Problem Set 8: Short-Run Equilibrium

You should be able to do exercise 8.1 after Wednesday’s lecture, and 8.2 and 8.3 after Friday.

In addition to these questions, I’ve published a few old exam questions in the Canvas assignment for this week. You may hand these in for credit as well! Each one is worth 3 points, just like a homework problem.

Partial Equilibrium

In this problem set we’ll think about a market with multiple types of consumers and firms. We’re concerned here with good 1; good 2 is a composite good (e.g., “money spent on other things”) with price normalized to 1.

There are 3 types of consumers:

• There are 10 consumers with the utility function $u(x_1,x_2) = 20\sqrt{x_1} + x_2$.
• There are 9 consumers with the utility function $u(x_1,x_2) = 40\sqrt{x_1} + x_2$.
• There are 6 consumers with the utility function $u(x_1,x_2) = 60\sqrt{x_1} + x_2$.

Assume each consumer has enough income that there isn’t a corner solution for any price in this question.

There are two types of firms producing good 1. While each firm has the production function $q_1 = \sqrt{K L}$, they have different amounts of (fixed) capital:

• There are 5 firms that each have $\overline K = 12$.
• There are 5 firms that each have $\overline K = 20$.

(This question only considers the short run.) Firms buy labor at wage rate $w = 1$. Assume capital is free (or, if you prefer, that it’s sunk cost and nonrecoverable).

Derive the individual demand curves for each type of consumer, and the market demand curve.

Derive the individual supply curves for each type of firm, and the market supply curve.

Plot the market demand and market supply curves. Find the equilibrium price and quantity in the market, and add that point to your diagram.

Welfare Analysis

Now let’s look at the consumer and producer surplus for the agents in the first question.

For each type of consumer, plot the individual demand curve for a typical consumer, and shade their area of consumer surplus in equilibrium. How many units of good 1 will each consume in equilibrium? What will their consumer surplus be? What is total consumer surplus in the market?

For each type of firm, plot the individual supply for a typical firm, and shade its area of producer surplus in equilibrium. How many units of good 1 will each produce in equilibrium? What will their producer surplus be? What is total producer surplus in the market?

The Social Planner’s Problem

We could, in fact, model this economy as a single consumer and a single firm.

Look at the demand functions you found; if you modeled this economy as a single consumer with the utility function $u(x_1,x_2) = a \sqrt{x_1} + x_2$, what would the value of $a$ be? In other words, for what value of $a$ would this (one) consumer’s demand function be the same as the market demand you found in Exercise 8.1(a)?

Similarly, if you modeled this economy as a single firm with capital $\overline K$, what level of capital would that be? In other words, for what value of $\overline K$ would this (one) firm’s supply function be the same as the market supply you found in Exercise 8.1(b)?

Suppose you modeled the economy as a single consumer and a single firm. If the firm produced $Q$ units, what would its total cost be? If the consumer consumed $Q$ units, what would its total benefit be (ignoring good 2)? Plot these total cost and total benefit curves on a graph with $Q$ on the horizontal axis and dollars on the vertical axis. Then, take the derivative of each with respect to $Q$ and plot marginal benefit vs. marginal cost on a graph with $Q$ on the horizontal axis and dollars per unit on the vertical axis.

Solve for the optimal quantity $Q^*$ that maximizes total benefit minus total cost.

Confirm the following:

This total quantity is the same as the equilibrium quantity you found in 8.1(c).

At that quantity, the marginal benefit and marginal cost are both equal to the equilibrium price you found in 8.1(c).

At that quantity, total benefit minus total cost equals the sum of consumer surplus from 8.2(a) and producer surplus from 8.2(b).

Roses and Pho (50Q only)

Minh loves red roses (good 1) and pho (good 2). Her preferences may be represented by the Cobb-Douglas utility function \(u(x_1,x_2) = x_1^{1 \over 3}x_2^{2 \over 3}\)

Find Minh’s Ordinary (“Marshallian”) demand functions $x_1^\star(p_1,p_2,m)$ and $x_2^\star(p_1,p_2,m)$ and indirect utility function $V(p_1,p_2,m)$. Feel free to use the Cobb-Douglas trick!

Find Minh’s expenditure function $E(p_1,p_2,U)$ and Compensated (“Hicksian”) demand functions $h_1(p_1,p_2,U)$ and $h_2(p_1,p_2,U)$, using the process we developed in Exercise 4.6.

Now suppose Minh’s income is $m = 144$ and the price of red roses is $p_1= 6$. Tragically, due to higher tariffs on beef imports, the price of pho is set to rise from $p_2 = 4$ to $p_2 = 32$!

In a budget-line/indifference curve diagram, draw:

• Her budget lines before and after the price change, labeled $BL_1$ and $BL_2$.

• Her utility-maximizing bundles before and after the price change, labeled $A$ and $C$.

• Her “Hicks decomposition bundle,” which is the cost-minimizing way to achieve her initial utility (i.e. before the price change) at the new prices. (To find this bundle, calculate her initial utility at bundle $A$, and plug that utility along with the new prices into the Hicksian demand functions you derived above.) Label this bundle $B$.

• Her “Equivalent bundle,” which is the cost-minimizing way to achieve her new utility (i.e. after the price change) at the old prices. (To find this bundle, calculate her new utility at bundle $C$, and plug that utility along with the initial prices into the Hicksian demand functions you derived above.) Label this bundle $E$.

• The indifference curves passing through $A$, $B$, $C$, and $E$.

What is the compensating variation of the price change for Minh? In other words, by how much would her income need to increase in response to the price increase, for her to be as happy as she was before the price change?

What is the equivalent variation of the price change for Minh? In other words, how much income would Minh be willing to give up in order for the price not to have changed at all?

Finally, what is the change in consumer surplus Minh experiences due to this price change? To find this, take the integral of her Marshallian demand for good 2 between the old and new prices. How does this compare to the values for CV and EV you found above?

Exercises 8.5-8.8 are old exam questions; they may be found on Canvas.