Problem 2
Assume our production function is given by
q(E, K) = \sqrt{EK} = E^{\frac{1}{2}}K^{\frac{1}{2}}
where $MP_E = \frac{K}{2E^{\frac{1}{2}}}$ and $MP_K = \frac{E}{2K^{\frac{1}{2}}}$. The market price for labor and capital is $10 and $40
respectively.
a) Assume p = 80. Define the profit function.
b) In the short-run, capital is fixed at 80 (ie K = 80). Based on that, what is the optimal
level of employment? What is the firm's profit at this level?
c) Now assume that the firm is in the long-run and wants to produce 400 units. Define the cost
minimization problem.
d) What is the optimal consumption of capital and employment when q = 400? What is the
firm's cost at this level of production?
Problem 3
Assume that labor demand is defined as $L_D = 15 - \frac{w}{4}$, and labor supply is defined as $L_S = w$.
a) Find the optimal level of employment ($L^*$) and wages ($w^*$).
b) Assume the government implements a new minimum wage of $20. Find the new labor de-
mand ($L_{MD}$) and supply ($L_{MS}$). How many workers are displaced by this change?
c) Graph $L^*$, $w^*$, $L_{MD}$, and $L_{MS}$. Be sure to label the graph correctly using the answers found
above.
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