Technology

Context

• The sustained increase in living standards is primarily due to technological progress (e.g., the industrial revolution, the information revolution, etc.).

• However, not all technological changes favor everyone in the economy.
  • How is technological progress incorporated into firm production?
  • How does it affect the scaling of production?
  • How does it affect the allocation of production resources?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• The Flying Shuttle (Case Study)
• Basic Concepts
• A Cobb-Douglas Example
• Current Field Developments

Learning Objectives

• Explain how economists describe technological progress in modern applications.
• Describe the relationship between production inputs induced by various technologies.
• Analyze the effect of scaling inputs on production output.

The Flying Shuttle

Textile Technology

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

Spinsters

Weavers

The Industrial Revolution Leap

What was the Impact of the Flying Shuttle?

• 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)

Production Output and Factors

• The production output is the result of a production process (Typically a commodity or a service produced by a firm).

• The production factors (or input factors) are the resources used in a production process.

• Which are the most common production factors?

  • Labor
  • Human Capital
  • Physical Capital
  • Financial Capital
  • Land

• How do we measure the production factors and output?

  • Units per time (e.g., hours worked per week, cars produced per year, etc.).

Technological Feasibility

Production Function

• A function that transforms amounts of input factors to the maximum amount of output that can be produced for a given technology is called a production function.

  • Cobb-Douglas production: \[f(K, L) = 10 K^{0.5}L^{0.5}\]
  • Fixed proportions production: \[f(K, L) = \min\left\{K, L\right\}\]
  • Perfect substitutes production: \[f(K, L) = 3 K + 2 L\]
  • Root production: \[f(L) = 5 \sqrt{L}\]

Production Properties

• 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.

Marginal Product

• How does output change when some input changes?
• The marginal product is the rate of change of the production output with respect to an input factor.
• For a production function \(f\), the marginal product is the derivative \(f’\).

The “Law” of Diminishing Marginal Product

Marginal Product Example

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

Returns to Scale

• What happens to production if we double all the input factors?
  • We check what happens to the isoquant.

• An isoquant is a locus of points showing all the technologically efficient ways of combining production factors to produce a fixed level of output.

Increasing Returns to Scale

Decreasing Returns to Scale

Constant Returns to Scale

Returns to Scale Example

• 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 \(\alpha+\beta = 1\).

Marginal Rate of Technical Substitution

• How much more capital do we need to keep the output constant if we decrease labor by one unit?
  • We check what happens to the marginal rate of technical substitution (MRTS).

• The marginal rate of technical substitution is the rate of change of an input factor with respect to changes of another input factor while production is kept constant.

Diminishing Marginal Rate of Technical Substitution

Marginal Rate of Technical Substitution Example

• 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
  • \(\mathrm{MRTS}(K,L) = \frac{\mathrm{MP}(K;L)}{\mathrm{MP}(L;K)} = \frac{2}{3}\frac{L}{K}\).

Current Field Developments

• Technology affects productivity
• How can we measure productivity?
  • Take log in the Cobb-Douglas production function. \[ \log q = \log A + \alpha \log K + \beta \log L\]
  • If we regress \(\log q\) on \(\log K\) and \(\log L\) (which we observe), the estimation constant gives us an estimate of \(\log A\).
  • Initially performed at a macroeconomic level.
  • More advanced methods try to estimate \(A\) at an industry (even gender) specific level.

Comprehensive Summary

• Technology governs
  • how economical is a production process,
  • how different production factors substitute each other (MRTS), and
  • how easily production can be scaled (returns to scale).
• Different production functions describe different technologies.
• In many cases, production output is increasingly more difficult to produce (“Law” of diminishing marginal product).
• In most cases, reducing one input factor (e.g., labor) requires increasing another factor (e.g., capital) to keep output constant (Diminishing MRTS).

Further Reading

• Varian, 2010, Chapter 19
• CORE-Team, 2017, secs. 2.4,7.2

Mathematical Details

Production Function

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 \[f(x_{1}, x_{2}) = \max \left\{ q \ge 0 \colon \left(x_{1}, x_{2}, q \right) \in \mathcal{Q} \right\}.\]

Some Usual Production Functions

Root Production Function (1D Factor Space)

Consider the production set \[\mathcal{Q} =\left\{\left(x, q \right) \in \mathbb{R}^2_{\ge 0} \colon 0 \le q \le A x^{r} \right\}\] where \(A > 0,\ r > 1\). The production function resulting from \(\mathcal{Q}\) is \[f(x) = A x^{r}.\] The parameter \(A\) scales production uniformly, while \(r\) controls the production function’s curvature.

Constant Elasticity of Substitution Production Function (nD Factor Space)

The constant elasticity of substitution production function is given by \[ f\left(x_{1}, x_{2}\right) = A \left(\alpha x_{1}^{\rho} + \left(1 - \alpha\right) x_{2}^{\rho}\right)^{\frac{r}{\rho}} \] where \(A > 0,\ r > 0,\ \alpha \in (0, 1),\ \rho \le 1\). Essentially, production output is modeled as a generalized mean of input factors. The parameter \(A\) is interpreted as productivity, \(r\) controls the returns to scales, \(\alpha\) controls the shares of the factor in production, and \(\rho\) the elasticity of substitution between the factors. Three special cases can be obtained by changing the value of \(\rho\).

• Fixed Proportions

  For \(\rho\to -\infty\), we get \(f\left(x_{1}, x_{2}\right) \to A \min\left\{x_{1}, x_{2}\right\}^{r}\)

• Perfect Substitutes

  For \(\rho=1\), we get \(f\left(x_{1}, x_{2}\right) = A \left(\alpha x_{1} + \left(1 - \alpha\right) x_{2}\right)^{r}\)

• Cobb-Douglas

  Lastly, for \(\rho\to 0\), we get \(f\left(x_{1}, x_{2}\right) \to A x_{1}^{\alpha r} x_{2}^{\left(1 - \alpha\right)r}\)

Returns to Scale

Increasing Returns to Scale

A production function \(f\) exhibits increasing returns to scale if, for any \(t>1\), it holds \[f(t x_{1}, t x_{2}) > t f(x_{1}, x_{2}).\] Output is scaled more than the amount with which the input factors are scaled.

Decreasing Returns to Scale

A production function \(f\) exhibits decreasing returns to scale if, for any \(t>1\), it holds \[f(t x_{1}, t x_{2}) < t f(x_{1}, x_{2}).\] Output is scaled less than the amount with which the input factors are scaled.

Constant Returns to Scale

A production function \(f\) exhibits constant returns to scale if, for any \(t>0\), it holds \[f(t x_{1}, t x_{2}) = t f(x_{1}, x_{2}).\] Output is scaled exactly as the amount with which the input factors are scaled.

What Happens when some Factors are Fixed?

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 \[ f_{2}(x_{1}) := f_{1}(x_{1}, \hat{x_{2}}). \]

Usually Assumed Properties

Monotonicity

A production function is monotonic, if whenever inputs increase, then the outputs do not decrease. \[x_{1} \ge x_{2} \implies f(x_{1}) \ge f(x_{2})\] If \(f\) is differentiable, then \(f’\ge 0\), if and only if \(f\) is monotonic.

Free Disposal

A technology is characterized by free disposal if inputs can be discarded at no cost. If a technology exhibits free disposal, its production function is monotonic (why?).

Production Set Convexity

A technology is convex when mixtures of feasible combinations are also feasible. If technology is convex, then the corresponding production function is concave. A (production) function \(f\) is concave if for any \(\lambda\in[0,1]\) we have \[f(\lambda x_{1} + (1-\lambda)x_{2}) \ge \lambda f(x_{1}) + (1-\lambda)f(x_{2}).\] If \(f\) is differentiable, then \(f’’\le 0\) if and only if \(f\) is concave.

Marginal Product

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.

Marginal Rate of Technical Substitution

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.

Elasticity of 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.

Exercises

Group A

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.

Group B

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.

Group C

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*}

References

References

Topic's Concepts

• marginal rate of technical substitution
• constant returns to scale
• decreasing returns to scale
• increasing returns to scale
• isoquant
• law of diminishing marginal product
• marginal product
• free disposal
• monotonic
• root production
• perfect substitutes production
• fixed proportions production
• cobb douglas production
• production function
• production set
• feasible allocation
• input factors
• production factors
• production output