= 243. \end{align*}

• In the competitive equilibrium, profit is

  \begin{align*} \pi_{c} &= p_{c} q_{c} - c\left(q_{c}\right) \\ &= 400 - (200 + 2) \\ &= 198. \end{align*}

• The consumers' surplus is the consumer benefit from buying a commodity or service at a price lower than her reservation value.
• The producers' surplus is the producer's benefit stemming from selling a commodity or service at a price greater than her reservation value.
• The total welfare (or economic surplus) is the sum of consumers' and producers' surplus.

• We can compare the total welfare of the two market structures to examine which one is economically more efficient.
• The deadweight loss (or excess burden) is a measure of lost economic efficiency compared to when the socially optimal quantity of a commodity or a service is produced.
• The deadweight loss of a monopoly is the difference between the competition and the monopoly’s total welfare.

• The minimum efficient scale is the lowest production scale where a firm minimizes its average cost.

Using the cost function, the profit maximization problem of a monopolist is given by \[\max_{p} \left\{ p d(p) - c\left(d(p)\right) \right\}.\] The cost function incorporates the technological constraints faced by the firm. The profit can be decomposed into two parts, namely

• revenue \(p d(p)\) and
• cost \(c\left(d(p)\right)\).

The difference with the competition case stems from the dependence of prices on the supplied quantity via the demand function \(d(p)\).

As long as demand is a one-to-one function (i.e., there is a single price for each demanded quantity and vice versa), maximizing over quantities or prices does not affect the optimization outcome. This condition will be satisfied in most healthy markets. The problem is then written using quantities as an optimization control as \[\max_{q} \left\{ p(q) q - c\left(q\right) \right\}.\]

For non-boundary solutions, the marginal revenue and marginal cost are equalized at the maximum (why?). With price as the control, this means \[d(p) + p d'(p) = c'\left(d'(p)\right) d'(p).\] With quantity as the control, the variational condition becomes \[p_{d}'(q) q + p_{d}(q) = c'\left(q\right),\] where \(p_{d}\) is the inverse demand. The variational condition is necessary. It gives a maximum if \[p_{d}''(q) q + 2 p_{d}'(q) - c''\left(q\right) < 0.\]

The minimum efficient scale is the level of output that minimizes average cost, namely \[\mathrm{MES} = \mathrm{arg\,min}_{q} \bar{c}(q).\] The greater is the ratio of the minimum efficient scale to the demanded quantity corresponding to the price being equal to the minimum average cost, i.e., \[\frac{\mathrm{MES}}{d(\bar{c}(\mathrm{MES}))},\] the more possible become monopolistic structures.

For this example let the cost function be given by \[c(q) = c_{1} q + c_{0} \quad\quad (c_{1},c_{0} > 0).\] Let also the inverse demand function be \[p(q) = p_{0} + p_{1} q \quad\quad (p_{0} > 0, p_{1} < 0).\]

The calculations of this example are used for producing the figures of sections 8.10.1 and 8.10.2. Let the cost function be given by \[c(q) = q^{3} - 2 q_{2} q^{2} + (q_{2}^{2} + q_{1}) q + q_{0} \quad\quad (q_{2},q_{1},q_{0} > 0).\] Let also the inverse demand function be \[p(q) = p_{1} q + p_{0} \quad\quad (p_{0} > 0, p_{1} < 0).\]

1. Consider a monopolist with cost function \(c(q) = 2 q\). Suppose that market demand is given by \(d(p) = 100 - 2p\). Calculate the monopolist's price, quantity, and profit. In addition, calculate the deadweight loss.

   The monopolist solves \[ \max_{p}\left\{ p d(p) - c\left(d(p)\right) \right\}. \] The first order condition of the problem is \[ d(p) + p d'(p) - c'\left(d(p)\right) d'(p) = 0. \] Substituting the exercise's functional forms and calculating gives \(p_{m} = 26\). Substituting back to demand gives \(q_{m} = d(26) = 48\). The monopolist achieves profit \[ \pi(26) = 2 d(26) - c\left(d(26)\right) = 26 \cdot 48 - 2 \cdot 48 = 1152. \] When demand is affine, the deadweight loss can be calculated using a geometric argument and the formula for the area of a triangle. For this, we need the perfect competition price and quantity. The perfect competition price is given by \(p_{c} = 2\) (why?) and, the perfect competition quantity is \(q_{c} = 96\). One then calculates \[ D\left(q_{c}, q_{m}\right) = \frac{1}{2} \left(q_{c} - q_{m}\right) \left(p_{m} - p_{c}\right) = \frac{1}{2} (96 - 48)(26 - 2) = 576 \]

2. Consider a monopolist with cost function \(c(q) = q^{2}\). Suppose that market demand is given by \(d(p) = 100 / p\). Calculate the quantity that the monopolist would like to produce.

   There are two differences compared to exercise 1. Firstly, demand is hyperbolic instead of affine, and secondly, the cost function is quadratic instead of linear. In this case, the maximization problem of the monopolist is \[ \max_{p} \left\{ p \frac{100}{p} - \frac{100^{2}}{p^{2}} \right\} = \max_{p} \left\{ 100 - \frac{100^{2}}{p^{2}} \right\}. \] The derivative of the objective is \(2 \cdot 10^{4} / p^{3}\), which is positive for all positive prices. This implies that the objective is increasing and, therefore, the maximization problem does not have a solution. The monopolist would like to let prices go to infinity. Taking this limit in the demand equation, we see that the monopolist would like to abstain from producing, i.e., \[ \lim_{p\to\infty} d(p) = \lim_{p\to\infty} \frac{100}{p} = 0. \]

3. Consider a monopolist operating in a market where the commodity's demand depends on a qualitative attribute of the product besides its price. Specifically, demand is given by \(d(s, p) = 3 \sqrt{s} / p^{3}\), where \(s\) is the qualitative attribute of the market's commodity. The cost function of the monopolist also directly depends on the qualitative attribute, i.e., \(c(s,q) = s/2 + 2 q\). The monopolist's goal is to maximize profit by simultaneously choosing both the qualitative attribute level and the commodity's price. Calculate the monopolist price, the optimal level of the qualitative attribute, the output level, and the profit.

   The problem of the monopolist is \[ \max_{s, p} \left\{ p d(s, p) - \frac{s}{2} - 2 d(s, p) \right\} = \max_{s, p} \left\{ 3 \sqrt{s} \left( \frac{1}{p^{2}} - 2 \frac{1}{p^{3}} \right) - \frac{s}{2} \right\}. \] The first order conditions of the problem are

   \begin{align*} -2 \frac{1}{p^{3}} + 6 \frac{1}{p^{4}} &= 0, \\ \frac{3}{2} \frac{1}{\sqrt{s}} \left( \frac{1}{p^{2}} - 2 \frac{1}{p^{3}} \right) - \frac{1}{2} &= 0. \end{align*}

   The first condition involves only \(p\), and the monopolistic price can be calculated from it. Solving for prices gives \(p=3\). By substituting \(p\) in the second condition, one can then solve for the qualitative attribute to get \(s=1/81\). Using the demand function, the output quantity level that the monopolist produces is \(q = 1/81\). Finally, the profit is \[ \pi = 3 \sqrt{\frac{1}{81}} \left( \frac{1}{9} - 2 \frac{1}{27} \right) - \frac{1}{2}\frac{1}{81} = \frac{1}{162} . \]

4. Can the monopolist select a price level for which demand is inelastic (i.e., \(0 > e_{d}>-1\))?

   No. By the monopolistic pricing condition, one can see that the monopolist would like to set a negative price in such a case. Specifically,

   \begin{align*} p = \frac{c'\left(d(p)\right) e_{d}}{e_{d} + 1} < 0. \end{align*}

5. Suppose that the elasticity of demand \(e_{d}<-1\) and the marginal production cost \(c_{1}>0\) are constant. Calculate the marginal effect on prices of an improvement in the production technology reducing the marginal cost in a monopolistic market.

   The monopolistic pricing condition is

   \begin{align*} p = \frac{c_{1} e_{d}}{e_{d} + 1}, \end{align*}

   where elasticity does not depend on prices by the assumptions of the exercise. We can calculate

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c_{1}} &= \frac{e_{d}}{e_{d} + 1}. \end{align*}

   Since \(e_{d} < -1\), then \(\mathrm{d}\, p / \mathrm{d}\, c_{1} > 0\). This implies that whenever a technological improvement reduces marginal cost, the monopolistic price in the market falls.

1. Consider a monopolist with cost function \(c(q) = c_{1} q\). Suppose that market demand is given by \(d(p) = a / p^{b}\) for \(a>0\) and \(b \ge 1\). Calculate the monopolist's price, quantity, and profit. In addition, calculate the deadweight loss for \(c_{1}=2\), \(a=16\), and \(b=2\).

   The difference with exercise 1 is that demand, in this case, is hyperbolic instead of affine. The first order condition of this problem yields \[ \frac{a}{p^{b}} + p \frac{-a b}{p^{b+1}} - c_{1} \frac{-a b}{p^{b+1}} = 0, \] which gives \(p_{m}= c_{1}b / (b-1)\) and \(q_{m} = a (b-1)^{b} / (c_{1}b)^{b}\). The monopolist makes profit \[ \pi_{m} = \frac{c_{1}b}{b-1} \frac{a (b-1)^{b}}{(c_{1}b)^{b}} - c_{1} \frac{a (b-1)^{b}}{(c_{1}b)^{b}} = \frac{c_{1} a (b-1)^{b-1}}{(c_{1}b)^{b}}. \] The competition price is \(p_{c}=c_{1}\) and the competition quantity is \(q_{c} = a / c_{1}^{b}\). The deadweight loss can be calculated by \[ D\left(q_{c}, q_{m}\right) = \int_{q_{m}}^{q_{c}} \left(p(z) - c'(z)\right) \mathrm{d} z, \] where \(p(q) = (a / q)^{1/b}\) is the inverse demand function. For the given parameter values, we get \(q_{c} = 4\), \(q_{m} = 1\), and

   \begin{align*} D\left(q_{c}, q_{m}\right) &= \int_{q_{m}}^{q_{c}} \left(a^{1/b} z^{-1/b} - c_{1} \right) \mathrm{d} z \\ &= \int_{1}^{4} \left(4 z^{-1/2} - 2 \right) \mathrm{d} z \\ &= \left. 8 z^{1/2} - 2 z \right|_{1}^{4} \\ &= 16 - 8 - (8 - 2) \\ &= 2 \end{align*}

2. Show that if demand is affine and the marginal cost is constant, the rate of change of prices with respect to marginal cost in a monopolistic market is \(1/2\).

   Suppose that marginal cost is equal to \(c\). The monopolistic pricing condition is

   \begin{align*} p = \frac{c e_{d}(p)}{e_{d}(p) + 1}. \end{align*}

   Therefore, we can calculate

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p)}{e_{d}(p) + 1} + \frac{c}{\left(e_{d}(p) + 1\right)^{2}} \left(e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c} \left(e_{d}(p) + 1\right) - e_{d}(p) e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c}\right) \\ &= \frac{e_{d}(p)}{e_{d}(p) + 1} + \frac{c}{\left(e_{d}(p) + 1\right)^{2}} e_{d}'(p) \frac{\mathrm{d}\, p}{\mathrm{d}\, c}, \end{align*}

   from which we get

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p) \left(e_{d}(p) + 1\right)}{\left(e_{d}(p) + 1\right)^{2} - c e_{d}'(p)}. \end{align*}

   Thus, we also need to calculate

   \begin{align*} e_{d}'(p) &= \frac{\mathrm{d}\, \left(d'(p)\frac{p}{d(p)}\right)}{\mathrm{d}\, p} \\ &= d''(p)\frac{p}{d(p)} + d'(p)\frac{1}{\left(d(p)\right)^{2}} \left(d(p) - p d'(p) \right) \\ &= d''(p)\frac{p}{d(p)} + \frac{1}{p} d'(p)\frac{p}{d(p)} \left(1 - d'(p)\frac{p}{d(p)} \right) \\ &= d''(p)\frac{p}{d(p)} + \frac{1}{p} e_{d}(p) \left(1 - e_{d}(p) \right) . \end{align*}

   Firstly, since demand is affine, we have \(d'' = 0\) and, secondly, by the monopolist's pricing condition, we can rewrite the last expression as

   \begin{align*} e_{d}'(p) &= \frac{e_{d}(p) + 1}{c} \left(1 - e_{d}(p) \right) = \frac{1 - \left(e_{d}(p)\right)^{2}}{c} . \end{align*}

   Substituting this result in the expression for \(\mathrm{d}\, p / \mathrm{d}\, c\), we obtain

   \begin{align*} \frac{\mathrm{d}\, p}{\mathrm{d}\, c} &= \frac{e_{d}(p) \left(e_{d}(p) + 1\right)}{\left(e_{d}(p) + 1\right)^{2} - 1 + \left(e_{d}(p)\right)^{2}} = \frac{1}{2}. \end{align*}

• Marseille:
  • Sellers have stands.
  • Buyers approach the stands.
  • Prices are pairwise agreed.
• Ancona:
  • Fish crates go down a conveyor belt.
  • A screen displays the initial price. An auctioneer decides the initial price.
  • The price declines incrementally until a buyer pushes a button to buy.
• Gallegati, Giulioni, Kirman, & Palestrini, 2011 studied prices in these two markets.

• Individual vendors vary prices of the same kind of fish to different buyers by up to \(30\%\).
• Some buyers are loyal to certain sellers, while others circulate.
• More loyal customers pay slightly higher prices than their less loyal counterparts.
• Sellers prioritize the demand of more loyal customers.
• This market structure leads to higher profits for sellers and higher payoffs (counting demand satisfaction) for \(90\%\) of loyal customers.

• The classic typology from Pigou, 1920, distinguishes three price discrimination classes.

• Their names are not very helpful: first, second, and third price discrimination degree.
• We follow the naming convention of Belleflamme & Peitz, 2010.
  • Group Pricing (aka third-degree price discrimination)
  • Personalized pricing (aka first-degree or perfect price discrimination)
  • Menu Pricing (aka second-degree price discrimination, or nonlinear pricing)

• Group pricing occurs when a seller offers different prices to different groups of consumers.
• This is perhaps the most common form of price discrimination.
• Example: Student discounts.

• Menu pricing occurs when a seller offers different prices depending on the characteristics of the enclosing package bought, but not across consumers.
• For instance, packages (or menus) may differ in the number of units of the good they contain.
• Quantity discounts or premia are the most prominent examples, but there are also other cases…
• Some airlines add surcharges for one-way tickets.

• Personalized pricing occurs when a seller sets the price of each sold unit equal to the maximum amount that the consumer who is buying it is willing to pay for that unit.
• The reservation value of each purchased unit must be known to the seller.
• In reality, impractical due to the enormous amount of required information.
• It can be thought of as an extreme case of group pricing.

Perfect price discrimination can increase the total welfare in cases of monopolies. Not everyone is doing better, though. The monopoly extracts the whole welfare and consumers nothing.

Monopolistic competition is a structure describing a market with many firms offering similar, but not perfectly identical, commodities or services. Firms can affect prices but not as effective as a monopolist.

1. Suppose that a monopolist can adopt a group pricing strategy in a market with two groups. Both groups' demands have constant price elasticities correspondingly equal to \(e_{d,1}\) and \(e_{d,2}\). The marginal production cost is constant at \(c_{1}\), and there are no fixed cost. What price does the monopolist charge to each group?

   The monopolist solves \[ \max_{p_{1}, p_{2}} \left\{ p_{1} d_{1}(p_{1}) + p_{2} d_{2}(p_{2}) - c_{1}\left(d_{1}(p_{1}) + d_{2}(p_{2})\right) \right\}. \] The optimality conditions for this problem are

   \begin{align*} d_{i}(p_{i}) + p_{i} d_{i}'(p_{i}) - c_{1} d_{i}'(p_{i}) &= 0 \qquad (i=1,2). \end{align*}

   From them, we deduce that

   \begin{align*} p_{i} = \frac{c}{\frac{1}{e_{d,i}} + 1} \qquad (i=1,2). \end{align*}

• In the games that we will examine, the utilities and the choices of players are common knowledge.
• Common knowledge is information that is known and understood in the same way by all the players of a game.
• There is an element of infinite recursion in the idea of common knowledge.

• The agents of a game have common knowledge of a property \(P\) when they all know \(P\), they all know that they know \(P\), they all know that they all know that they know \(P\), etc.

• Each player in a game has one or more decisions to make.

• A single choice made at a particular decision node is called an action.
• The collection of all actions of a player in a game is called a pure strategy (or simply strategy when it is understood from context that it is pure).
• In games where a player has a single decision to make, her actions and strategies coincide.

• Simple games can be represented using a table documenting the primitives of the game.
• This representation is called the normal form of a game.

• The extensive form of a game is a representation in terms of a tree.

• The extensive form depicts more information than the normal form:

  • It illustrates the order in which the players act.
  • It illustrates the information available to each player.

• If a player has the same information at two (or more) nodes, the nodes are connected with a dotted line.
• An information set is a collection of decision nodes that the player making decisions cannot distinguish at the time of decision.
• The player knows that she is located at one of the nodes of the information set, but she does not know at which one of them.

• On some occasions, a player can choose a strategy that makes her better off irrespective of the strategies chosen by other players.
• A strategy that, for all strategies other players can choose, gives a higher payoff to a player compared to every other strategy available to her is called a dominant strategy.

• \(Left \succ_{B} Right\)

• Nash equilibria do not say how, why, or whether these strategies are reached in a game.
• The definition of Nash equilibria suggests that if they are reached, then there is no incentive for anyone to change her behavior.

• Nash equilibria are not unique.
• Multiple situations in a game may constitute points from which no one wants to deviate.
• The payoffs of the players in different Nash equilibria can be significantly different.

• \(\mathrm{NE} = \left\{ \left\{ Bottom, Left \right\}, \left\{ Top, Right \right\} \right\}\)

• No. Nash equilibria can be Pareto inefficient.
• A classic example is the prisoner's dilemma.

• If firm \(2\) chooses \(x_{2} > \frac{1}{2}\), firm \(1\) would like to undercut by a small amount and set \(x_{1} = x_{2} - \varepsilon > \frac{1}{2}\).
• Then firm \(2\) has a profitable deviation by changing to \(x_{2} = \frac{1}{2}\).
• Thus any \(x_{2} > \frac{1}{2}\) cannot be an equilibrium.
• Similarly, any \(x_{2} < \frac{1}{2}\) cannot be an equilibrium.
• Analogous arguments hold for firm \(1\) because of symmetry.

• Therefore, the only possible equilibrium is \(x_{1} = \frac{1}{2} = x_{2}\).
• Firms split the market and make equal profits.
• Any deviation leads to fewer profit for the firm that moved.

• \(\mathrm{NE} = \left\{ \left\{Bottom, \left(Left', Right \right)\right\}, \left\{Bottom, \left(Right', Right \right)\right\}\right\}\)

Pure strategies are collections of actions, one for each decision to be made. Sometimes the players prefer to choose strategies based on some randomization rule. Players can randomize by assigning the probabilities (weights) with which they use their pure strategies. A distribution over the player's pure strategies is called a mixed strategy.

For example, suppose that player \(A\) has two pure strategies (\(Top\) and \(Bottom\)). The mixed strategy \((p, 1-p)\) assigns probability \(p\) to choosing \(Top\) and probability \(1-p\) to choosing \(Bottom\).

Finite games always have at least one Nash equilibrium in mixed strategies (Nash, 1950). A game is finite if the number of players, actions, and decision nodes are finite.

Going back to the Street fighter mechanics, suppose that the payoffs in the interaction between a grappler and a defender are given by the following table.

How can we calculate the mixed strategy Nash equilibrium in this case? The grappler has two pure strategies (\(Command\ Grab\) and \(Normal\ Grab\)). The mixed strategy \((p, 1-p)\) assigns probability \(p\) to choosing \(Command\ Grab\) and probability \(1-p\) to choosing \(Normal\ Grab\). The grappler chooses these probabilities so that it makes the defender indifferent between \(Neutral\ Jump\) and \(Tech\) (why?). \[\underbrace{2p + (-2)(1-p)}_{\substack{\text{Expected payoff when}\\ \text{the defender chooses }\\ Neutral\ Jump}} = \underbrace{(-3)p + 0(1-p)}_{\substack{\text{Expected payoff when}\\ \text{the defender chooses }\\ Tech}} \implies p = \frac{2}{7}.\] Similarly the defender mixes \(Neutral\ Jump\) with \(q\) and \(Tech\) with \(1-q\), such that \[\underbrace{(-2)q + 3(1-q)}_{\substack{\text{Expected payoff when}\\ \text{the grappler chooses }\\ Command\ Grab}} = \underbrace{2q + 0(1-q)}_{\substack{\text{Expected payoff when}\\ \text{the grappler chooses }\\ Normal\ Grab}} \implies q = \frac{3}{7}.\] The Nash equilibrium of the grappler game is \[\left\{\left(\frac{2}{7}, \frac{5}{7}\right), \left(\frac{3}{7}, \frac{4}{7}\right)\right\}.\]

There is no pure strategy Nash equilibrium. However, there is a symmetric mixed strategy equilibrium in which both players randomize using \(\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right)\).

Repeated games are games played at a finite or infinite number of dates, and at each date, the same strategic interaction is repeated. The strategic interaction repeated at each date is called a stage game. When there is a future, threats can be used to sustain cooperation.

For example, suppose that the prisoner's dilemma is used as the stage game of a repeated game with an infinite time horizon. The payoff of all future dates is discounted by \(0 <\delta < 1\). Is there any room for coordination in this case? Players can coordinate by using trigger strategies: If at all previous dates the other player has denied, then deny. Otherwise, confess. If players coordinate, then their payoffs are \[u^{c,i} = -\frac{1}{1 - \delta}\] If player \(i\) deviates at the current date, then her payoff is \[u^{d,i} = -3\frac{\delta}{1 - \delta}\] Coordination can be supported if \[u^{c,i} \ge u^{d,i} \iff \delta \ge \frac{1}{3}\]

1. Consider the following game.

   1. Find all the pure strategy Nash equilibria.
   2. How do you interpret the objective of the players in this game?
   1. There are two Nash equilibria in this game, namely \(\{Foo, Foo\}\) and \(\{Bar, Bar\}\).
   2. The objective of the players is to coordinate. The players would like to choose the same action. Games of this type are called coordination games.

2. Suppose that two players play a game, and it is known with certainty that player \(A\) is not choosing a Nash equilibrium strategy of a game. Should player \(B\) choose her Nash equilibrium strategy?

   Not necessarily. In general, player \(B\) maximizes her payoff by choosing the best response strategy to player \(A\)'s strategy. If the Nash equilibrium strategy of player B is a dominant strategy, then she should choose it. Dominant strategies are best responses irrespective of the choice of other players. Therefore, if her Nash equilibrium strategy is a dominant strategy, she will still maximize her payoff by choosing it. For example, if player \(A\) chooses \(Deny\) in a prisoner's dilemma game, since the Nash equilibrium strategy of player \(B\), namely \(Confess\), is a dominant strategy, she maximizes her payoff by choosing it.

   In other games, in which her Nash equilibrium strategy is not dominant, the best response to the choice of player \(A\) can differ from the Nash equilibrium strategy and, therefore, player \(B\) might switch to a non Nash equilibrium strategy. For example, consider the game obtained by modifying the actions' labels and payoffs of the prisoner's dilemma in the following way.

   In this game, if player \(A\) chooses \(Bottom\), the best response of player \(B\) is \(Right\). As is to be expected, \(Left\) is not a dominant strategy.

3. Find the equilibrium of the following game using backward induction.

   The Nash equilibrium obtained by backward induction is \[\left\{\left(Bottom, Up', Up \right), \left(Right', Right\right)\right\}\]

4. Consider the game

   1. Find all the pure strategy Nash equilibria of the game.
   2. Suppose that player A plays first. Draw the extensive form of the game.
   3. Solve the sequential game using backward induction.
   4. Are all Nash equilibria of the static game present in the sequential game? Why, or why not?
   1. There are two Nash equilibria given by \(\{Top, Left\}\) and \(\{Bottom, Right\}\).
   2. The extensive form of the sequential game is
   3. The Nash equilibrium obtained by backward induction is \(\{Top, \left(Left', Right\right)\}\).
   4. Nash equilibria in which player \(A\) plays \(Bottom\) cannot be obtained by backward induction in the sequential game. This happens because player \(A\) prefers choosing \(Top\) at date \(1\) as she receives a higher payoff. Even if player \(B\) attempts to convince player \(A\) to choose \(Bottom\) by threatening that she will play \((Right',Right)\), her statement is not credible. Once player \(A\) chooses \(Top\), player \(B\) receives a greater payoff by choosing \(Left'\). In general, dynamic aspects in a game can affect the resulting the credibility of some Nash equilibria. Therefore, when analyzing strategic interactions for which time is a central component, it is important not to neglect it because doing so might lead to less refined equilibrium outcomes.

1. Consider the game

   1. Does the game have any pure strategy Nash equilibria?
   2. Find all the Nash equilibria of the game.
   1. No. There is no set of strategies that constitute best responses to one another.

   2. Suppose that player \(A\) randomizes by playing \(Top\) with probability \(p > 0\) and \(Bottom\) with probability \(1-p\). If this mixed strategy is a best response, it must equalize the expected payoff of the strategies of player \(B\) because otherwise, player \(B\) would choose either \(Left\) or \(Right\). The unique best response of player \(A\) to \(Left\) is \(Bottom\), which means that the mixed strategy \((p, 1-p)\) with \(p>0\) cannot be the best response to player \(B\)'s choice of \(Left\). Similarly, we can exclude the case of player \(B\) choosing \(Right\).

      Therefore, for the mixed strategy to be a best response, it should hold \[ \underbrace{p \cdot 2 + (1-p) \cdot 0}_{\text{Player B's expected payoff for $Left$}} = \underbrace{p \cdot 0 + (1-p) \cdot 2}_{\text{Player B's expected payoff for $Right$}}. \] This implies that \(p= 1/2\).

      Suppose that player B randomizes by playing \(Left\) with probability \(q > 0\) and \(Right\) with probability \(1-q\). Using similar arguments as above, the randomization of player B should equalize the payoff of the strategies of player A, namely \[ \underbrace{q \cdot 0 + (1-q) \cdot 2}_{\text{Player A's expected payoff for $Top$}} = \underbrace{q \cdot 2 + (1-q) \cdot 0}_{\text{Player A's expected payoff for $Bottom$}}. \] The last condition implies that \(q= 1/2\).

      Combining the above, the unique Nash equilibrium of the game is given by the mixed strategies \[ \left\{\left(p = \frac{1}{2}, 1-p = \frac{1}{2}\right), \left(q = \frac{1}{2}, 1-q = \frac{1}{2}\right)\right\}. \]

2. We consider the prisoner's dilemma as the stage game of a repeated game.

   1. Suppose that the stage game is repeated twice. Find the unique Nash equilibrium of the game.
   2. Suppose that the stage game is repeated an arbitrary finite number of times (say \(n=10^{10}\)). Find the unique Nash equilibrium of the game.
   3. Why is it impossible for the players to coordinate and play the Pareto efficient strategies in the above cases?
   1. We can find the Nash equilibrium by backward induction. At date \(2\), the unique Nash equilibrium of the stage game is obtained when both players choose \(Confess\). Since there are no further dates in which the stage game is repeated, the players cannot coordinate in the Pareto efficient outcome by promising cooperation in the future. At date \(1\), the unique Nash equilibrium of the stage game is obtained once more for both players choosing \(Confess\). Since the only Nash equilibrium of the game at date \(2\) is the pair \(\{Confess, Confess\}\), players cannot coordinate at the Pareto efficient outcome at date \(1\) by threatening to punish non-cooperation in the future. Therefore, the unique Nash equilibrium of the game is obtained by \(\{(Confess, Confess), (Confess, Confess)\}\).
   2. If the game is repeated a finite number of times, the unique Nash equilibrium is obtained when both players choose \(Confess\) at every date. This result can be obtained by backward induction. At the final date of the game, the unique Nash equilibrium of the stage game is \(\{Confess, Confess\}\), justified as in the case of the two repetitions. Suppose that we have established that the unique Nash equilibrium of the last \(k\) repetitions of the game is \[ \left\{\underbrace{(Confess, ..., Confess)}_{k-times}, \underbrace{(Confess, ..., Confess)}_{k-times}\right\}. \] With the argument used for date 1 in the case of two repetitions, one concludes that the unique Nash equilibrium of the \((n-k)\) -th date of the game is \(\{Confess, Confess\}\). Continuing in this respect, we conclude that the unique Nash equilibrium of the repeated game is obtained when both players choose \(Confess\) at every date.
   3. When a stage game with a unique Nash equilibrium is repeated a finite number of times, the players cannot establish credible promises or threats. However, if the repeated game has an infinite horizon, then coordination strategies that deviate from stage game Nash equilibria are feasible, even if there is only a unique Nash equilibrium in the stage game. An example of this is the infinitely repeated prisoner's dilemma. Promises and threats can be established when a stage game is repeated only a finite number of times, only if the stage game has multiple Nash equilibria, one of which can be used in coordination and one in non-coordination.

• At the end of the 1980s, the world lysine industry consisted of three significant sellers:
  • Ajinomoto,
  • Kyowa, and
  • Sewon.
• The three largest consumption regions were Japan, Europe, and North America.
• Most production took place in Japan, but it was based on imports of US dextrose.
• Ajinomoto had the largest share of the world market.

• In February 1991, Archer Daniel Midland Co. (ADM) entered the market and built by far the world's largest lysine plant in the US.

• ADM hired biochemist Mark Whitacre, Ph.D., as head of the new division.
• ADM's plant was three times the size of Ajinomoto's largest plant.
• ADM gave Ajinomoto and Kyowa executives an unrestricted tour to show its production capacity.
• Companies engaged in a price war.
  • Three months before ADM's entry, the average US lysine price was \($1.22\) per pound.
  • After an 18-month price war, the US price averaged \($0.68\) per pound.
  • ADM's share of the US market reached \(80\%\).

• After the price war, ADM was willing to soften competition.

• In 1992, Mark Whitacre and his boss Terrance Wilson met with top Ajinomoto and Kyowa managers.
• Wilson proposed forming a world lysine association that would regularly meet.
• The new association would collect and distribute market information.
• Wilson also suggested that the new association could provide a convenient cover for illegal price-fixing discussions! (Connor, 2001)
• After a year, the lysine association was founded, met quarterly, and performed the two functions that Wilson proposed.

WILSON: The only thing we need to talk here because we are gonna get manipulated by these God damn buyers, they're sh, they can be smarter than us if we let them be smarter.

MIMOTO: (Laughs).

WILSON: Okay?

MIMOTO: (ui).

WILSON: They are not your friend. They are not my friend. And we gotta have 'em. Thank God we gotta have 'em, but they are not my friends. You're my friend. I wanna be closer to you than I am to any customer. 'Cause you can make us, I can make money, I can't make money. At least in this kind of a market. And all I wanna is ta tell you again is let's-let's put the prices on the board.

Co-conspirator explains how end-of-year compensation scheme eliminates incentives to cheat on cartel

• In 1992, Mark Whitacre became an FBI whistle-blower. He is the highest-level corporate executive to ever have done so.
• Whitacre's wife pressured him into becoming a whistle-blower. She threatened to inform the FBI herself, if Whitacre wouldn't do it.
• Whitacre informed the FBI that he and other ADM executives performed illegal price-fixing operations.
• Over the next three years, Whitacre collected information and recorded conversations with ADM executives and competitors.
• ADM settlement involved the greatest US federal charges at the time.
• Whitacre was convicted for embezzling and money laundering, which he performed while cooperating with the FBI in the price fixing case.

• Why did ADM initially engage in a price war?

• Why did it initiate the price fixing discussions afterward?
• Why was there so much distrust among companies?

• Oligopolies may compete using pricing strategies or by choosing quantities.
• Different means of competition strategies crucially affect the market outcome.
• The means of competition are decisive components of the market structure.

• Oligopolies have used explicitly collusive strategies in the past (e.g., cartels).
• Nowadays, collusion is usually illegal.
• Instances of tacit collusion have also been documented.
• Tacit collusion strategies do not explicitly require the exchange of information.

• The necessary condition for each firm is

\[28 - 2 q_{j} - 4 q_{i} = 4\]

• Solving for \(q_{i}\) gives the best response of firm \(i\)

\[q_{i} = \frac{24 - 2 q_{j}}{4}.\]

• Combining the two best responses gives the Nash equilibrium

\[q_{i} = 4 = q_{j}.\]

• Profits are then

\[\pi_{i} = 32.\]

• Suppose that firm \(j\) sets a price \(p_{j}\) that is greater than the marginal cost of firm \(i\) (i.e., \(4\)).
• Firm \(i\) can undercut by a small amount and grab all the market. For instance, set price \(p_{i} = \frac{p_{j} + 4}{2}\).
• Thus, firm \(i\) can only set a price equal to firm \(j\)'s marginal cost.
• Analogous arguments hold for firm \(j\)'s strategy.

• The only possible equilibrium is to set a price equal to the (common) marginal cost.
• Firms do not have any incentive to deviate.
• Setting lower prices leads to losses.
• Setting higher prices leads to zero profit.
• Even with two firms, price competition leads to price setting similar to the competitive equilibrium case.

Firms choose their strategies at the same time. Both firms choose their supplied quantities. This market structure is known as the Cournot model of competition.

We extend the problem by allowing \(n>2\) firms that simultaneously choose their strategies. All firms choose their supplied quantities.

How is equilibrium affected when more than two firms are in the market? Each firm solves \[\max_{q_{i}} \left\{ p\left( \sum_{j=1}^{n} q_{j} \right) q_{i} - c(q_{i}) \right\}.\] Analogously to the two-firm case, we obtain \(n\) best response functions \[q_{i} = b_{i}\left((q_{j})_{j\neq i}\right) \quad\quad (i = 1,... , n).\] Solutions to the system of best responses (if any) are the Nash equilibrium of this oligopoly model.

1. Suppose that there are two firms in a market with an affine inverse demand function \(p(q) = p_{0} + p_{1}q\), where \(p_{0}>0\) and \(p_{1}<0\). The firms compete by simultaneously choosing quantities and have constant marginal costs equal to \(c_{1}\). Production has no fixed cost. Find the equilibrium outputs, price, and profits.

   Let \(i\neq j\) for \(i,j=1,2\). Firm \(i\) chooses its strategy by solving \[ \max_{q_{i}} \left\{ p(q_{i} + q_{j}) q_{i} - c_{1} q_{i} \right\}. \] For interior solutions, the firm sets

   \begin{align*} p_{0} + 2p_{1}q_{i} + p_{1}q_{j} - c_{1} = 0, \end{align*}

   which gives the best response function

   \begin{align*} b_{i}(q_{j}) = \frac{c_{1} - p_{0} - p_{1}q_{j}}{2p_{1}} . \end{align*}

   Since the problem is symmetric, we can set \(q_{i}=q_{j}\) in the first order condition to easily obtain the equilibrium output

   \begin{align*} q_{i} = \frac{c_{1} - p_{0}}{3p_{1}} . \end{align*}

   Thus, the total market quantity is

   \begin{align*} q = q_{1} + q_{2} = 2\frac{c_{1} - p_{0}}{3p_{1}}, \end{align*}

   which implies that the equilibrium price is

   \begin{align*} p(q) = \frac{p_{0} + 2 c_{1}}{3}. \end{align*}

   Finally, each firm \(i\) makes profit \[ \pi_{i}= (p(q) - c_{1}) q_{i} = - \frac{\left(c_{1} - p_{0}\right)^{2}}{9p_{1}} . \]

2. Can an oligopolistic market structure result in an efficient level of output?

   Yes. The simultaneous price competition with homogeneous products leads firms to choose the competitive price, which, in turn, results in the production of the efficient output level. In contrast, competition in quantities results in deadweight losses.

3. Consider a market with inverse demand function \(p(q) = 100 - 2q\). The total cost function for any firm in the market is given by \(c(q) = 4q\).

   1. What is the marginal cost for any firm in the market?
   2. Calculate the perfect competition market quantity and price.
   3. Suppose that the market consists of two firms that compete by simultaneously choosing their supplied quantities. Find the best responses and the equilibrium market output and price?
   4. Draw the two best responses in a single graph to indicate the equilibrium point.
   5. Suppose that the two firms collude by maximizing and splitting their joint profit. Find market output and the market price. Compare them with the non-collusive market output and price.
   6. Suppose that a firm decides to deviate from the collusive output, while the other one keeps its collusive strategy unchanged. What are the firms' output levels and profits?
   7. Is the collusion strategy sustainable in this game?
   1. \(c'(q) = 4\).

   2. The perfect competition price is equal to the marginal production cost, i.e., \(p_{c}=4\). By inverting the inverse demand and substituting the competitive price, we obtain

      \begin{align*} q_{c} = \frac{100 - p_{c}}{2} = 48. \end{align*}

   3. Each firm solves

      \begin{align*} \max_{q_{i}} \left\{ p\left( q_{i} + q_{j} \right) q_{i} - c(q_{i}) \right\}, \end{align*}

      with first order condition

      \begin{align*} 100 - 4q_{i} - 2 q_{j} -4 = 0. \end{align*}

      Therefore, the best response of each firm is

      \begin{align*} b_{i}(q_{j}) = \frac{48 - q_{j}}{2} \end{align*}

      and the firm's optimal quantity is \(q_{i} = 16\). The market quantity is \(q_{s} = 32\), and the market price is \(p_{s} = 36\).

   4. The intersection of the best responses indicates the equilibrium quantities.

   5. In this case, the two firms solve the monopolist problem and split the profit. The monopolistic quantity is given by

      \begin{align*} q_{m} = \frac{100 - 4}{4} = 24, \end{align*}

      in which case, the market price is \(p_{m}=52\). With collusion, the market power of the firms increases, the produced output level decreases (\(q_{m} = 24 < 32 =q_{s}\)), while the market price increases (\(p_{m} = 52 > 36 =p_{s}\)).

   6. Suppose that firm \(i\) deviates from collusion, while firm \(j\) keeps producing half of the monopolistic quantity. The best deviation of firm \(i\) can be calculated from its best response function, namely

      \begin{align*} q_{b} = b_{i}\left(\frac{q_{m}}{2}\right) = \frac{48 - 12}{2} = 18. \end{align*}

      The best deviation for the firm is to produce at an output level greater than the level of the simultaneous quantity competition (i.e., \(q_{b} = 18 > 16 = q_{s}/2\)). Firm \(i\) takes advantage of the fact that firm \(j\) is not producing at its best response output level and produces more to grab a greater share of the market. The market quantity is

      \begin{align*} q_{w} = q_{b} + \frac{q_{m}}{2} = 18 + 12 = 30. \end{align*}

      Then, the price is \(p_{w} = 40\), and the firms' profits are

      \begin{align*} \pi_{i} &= (p_{w}-c)q_{b} = 36 \cdot 18 = 648 \\ &> 576 = 48 \cdot 12 = (p_{m}-c)\frac{q_{m}}{2} = \frac{\pi_{m}}{2} \end{align*}

      and

      \begin{align*} \pi_{j} &= (p_{w}-c)\frac{q_{m}}{2} = 36 \cdot 12 = 432 < 576 = \frac{\pi_{m}}{2} . \end{align*}
   7. No. Each firm has a profitable deviation from producing half of the monopolistic output in this static game. More than a single date is needed for firms to be able to collude successfully.

1. Give an example in which there is no equilibrium in a duopoly market in which firms simultaneously compete using quantities. Draw the best responses of the firms in this market.

   Suppose that the inverse demand function of the market is given by \(p(q) = p_{0} + p_{1} q = 25 - 4 q\). Let the marginal cost of firm 1 be \(c_{1}=15\) and that of firm 2 to be \(c_{2}=1\). For these values, if there was an equilibrium in this market, according to the best response functions, firm 1 should produce

   \begin{align*} q_{1} = \frac{2c_{1} - c_{2} - p_{0}}{3 p_{1}} = -\frac{30 - 1 - 25}{12} < 0. \end{align*}

   Therefore, this market does not have an equilibrium. Visually, this results in two best response functions that do not cross each other.

2. Suppose there are \(n\) identical firms in a market competing by simultaneously choosing quantities. Show that the elasticity of the market demand must be less than \(-1/n\) for an equilibrium to exist.

   This result is the analog of the upper bound for the elasticity of monopolistic markets discussed in exercise 4. In particular, the exact result of the monopolistic case is obtained for \(n=1\). To simplify notation, let \(q = \sum_{j=1}^{n} q_{j}\). Each firm solves

   \begin{align*} \max_{q_{i}} \left\{ p\left( q \right) q_{i} - c(q_{i}) \right\}, \end{align*}

   with the first order condition of firm \(i\) being

   \begin{align*} p'\left(q\right) q_{i} + p\left(q\right) - c'(q_{i}) = 0. \end{align*}

   By the inverse function theorem, we can rewrite the above condition as

   \begin{align*} 0 &= \frac{1}{d'\left(p(q)\right)} q_{i} + p\left(q\right) - c'(q_{i}), \end{align*}

   where \(d\) is the demand function, and, then, express it in terms of the demand elasticity as

   \begin{align*} \frac{c'(q_{i})}{p\left(q\right)} &= \frac{1}{d'\left(p(q)\right)} \frac{q}{p(q)} \frac{q_{i}}{q} + 1 = \frac{1}{e_{d}\left(p(q)\right)} \frac{q_{i}}{q} + 1. \end{align*}

   Since all firms are identical, the equilibrium, if it exists, is symmetric. This means that \(q_{i} = q / n\). Substituting the symmetry condition in the first order condition gives

   \begin{align*} p\left(q\right) &= \frac{c'(q_{i})}{\frac{1}{e_{d}\left(p(q)\right)} \frac{1}{n} + 1}. \end{align*}

   Since the marginal cost is positive, positive prices require that

   \begin{align*} \frac{1}{e_{d}\left(p(q)\right)} \frac{1}{n} + 1 > 0, \end{align*}

   which is equivalent to \(e_{d}\left(p(q)\right) < - 1 / n\).

• In the early 1960s, the Soviet government planned for cotton, or 'white gold', to become a major exporting industry.

• Amu Darya and Syr Darya rivers were diverted from feeding the Aral sea to irrigating the desert.
• The amount of water taken from the rivers doubled between 1960 and 2000.
• The plan was a success.
• In 1988, Uzbekistan was the largest exporter of cotton in the world.
• In 2006, \(17\%\) of Uzbekistan's exports came from cotton.

• A positive production externality is an externality for which a production process is boosted by the actions of external economic agents that are not beneficiaries of the profit derived from the process.
• A negative production externality is an externality for which a production process is hindered by actions of economic agents that are not beneficiaries of the profit derived from the process.

• Suppose there are two markets with one profit maximizing firm in each of them.
• The firm of market \(1\) produces cotton and uses a river for irrigation. The more water it draws from the river, the lower the production cost becomes.
• The firm of market \(2\) is a fishery. It does not control the level of water in the river. However, the less water there is in the river, the more costly it is to catch fish.
• Let \(p_{c}=5\) be the price of cotton and \(p_{f}=2\) be the price of fish. The water level is not priced in the market.
• Let \(\xi\in[0,1]\) be a variable that measures how much water of the river is used for irrigation as a percentage of its full capacity.

• The cotton producer has production cost \[c_{c}(q_{c}, \xi) = q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right)\]
• The cost is decreasing in the percentage of water used in irrigation.
• The fishery cost function is \[c_{f}(q_{f}, \xi) = q_{f}^{2}\left(1 + \xi^{2}\right)\]
• The cost is increasing in the percentage of water used in irrigation (externality).

• The cotton producer solves \[\max_{q_{c}, \xi} \left\{ p_{c}q_{c} - q_{c}^{2} \left(5 + \left(1 - \xi\right)^{2}\right) \right\}\]
• The fishery problem is \[\max_{q_{f}} \left\{ p_{f}q_{f} - q_{f}^{2}\left(1 + \xi^{2}\right) \right\}.\]
• The fishery does not control the level of externality, but its cost function is increasing in it.

• The cotton producer would like to choose \(q_{c}\) and \(\xi\) such that

  \begin{align*} 5 &= 2 q_{c} \left(5 + \left(1 - \xi\right)^{2}\right) \\ 0 &= -2 q_{c}^{2} \left(1 - \xi\right) \end{align*}
• This gives \(\xi = 1\) and \(q_{c}=\frac{1}{2}\).
• The fishery chooses \[2 = 2 q_{f} \left(1 + \xi^{2}\right)\]
• This gives (for \(\xi = 1\)) \(q_{f}=\frac{1}{2}\).

• The river is dried out (\(\xi = 1\)).
• The cotton producer has profit \[\pi_{c} = 5 \frac{1}{2} - \frac{1}{4} 5 = \frac{5}{4}\]
• The fishery has profit \[\pi_{f} = 2 \frac{1}{2} - \frac{1}{4} 2 = \frac{1}{2} \]

• Consider a merged, profit maximizing firm that produces in both markets.
• The merged firm solves \[\max_{q_{c}, q_{f}, \xi} \left\{ p_{c}q_{c} + p_{f}q_{f} - q_{c}^{2} \left(1 + \left(1 - \xi\right)^{2}\right) - q_{f}^{2} \xi \right\} .\]

• The merged firm chooses

  \begin{align*} 5 &= 2 q_{c} \left(5 + \left(1 - \xi\right)^{2}\right) \\ 2 &= 2 q_{f} \left(1 + \xi^{2}\right) \\ q_{c}^{2} 2 \left(1 - \xi\right) &= q_{f}^{2} 2 \xi \end{align*}
• This gives approximately \(\xi = 0.17\), \(q_{c}=0.44\), and \(q_{f}=0.97\).

• The externality is internalized.
• The river is not dried out (\(\xi \approx 17\%\)).
• Cotton production is reduced (\(0.44\) instead of \(\frac{1}{2}\)).
• Fish production is increased (\(0.97\) instead of \(\frac{1}{2}\)).
• The merged firm's profit is \[\pi_{m} \approx 2.07 > \frac{5}{4} + \frac{1}{2} = 1.75\]
• The profit of the merged firm is greater than the sum of the profits of the market with two firms
• This shows that the market solution is not efficient.

• The model shows that internalizing the externality is economically more efficient than the market solution.

• In the case of the Aral Sea, the externality was not internalized, despite that the Soviet Union's economy was centrally planned.
• Why did this happen?
• Did we account for all the costs in the model? Did the Soviet regime do?

• The combination of non-excludability and non-rivalry makes the market-based provision of public goods impossible.
• Non altruistic agents avoid contributing to the public good production and attempt to free-ride.
• This is typically Pareto inefficient.

• Suppose that there are \(2\) firms in a market.
• Each firm has a budget equal to \(100\) Euros.
• They decide how much to contribute to the production of a network infrastructure (say \(x_{1}\) and \(x_{2}\)).
• The sum of the contributions is used in the production of the infrastructure. \[q = 3\frac{x_{1} + x_{2}}{2}.\]
• The produced output (infrastructure capacity) is equally split.

• The firms' profits are \[\pi_{i}\left(x_{1}, x_{2}\right) = 3\frac{x_{1} + x_{2}}{4} + 100 - x_{i}.\]

• The profit of firm \(1\) is strictly decreasing in its contribution. \[\frac{\partial \pi_{1}}{\partial {x_{1}}}\left(x_{1}, x_{2}\right) = \frac{3}{4} - 1 < 0.\]
• Therefore, \(x_{1} = 0\) and \[\pi_{1}\left(0,x_{2}\right) = 3\frac{x_{2}}{4} + 100.\]
• Since the game is symmetric, the same argument is valid for agent \(2\).
• Therefore the Nash equilibrium of the game is \(x_{i} = 0\), with payoffs \(\pi_{i} = 100\) for every agent \(i\).
• The total welfare, in this case, is equal to \(\pi_{1} + \pi_{2} = 200\).
• Both firms try to free-ride, and no production takes place.

• The total welfare in the market is determined by

  \begin{align*} W\left(x\right) = 3\frac{x}{2} + 200 - x. \end{align*}
• The total welfare is increasing in \(x\). Therefore, it is maximized for \(x = 200\).
• The maximized total welfare is \(W(200) = 300\).
• This solution is Pareto dominates the market solution \(W(200)>\pi_{1} + \pi_{2}\).

Suppose there are two markets with one profit maximizing firm in each of them. The emitter is a firm in the first market producing an externality that negatively affects the production cost of the second market. The receiver is a firm in the second market that cannot control the externality, yet this externality positively affects its production cost.

A public good is a good that is non-excludable and non-rivalrous. The consumption of a commodity is non-rivalrous when its consumption from an agent does not affect the availability of the commodity for other agents. The consumption of a commodity is non-excludable when it is impossible to exclude agents from its consumption.

1. Consider two monopolistic markets intermingled by an externality. In the first market, the firm's profit as a function of output is \(\pi_{1}(q_{1}) = 48 q_{1} - q_{1}^{2}\). The output of the first monopolist affects the profit of the monopolist in the second market as an externality. The second firm's profit as a function output is \(\pi_{2} = (60 - q_{1}) q_{2} - q_{2}^{2}\).

   1. Suppose that each firm independently maximizes its profit. Calculate the optimal quantity and profit of each market. Moreover, calculate the total profit for both markets.
   2. Suppose that the firm in market \(1\) is not allowed to produce any output. Calculate the optimal quantity and profit in market \(2\).
   3. Suppose that firm \(1\) has to pay a transfer to firm \(2\), equal to the damages caused by the production externality. How do the profit functions of the two firms change? Calculate the optimal quantity and profit of each market. Calculate also the total profit in both markets.
   4. Suppose that the two firms merge. What is the resulting profit function of the merged firm? Calculate the optimal quantity and profit.
   5. Compare the profits of the above cases. Which case is the most efficient?

   1. Firm 1 solves

      \begin{align*} \max_{q_{1}} \left\{ 48 q_{1} - q_{1}^2 \right\}, \end{align*}

      which results in \(q_{1} = 24\). Firm 2 solves

      \begin{align*} \max_{q_{2}} \left\{ (60 - q_{1}) q_{2} - q_{2}^2 \right\}, \end{align*}

      which gives

      \begin{align*} q_{2} = 30 - \frac{q_{1}}{2} = 30 - 12 = 18. \end{align*}

      Then, we can calculate the profits

      \begin{align*} \pi_{1} &= 48 \cdot 24 - 24^{2} = 576, \\ \pi_{2} &= (60 - 24) 18 - 18^{2} = 324, \\ \pi &= \pi_{1} + \pi_{2} = 900. \end{align*}

   2. In this case, \(q_{1}=0\) and the quantity that maximizes profit in the second market is

      \begin{align*} q_{2} = 30 - \frac{q_{1}}{2} = 30. \end{align*}

      The firm's profit is

      \begin{align*} \pi = \pi_{2} = (60 - 0) 30 - 30^{2} = 900. \end{align*}

   3. With the transfer, firm 1 solves

      \begin{align*} \max_{q_{1}} \left\{ 48 q_{1} - q_{1}^2 - q_{1}q_{2}\right\}, \end{align*}

      which results in a best response function

      \begin{align*} q_{1} = 24 - \frac{q_{2}}{2}. \end{align*}

      Firm \(2\) solves

      \begin{align*} \max_{q_{2}} \left\{ 60 q_{2} - q_{2}^2 \right\}, \end{align*}

      which gives \(q_{2} = 30\). Substituting into the best response of the first firm, we get \(q_{1} = 9\). Then, the profits are given by

      \begin{align*} \pi_{1} &= (48 - 30) \cdot 9 - 9^{2} = 81, \\ \pi_{2} &= 60 \cdot 30 - 30^{2} = 900, \\ \pi &= \pi_{1} + \pi_{2} = 981. \end{align*}

   4. The merged firm maximizes

      \begin{align*} \max_{q_{1}, q_{2}} \left\{ 48 q_{1} - q_{1}^2 + 60 q_{2} - q_{2}^{2} - q_{1}q_{2}\right\}. \end{align*}

      The first order conditions of the problem are

      \begin{align*} 48 - 2 q_{1} - q_{2} &= 0, \\ 60 - 2 q_{2} - q_{1} &= 0. \end{align*}

      Solving the above system gives \(q_{1}=12\) and \(q_{2}=24\). The profit of the merged firm is

      \begin{align*} \pi = 48 \cdot 12 - 12^2 + 60 \cdot 24 - 24^{2} - 12 \cdot 24 = 1008. \end{align*}

   5. The merged case is the Pareto efficient structure because the externality is internalized.

      Case	Structure	\(\pi_{1}\)	\(\pi_{2}\)	\(\pi\)
      1	Externality in the second market	576	324	900
      2	First market shuts down	0	900	900
      3	Externality in the first market	81	900	981
      4	Internalized externality			1008

2. Suppose there are two individuals, \(1\) and \(2\), each consuming one private good in the amounts \(x_{1}\) and \(x_{2}\), respectively. The price of this private good is \(p\). In addition, they consume a public good \(G\). The marginal cost of production of \(G\) is equal to one. The amount of the public good is determined by \(G = g_1 + g_2\), where \(g_{1}\) and \(g_{2}\) are the individual contributions of the players. Individual \(i\)'s preferences are represented by a Cobb-Douglas utility function \(u(x_{i}, G) = x_{i}^{\alpha}G^{\beta}\). Both players have a budget of \(B\).

   1. Explain why in this case, \(G\) can be considered a public good.
   2. Set up individual \(i\)'s budget constraint.
   3. Find individual \(i\)'s best response contribution \(g_{i}\).
   4. Compute the Nash equilibrium for the contributions \(g_{1}\) and \(g_{2}\).
   5. Does the Nash equilibrium allocation entail more or fewer resources directed to the public good than the Pareto efficient allocation?
   6. Argue that the Nash equilibrium is not Pareto-efficient.
   1. The good \(G\) is non-rivalrous, as both players consume the exact same good \(G\), and non-excludable as the good is not priced in the market.
   2. The budget of individual \(i\) is \(p x_{i} + g_{i} \le B\).

   3. Individual \(i\)'s optimization problem is

      \begin{align*} \max_{x_{i}, g_{i}} \left\{ u(x_{i}, G) + \lambda(B - p x_{i} - g_{i}) \right\}, \end{align*}

      where \(\lambda\) is the Lagrange multiplier. The first order conditions are

      \begin{align*} \alpha \frac{u(x_{i}, G)}{x_{i}} &= \lambda p, \\ \beta \frac{u(x_{i}, G)}{G} &= \lambda, \end{align*}

      which imply

      \begin{align*} \frac{\alpha}{\beta}G &= px_{i}. \end{align*}

      Combining the last condition with the budget constant, we get

      \begin{align*} \frac{\alpha}{\beta}G &= B - g_{i}, \end{align*}

      from which we conclude that the best response is

      \begin{align*} g_{i} &= \frac{\beta}{\alpha + \beta}B - \frac{\alpha}{\alpha + \beta}g_{j}. \end{align*}

   4. Due to symmetry, the Nash equilibrium can be easily calculated by setting \(g_{i}=g_{j}\), which results in

      \begin{align*} g_{n} &= \frac{\beta}{2\alpha + \beta}B. \end{align*}

   5. The Pareto efficient solutions are obtained by solving the joint maximization problem

      \begin{align*} \max_{x_{1}, x_{2}, G} \left\{ u(x_{1}, G) + u(x_{2}, G) + \mu(2B - p (x_{1} + x_{2}) - G) \right\}, \end{align*}

      where \(\mu\) is the Lagrange multiplier of this problem. The first order conditions are

      \begin{align*} \alpha \frac{u(x_{1}, G)}{x_{1}} &= \mu p, \\ \alpha \frac{u(x_{2}, G)}{x_{2}} &= \mu p, \\ \beta \frac{u(x_{1}, G) +u(x_{2}, G)}{G} &= \mu. \end{align*}

      The first two conditions and the budget constraint imply

      \begin{align*} \alpha \left( u(x_{1}, G) + u(x_{2}, G) \right) &= \mu p \left( x_{1} + x_{2} \right) = \mu \left( 2B - G \right). \end{align*}

      Combining with the third first order condition gives

      \begin{align*} G_{p} &= \frac{\beta}{\alpha + \beta}2B. \end{align*}

      For \(\alpha \neq 0\), \(G_{p} \neq 2 g_{n}\), from which we conclude that the Nash equilibrium is not Pareto efficient.

   6. In the Nash equilibrium, fewer than the Pareto efficient resources are allocated in the production of the public good. Specifically, for all \(\alpha>0\), we have \(G_{p} > 2 g_{n}\).

1. Consider a public good game with two players whose payoffs are determined by \(\pi_{1}(q) = (1 + q) (B_{1} - q x_{1})\) and \(\pi_{2}(q) = (2 + q)(B_{2} - q x_{2})\). In these expressions, \(x_{1}\) and \(x_{2}\) represent the player specific, fixed contributions to producing the public good, and \(q\) the public good output. Players have a binary choice concerning the public good; they can commonly choose \(q = 1\) and purchase the public good, or \(q=0\) and abstain from it. Player 1 has budget \(B_{1}\), while player 2 has budget \(B_{2}\).

   1. What is the maximum amount that player 1 is willing to contribute to the public good?
   2. What is the maximum amount that player 2 is willing to contribute to the public good?
   3. Suppose that \(B_{1} = 100\) and \(B_{2} = 75\). By producing the public good, the players have a Pareto improvement over not producing it, as long as the production cost of the public good is no greater than \(\hat x\). Find \(\hat x\).
   4. Find a combination of specific contributions \(x_{1}\) and \(x_{2}\) whose sum is less than \(\hat x\), but for which letting the players individually maximize their payoffs does not result in a Pareto efficient allocation.

   1. Player 1 prefers consuming the public good if and only if

      \begin{align*} \pi_{1}(1) \ge \pi_{1}(0) \iff 2 (B_{1} - x_{1}) \ge B_{1} \iff x_{1} \le \frac{B_{1}}{2}. \end{align*}

   2. For player 2, an analogous approach gives

      \begin{align*} \pi_{2}(1) \ge \pi_{2}(0) \iff 3 (B_{2} - x_{2}) \ge 2 B_{2} \iff x_{2} \le \frac{B_{2}}{3}. \end{align*}

   3. The total production cost of the public good, say \(x\), has to be covered by the sum of the individual contributions, i.e., \(x_{1} + x_{2} \ge x\). Both players prefer covering the cost if

      \begin{align*} x &\le x_{1} + x_{2} \\ &\le \frac{B_{1}}{2} + \frac{B_{2}}{3} \\ &\le \frac{3B_{1} + 2B_{2}}{6} \\ &= \frac{300 + 150}{6} = 75 =: \hat x \end{align*}
   4. Suppose that \(x_{2}=50\) and \(x_{1}=20\). We then have \(x_{1} + x_{2} = 70 < 75 = \hat x\), but player 2 prefers not to contribute to the public good production because \(x_{2}>25\).

• If \(f\) is continuous at \(x_{0}\),
• then \({\displaystyle \lim _{x\to x_{0}}{f(x)}=f'(x)}\)

• If \({\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L}\) and \({\displaystyle f(x)\leq g(x)\leq h(x)}\),
• then \({\displaystyle \lim _{x\to c}g(x)=L}\).

• If \({\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty ,}\) and \({\displaystyle g'(x)\neq 0}\), and \({\displaystyle \lim _{x\to c}{\frac {f'(x)}{g'(x)}}}\) exists,
• then \({\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}\).

• \({\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}\)

• If \(f\) is a continuously differentiable function with \(f'(x_{0})\) at the point \(x_{0}\), then
  • \(f\) is invertible in a neighborhood of \(x_{0}\),
  • the inverse is continuously differentiable, and

  • the derivative of the inverse function at \(y_{0}=f(x_{0})\) is the reciprocal of the derivative of \(f\) at \(x_{0}\), i.e.

    \begin{align*} \left(f^{-1}\right)'(y_{0}) = \frac{1}{f'(x_{0})} = \frac{1}{f'\left(f^{-1}(x_{0})\right)}. \end{align*}

• Let \(f\) be a continuously differentiable function of two variables and a \((\hat x_{1}, \hat x_{2})\) be a point so that \(f(\hat x_{1}, \hat x_{2}) = \bar f\). If \(f_{x_{2}}(\hat x_{1}, \hat x_{2}) \neq 0\), then

  • there is a neighborhood of \((\hat x_{1}, \hat x_{2})\) and an implicit function \(g\) such that

    \begin{align*} f(x_{1}, g(x_{1})) = c \end{align*}

    for all \(x_{1}\) in the neighborhood, and

  • the derivative of the implicit function is given by

    \begin{align*} g'(x_{1}) = \frac{\mathrm{d} x_{2}}{\mathrm{d} x_{1}} = - \frac{f_{x_{1}}(x_{1}, g(x_{1}))}{f_{x_{2}}(x_{1}, g(x_{1}))} \end{align*}

    for all \(x_{1}\) in the neighborhood.

And more generally

• Indefinite integral (is a function) \[ f(x) = \int_{0}^{x} f'(z) \mathrm{d}\, z, \]
  • also usually denoted as \[ \int f'(x) \mathrm{d}\, z + K, \] where \(K\) is a constant.
• Definite integral (is a number) \[\int_{x_{0}}^{x_{1}} f'(x) \mathrm{d}\, x = f(x_{1}) - f(x_{0}).\]