Econ 50: Economic Analysis I

Problem Set 6: Production and Cost

Due Monday, November 3 at 11pm on Gradescope

For the next two problem sets, we’re going to use two production functions: \(f(L,K) = L\sqrt{K}\)and\(f(L,K) = L^{1 \over 3}K^{2 \over 3}\)In this problem set, we’ll investigate the costs associated with these production functions; next week we’ll look at how firms with these production functions maximize their profits.

You’ll do exercises 6.1 and 6.2 in section, and use the results of those problems in the remaining questions. Because of this (and also because it’s Halloweekend) the due date will be extended by 48 hours, to Monday night. The solutions will be posted at 8am on Tuesday morning; as usual, no homework will be accepted after the solutions are posted.

For these two problem sets, you are encouraged to use Desmos or an equivalent program to generate your graphs, rather than drawing them by hand.

The first two exercises derive the firms’ conditional demands for labor and capital in the long run.

Conditional Demands for Labor and Capital I

Consider a firm with the Cobb-Douglas production function \(q = f(L,K) = L\sqrt{K}\)

Derive expressions for the firm’s $MP_L$, $MP_K$, and $MRTS$.

Plot the isoquant for $q = 4$.

Suppose $w = 8$ and $r = 2$. How much would it cost to produce 4 units if the firm used $L = 1$ unit of labor? What about $L = 2$? Add the isocost lines corresponding to these two costs to your graph.

Use the Lagrange method to find the conditional demands for labor, $L^c(w,r,q)$ and $K^c(w,r,q)$. What are the values of $L^c$ and $K^c$ for $w = 8$, $r = 2$, and $q = 4$? How do those track with what you found above?

Conditional Demands for Labor and Capital II

Repeat the previous question for the production function \(f(L,K) = L^{1 \over 3}K^{2 \over 3}\)However, for part (c), please use $L = 1$ and $L = 4$.

Question 3 uses the conditional labor and capital demands to derive the long-run total cost, and relates total cost to the returns to scale.

Long-Run Total Costs and Returns to Scale

For each of the production functions in 6.1 and 6.2, find the long-run total cost of producing $q$ units of output if $w = 8$ and $r = 2$. (Note: for the first one, you’ll get a nasty number as a coefficient; that’s OK, you didn’t make a mistake!)

Are each of the production functions decreasing returns to scale, constant returns to scale, or increasing returns to scale? How is this reflected in the exponent on $q$ in the total cost functions you found above?

The last two questions derive and plot the short-run cost curves, and compare them to long-run cost.

Short-Run Costs I

Again consider a firm with the Cobb-Douglas production function \(q = f(L,K) = L\sqrt{K}\) The firm pays $w = 8$ per unit of labor hired, and $r = 2$ per unit of capital hired.

In the short run, assume the firm has a fixed capital stock of $\overline K = 4$.

Graphs! Note: you don’t have to graph these by hand! Feel free to use desmos.com or equivalent.

Now let’s see how the firm’s short-run costs that you calculated above compare to its long-run costs that you calculated question 6.3. Using Desmos or some other graphing calculator, plot the short-run total cost you calculated above, and the long-run total cost. What do you notice about their respective shapes? What does that tell you about returns to scale, and returns to labor input? Where are the two equal? Why?

Short Run Costs II

Repeat exercise 6.4 for the production function from exercise 6.2.