Homework Assignment 5

Intermediate Microeconomics (Econ 100A)
Kristian López Vargas
UCSC

Instructions:

You will scan your homework into a single PDF file to be upload.
Only legible assignments will be graded.
Late assignments will not be accepted.

Only two randomly-chosen questions will be graded.

Question 1:

Suppose a firm has the following production function: $$Q(L,K)=3K^{1/2}L^{1/2}$$ Recall that the isocost line is as follows: $$C=wL+rK$$

  1. What is the $($long run$)$ optimal choice of $L$ and $K$ for a given $Q$, $w$, and $r$? In other words, provide a formula for the optimal choice of labor $L^{*}(w,r,Q)$ and capital $K^{*}(w,r,Q)$ as a function of the parameters $Q$, $w$, and $r$.

  2. Given $Q=90$, $w=25$, and $r=9$, what are the optimal levels of labor and capital, $L^{*}$ and $K^{*}$? What is the cost of producing $Q=90$ at these input prices?

  3. Suppose now that you are in the short run, $Q=84$, $w=2$, $r=3$, and the capital level is fixed at $\bar{K}=16$. What is the optimal level of labor in the short run? What is the cost of producing $Q=84$ in the short run at these input prices?

Question 2:

Suppose a firm has the following production function: $$ Q(L,K)=\min \{K,2L\} $$ Recall that the isocost line is as follows: $$ C=wL+rK $$

  1. What is the $($long-run$)$ optimal choice of $L$ and $K$ for a given $Q$, $w$, and $r$? In other words, provide a formula for the optimal choice of labor $L^{*}(w,r,Q)$ and capital $K^{*}(w,r,Q)$ as a function of the parameters $Q$, $w$, and $r$.

  2. Given $Q=16$, $w=6$, and $r=2$, what are the optimal levels of labor and capital, $L^{*}$ and $K^{*}$? What is the cost of producing $Q=16$ at these input prices?

  3. Suppose now that you are in the short run, $Q=16$, $w=6$, $r=2$, and the capital level is fixed at $\bar{K}=20$. What is the optimal level of labor in the short run? What is the cost of producing $Q=16$ in the short run at these input prices? Would it be possible to meet $Q=16$ if $K=4$ in the short run?

Question 3:

Suppose a firm has the following production function: $$Q(L,K)=2L+K$$ Recall that the isocost line is as follows: $$C=wL+rK$$

  1. What is the $($ long-run $)$ optimal choice of $L$ and $K$ for a given $Q$, $w$, and $r$? In other words, provide a formula for the optimal choice of labor $ L^{*}(w,r,Q) $ and capital $ K^{*}(w,r,Q)$ as a function of the parameters $Q$, $w$, and $r$.

  2. Given $Q=20$, $w=6$, and $r=1$, what are $L^{*}$ and $K^{*}$?

Question 4:

A firm uses two inputs, capital and labor, to produce output. Its production function exhibits a diminishing technical rate of substitution $($e.g. Cobb-Douglas production function$)$.

  1. If the price of capital and the price of labor increase by the same percentage $($e.g., 40 percent$)$, what will happen to the cost-minimizing input quantities for a given output level?

  2. If the price of capital increases by 20 percent while the price of labor increases by 40 percent, what will happen to the cost-minimizing input quantities for a given output level?

Question 5:

Suppose a brewery uses a Cobb-Douglas production function for his production. He studies the production process and finds the following. An additional machine-hour of fermentation capacity would increase output by 600 bottles per day $(i.e. MP_K = 600)$. An additional man-hour of labor would increase output by 1200 bottles per day $(i.e. MP_L = 1200)$. The price of a man-hour of labor is ${$}$40 per hour. The price of a machine-hour of fermentation capacity is ${$}$8 per hour.

  1. Is the brewery currently minimizing its cost of production? Check using the minimization condition.

  2. It turns out, the brewery is not optimally choosing the factors of production. To lower its production cost, which factor of production should the brewery increase and which factor should he decrease?

  3. Suppose that the price of a machine-hour of fermentation capacity rises to ${$}$20 per hour. How does this change the answer from part 1?

Question 6:

For each of the total cost functions, write the expressions for the average cost, average fixed cost, average variable cost, and marginal cost:

  1. $TC\left(Q\right)=5Q$

  2. $TC\left(Q\right)=120+6Q$

  3. $TC\left(Q\right)=6Q^{2}$

  4. $TC\left(Q\right)=140+5Q^{2}$