Problem Set 7: Profit Maximization
Due Saturday, November 8 at 11pm on Gradescope
You should be able to do exercises 7.1 and 7.2 after Monday’s lecture; exercises 7.3, 7.4, and 7.5 after Wednesday’s; and the remainder after Friday.
The first two exercises use the production functions from the last problem set. You do not need to re-derive the cost functions; and again, do not plot the graphs here by hand — use Desmos or some other graphing calculator and screen shot them!
Revenue and Profit Maximization I
Consider the firm you analyzed in Exercises 6.1 and 6.4. Now assume it faces the demand function \(q(p) = 256p^{-2}\)
Derive the firm’s total revenue ($TR$). Plot it in a graph with the total cost curve you found previously.
Derive the firm’s marginal revenue ($MR$) and average revenue ($AR$). Plot it in a graph with the marginal cost ($MC$) and average cost ($AC$) you found previously.
Find the firm’s profit-maximizing quantity, the price they will set, and their profit when they profit-maximize. Indicate the profit or loss in each of your graphs.
Revenue and Profit Maximization II
Consider the firm you analyzed in Exercises 6.2 and 6.5. Now assume it faces the demand function \(q(p) = 56 - 2p\) Note: you might need to dust off your factoring skills for this one!
Derive the firm’s total revenue ($TR$). Plot it in a graph with the total cost curve you found previously.
Derive the firm’s marginal revenue ($MR$) and average revenue ($AR$). Plot it in a graph with the marginal cost ($MC$) and average cost ($AC$) you found previously.
Find the firm’s profit-maximizing quantity, the price they will set, and their profit when they profit-maximize. Indicate the profit or loss in each of your graphs.
Chain rules
Consider a chain of causality in which some variable $X$ influences some other variable $Y$, which in turn influences some third variable $Z$. (You can think, for example, of a rise in a resource like oil influencing the cost of producing a good, which in turn affects the equilibrium price of that good.)
To be specific, let’s suppose $y = f(x)$ and $z = g(y)$, where
\(f(x) = x^{-2}\)
and
\(g(y) = y^{\frac{5}{2}}\)
What is the elasticity of $Y$ with respect to $X$? In other words, how would a 1% increase in $X$ affect $Y$?
What is the elasticity of $Z$ with respect to $Y$?
What is the elasticity of $Z$ with respect to $X$?
In general, if $y = x^a$ and $z = y^b$, what is the elasticity of $Z$ with respect to $X$?
Elasticity and profit maximization
For exercises 7.1 and 7.2, calculate the price elasticity of demand at the optimal price/quantity combination at which the firm produces. Confirm that the formula for the IEPR holds: \(\frac{p - MC}{p} = \frac{1}{|\epsilon|}\) where \(\epsilon = \frac{dq}{dp} \times \frac{p}{q}\) is the price elasticity of demand at $(q,p)$. (Careful: remember $p^\prime(q) = dp/dq$…)
Profit maximization for a competitive firm
Consider a competitive firm with the production function $f(L,K) = L^\frac{1}{3}K^\frac{1}{6}$. Its capital is fixed at $\overline K = 64$, so its short-run production function is \(f(L) = 2L^\frac{1}{3}\) The firm buys labor and capital in competitive markets at prices $w=2$ and $r=4$, respectively, and sells its output in the competitive market at a price $p$.
Derive expressions for the firm’s total cost $TC(q)$, average cost $AC(q)$, and marginal cost $MC(q)$.
Calculate the firm’s profit as a function of $q$ and $p$. Then, take the derivative of this with respect to $q$, set it equal to zero, and solve to find the firm’s optimal quantity as a function of the market price, $q^\star(p)$.
Suppose the firm could sell its output at price $p = 192$. Plot one graph showing total cost and total revenue, and another showing marginal cost, marginal revenue, average cost, and average revenue. Indicate the firm’s optimal output $q^\star$. Finally, calculate the firm’s profit, and represent it in the graphs (as a distance in the first graph, and an area in the second).
Repeat part (c) if $p = 48$. What’s interesting about this price? What about the diagrams reflects this fact?
Output supply and labor demand
Consider a competitive profit-maximizing firm with the production function $f(L,K)=2 (LK)^\frac{1}{4}$. Its capital is fixed in the short run at $\overline K = 81$, so its short-run production function is \(f(L) = 6L^\frac{1}{4}\) It is a price taker in both input and output markets, paying $w$ for each unit of labor and $r$ for each unit of capital, and selling each unit of its output at price $p$.
Find the firm’s conditional demand for labor, $L^c(q)$. Use this to write the firm’s profit as a function of the quantity it produces, given the wage rate it pays and the price it sells output for: that is, $\pi(q\ |\ w,p)$. Then, take the derivative of this with respect to $q$ its profit-maximizing quantity as a function of $p$ and $w$, $q^\star(w,p)$.
Plug your expression for $q^\star(w,p)$ back into your conditional labor demand function to find the firm’s profit-maximizing demand for labor, $L^\star(w,p)$.
Now write the firm’s profit as a function of the amount of labor it employs, the wage rate, and the price: that is, $\pi(L\ |\ w,r)$. Then, take the derivative of this with respect to $L$ its profit-maximizing labor demand as a function of $p$ and $w$, $L^\star(w,p)$. Confirm that this is the same as your answer to (b)!
In the short run, what is the firm’s wage elasticity of labor demand (elasticity of $L$ with respect to $w$)? Its price elasticity of supply (elasticity of $q$ with respect to $p$)?
From Production to Costs and Back Again! (50Q only)
This is an exploratory exercise to show how you can use some of the mathematical lemmas we looked at in section can be applied to derive a firm’s production function from its total cost function.
We’re going to start by assuming we know a firm’s production function: specifically, that it’s given by: \(f(L,K) = 2 L K^2\) In the first part of the question, we’ll derive the firm’s total cost function in the same way we did in last week’s problem set. In the second part, we’ll go in reverse – we’ll derive the production function from the cost function!