Market Supply

Context

• Prices and traded quantities in any market are predominantly determined by demand and supply. Our understanding of supply is based on individual firm behavior.

• Nevertheless, most empirical estimations, analysis, and media discussions focus on markets instead of particular firms.
  • How do we consolidate individual firm production decisions into market supply?
  • Why do most popular discussions focus on a market level?
  • Why do we frequently rely on econometrically modeling markets as being competitive?

Course Structure Overview

Lecture Structure and Learning Objectives

Structure

• European Common Agriculture Policy (Case Study)
• An Aggregation Exercise
• Basic Concepts
• An Empirical Market Estimation Example
• Current Field Developments

Learning Objectives

• Describe how individual firms’ supplies are aggregated market supply.
• Illustrate aggregation with examples.
• Explain why successful policy interventions at a market level are challenging.
• Describe the ideal market structure of perfect competition.
• Provide an empirical example of using market models to estimate price effects on supply.

European Common Agriculture Policy

• In the late 1990s and early 2000s, the CAP aimed to maintain the status of European agriculture.

• It used as its main policy instrument the minimum intervention price.
• EU maintained high demand for agricultural products by guaranteeing suppliers minimum prices for their products.

Impact on the EU Markets

• 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.

Butter Mountains

wikipedia

• However, the EU had to purchase and store large quantities of produced agricultural products on many occasions.

• This led to the creation of butter mountains, milk lakes, and wine lakes.
• The alcohol/wine stock peaked at \(352\) million liters in 2007.

Impact Outside the EU

UNDP, 2003

• How to deal with the stock?

• Some products (e.g., diaries) were exported using subsidies to low-income countries.
• Farmers in developing countries could not keep up with cheap (subsidized) competition from the EU.
• The UNDP’s 2005 Human Development Report suggests that the main difficulty in the WTO agriculture negotiations was rich country subsidies.

The Animal Spirits

• 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.

A Practical Aggregation Example

• Suppose \(3\) firms are producing bicycles according to the supplies in the following table.
• What is the aggregate supply of bicycles in the market?

Market supply Aggregation

• For a given price level, the market supplied quantity is obtained by aggregating the supplied quantities of all firms in the market.
• The market supply function is the sum of the firms’ supply functions for each price level.
• The inverse market supply function is the inverse of the market supply. Namely, a mapping that gives a market price corresponding to each aggregate supplied quantity.

A Graphical Aggregation Example

Market Equilibrium

• We say that a market is in equilibrium if buyers and sellers have no incentive to deviate from their choices.
• There is no tendency to change either the price or the bought and sold quantities.

Competition and Equilibria

A competitive equilibrium is an equilibrium where

• there are many consumers willing to buy at the lowest price,
• there are many producers willing to sell at the highest price, and
• competition eliminates bargaining power.

• In other words, demand is equal to supply, and the market clears.

Non-Equilibrium States

• On some occasions markets may fail to clear due to either market failures or policy interventions.
• There two potential states that can occur.
  • The price is too high and supply is greater than demand.
  • The price is too low and demand is greater than supply.

Excess Supply

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

Excess Demand

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

The Welfare Properties of Competitive Equilibria

Market entry

• A market satisfies the free entry condition if no restrictions (legal, technological, etc.) prevent new firms from entering the market.
• A market has entry barriers if any form of restriction prevents new firms from operating in the market.

Profits in the Ideal Case of Free, Instantaneous Entry

• Suppose that there is a pool of potential firms that can enter a market.

• When does a potential firm have incentives to enter the market?
• Can firms with costlier technologies survive in the market in the long-run?
• When does an incumbent firm leave the market?
• An equilibrium in such a market (i.e., a situation for which there is no new entry or exit from the market) can only be achieved if
  • all firms produce at the same marginal cost (i.e., using the same technology) and
  • all firms have zero profits.

Perfect Competition

• A market is perfectly competitive if it has a large number of consumers and firms such that
  • the market does not suffer from any market failure (imperfect information, externalities, etc.),
  • there are no entry barriers in the market,
  • consumers and firms are price takers,
  • consumers try to maximize their utility,
  • firms produce a homogeneous commodity or service,
  • firms try to maximize their profits.

The Welfare Properties of Perfect Competition

Estimating Market Supply

Markets, 2021

• Suppose we consider estimating a market in equilibrium.
• In the data,
  • we observe traded quantities and market prices but
  • we do not observe demanded and supplied quantities.

Simultaneity

• 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.

Least Squares

• 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?

Full Information Maximum Likelihood

• 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.

Market Estimation Using R

• 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?

Market Visualization Using R

• 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?

Current Field Developments

• Methods for market estimation is an ongoing research topic.
• There are many methods focusing on estimating different market aspects and types, such as
  • complementary systems of demands (Deaton & Muellbauer, 1980, MicEconAids, 2017),
  • differentiated product markets (Berry, Levinsohn, & Pakes, 1995, BLPestimatoR, 2019),
  • shortages and surpluses (Maddala, 1986, Markets, 2021).
• Structural macroeconomic models frequently entail aggregate demand and supply for the national and, sometimes, international economies.

Comprehensive Summary

• Market supply consolidates the supplies of all firms in a market.
• Policy makers often target market supply.
• Controlling market forces can be challenging and have many unexpected side effects.
• Many estimation methods are available for estimating demand and supply.
• Competitive equilibria have are Pareto efficient.
• Perfect competition, although unrealistic, offers us a benchmark of how markets could ideally operate.

Further Reading

• Varian, 2010, secs. 24.1-4
• CORE-Team, 2017, secs. 8.2, 8.4, 8.5, 8.8

Mathematical Details

Suppose there are \(n\) firms in the market with supply functions \(s_{1}, …, s_{n}\). The market supply is given by

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

The rate of change of aggregate supply with respect to price

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.

\begin{align*} s’(p) = \sum_{i=1}^{n} s_{i}’(p) > s_{j}’(p) \quad\quad (\forall j). \end{align*}

Elasticity of aggregate supply

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

\begin{align*} e_{s}(p) &= s_{p}(p) \frac{p}{s(p)} \\ &= \left(\sum_{i=1}^{n} s_{i}’(p) \right) \frac{p}{s(p)} \\ &= \sum_{i=1}^{n} s_{i}’(p) \frac{p}{s_{i}(p)}\frac{s_{i}(p)}{s(p)} \\ &= \sum_{i=1}^{n} e_{s_{i}}(p) \frac{s_{i}(p)}{s(p)} \end{align*}

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

Exercises

Group A

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*}

References

References

Topic's Concepts

• perfectly competitive
• entry barriers
• free entry
• market shortage
• excess demand
• market surplus
• excess supply
• competitive equilibrium
• equilibrium
• inverse market supply
• market supply
• market supplied quantity