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\begin_body

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Prof.
 Marek Weretka's
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Econ 301 Intermediate Microeconomics 
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Problem Set 7
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 1 (Production Function) 
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Consider the following three production functions:
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\align center
\begin_inset Formula $f(K,L)=K²L$
\end_inset


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\align center
\begin_inset Formula $f(K,L)=K^{(1/3)}L^{(1/3)}$
\end_inset


\end_layout

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\align center
\begin_inset Formula $f(K,L)=2K+L$
\end_inset


\end_layout

\begin_layout Standard
a) In the 
\begin_inset Formula $(K,L)$
\end_inset

 space, sketch the map of isoquants for each of them.
 
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b) Give some economic interpretation for 
\begin_inset Formula $MPK$
\end_inset

 and 
\begin_inset Formula $MPL$
\end_inset

 (for an abstract production function).
 
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c) Find analitically 
\begin_inset Formula $MPK$
\end_inset

 for each production function above and plot it in the graph, with 
\begin_inset Formula $K$
\end_inset

 on the horizontal axis and 
\begin_inset Formula $MPK$
\end_inset

 on the vertical axis, assuming 
\begin_inset Formula $L=1$
\end_inset

.
 Is 
\begin_inset Formula $MPK$
\end_inset

 increasing, decreasing or constant? 
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d) Find 
\begin_inset Formula $MPL$
\end_inset

 for each of the four production functions and plot it on the graph, with
 
\begin_inset Formula $L$
\end_inset

 on the horizontal axis and 
\begin_inset Formula $MPL$
\end_inset

 on the vertical axis, assuming 
\begin_inset Formula $K=2$
\end_inset

.
 Is 
\begin_inset Formula $MPL$
\end_inset

 increasing, decreasing or constant? 
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e) Provide some economic intuition behind increasing, constant and decreasing
 returns to scale.
 Give an example of technology from real life that captures each of the
 three cases (one example per technology).
 
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f) For each of the three production functions above, show formally whether
 it exhibits increasing, decreasing or constant returns to scale.
 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 2 (Profit Maximization- Short run) 
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GMC is producing cars using machines 
\begin_inset Formula $(K)$
\end_inset

 and labor 
\begin_inset Formula $(L)$
\end_inset

.
 The technology is capital intensive; the production function is given by
\end_layout

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\noindent
\align center
\begin_inset Formula $F(K,L)=K^{(3/4)}L^{(1/2)}$
\end_inset


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The value of GMC physical capital (machines, real estate etc.) is equal to
 
\begin_inset Formula $K=$
\end_inset

$
\begin_inset Formula $16$
\end_inset

 billon (in calculations ignore billions).
 We analyze firm's behavior in the short run (note 
\begin_inset Formula $K$
\end_inset

 cannot be changed in the short run).
 Suppose the price of a car is equal to 
\begin_inset Formula $p$
\end_inset

 and the wage rate is 
\begin_inset Formula $w$
\end_inset

 (parameters).
 
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a) Write down the profit as a function of 
\begin_inset Formula $L$
\end_inset

.
 
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b) On a graph with 
\begin_inset Formula $L$
\end_inset

 on the horizontal axis and $ on the vertical axis, plot two components
 of profit function: total revenue (
\begin_inset Formula $pF(K,L)$
\end_inset

), and labor cost 
\begin_inset Formula $(wL$
\end_inset

) (when drawing, assume 
\begin_inset Formula $p=1$
\end_inset

 and 
\begin_inset Formula $w=2$
\end_inset

).
 On the graph, mark the profit level as the difference between the two lines
 (for any given L).
 
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c) In order to find 
\begin_inset Formula $x$
\end_inset

 that maximizes some function 
\begin_inset Formula $f(x)$
\end_inset

, we take the first derivative of the function with respect to 
\begin_inset Formula $x$
\end_inset

 and set it equal to zero (we call it a first order condition).
 Please explain intuitively why this method allows us to find the optimum.
 
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\begin_layout Standard
d) Set the derivative of your profit function with respect to L equal to
 zero and derive the secret of happiness (the equation that tells: 
\begin_inset Formula $MPL=(w/p)$
\end_inset

).
 Explain the economic intuition behind the latter condition.
 
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e) Find analitically the optimal level of labor 
\begin_inset Formula $L^{∗}$
\end_inset

 that maximizes the profit as a function of the real wage 
\begin_inset Formula $w/p$
\end_inset

.
 (we call it the firm's labor demand).
 Find the values of 
\begin_inset Formula $L$
\end_inset

 for the following values of parameters
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\begin_layout Standard
Plot your demand for labor on the graph with the real wage (
\begin_inset Formula $w/p$
\end_inset

) on the vertical axis and 
\begin_inset Formula $L$
\end_inset

 on the horizontal one.
 Using the table mark three points corresponding to the three values of
 (
\begin_inset Formula $w/p$
\end_inset

).
 
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f) What is the maximal profit for each of the three (
\begin_inset Formula $w/p$
\end_inset

) values? 
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\begin_inset Formula $\vphantom{}$
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\begin_layout Subsubsection*
Problem 3 (Labor Market) 
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\begin_layout Standard
\begin_inset Formula $\vphantom{}$
\end_inset


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\begin_layout Standard
a) Review Problem 3 from PS4 (you will need it for the midterm).
 In that problem, we found that Kate's labor supply is inelastic and equal
 to 
\begin_inset Formula $12h$
\end_inset

 (per day).
 Plot her labor supply in a graph with the real wage 
\begin_inset Formula $(w/p)$
\end_inset

 on the vertical axis and 
\begin_inset Formula $L$
\end_inset

 on the horizontal one.
 
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\begin_layout Standard
b) Suppose Kate works for GMC described in Problem 2 (in this PS).
 Add this company's labor demand curve to your graph in part (a).
 Find the equilibrium real wage rate (
\begin_inset Formula $w/p$
\end_inset

) that clears the labor market (do it both analitically and using the graph).
 
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c) Suppose the wage rate is above the equilibrium level you found in part
 (b).
 Do we observe unemployment or excess demand of labor on the market? What
 market forces drive the wage rate down? (give a newspaper story; why will
 the wage go down?) 
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d) Suppose Kate's preferences change so that now she is willing to supply
 only 
\begin_inset Formula $8h$
\end_inset

.
 On a graph show how this affects equilibrium real wage and the labor supply
 level.
 Find the two values analytically.
 
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e) Suppose the government passes the law that sets the minimal (real) wage
 rate equal to 
\begin_inset Formula $w/p=2$
\end_inset

.
 How does this affect the equilibrium on the labor market (assume Kate's
 supply is 
\begin_inset Formula $8h$
\end_inset

)? Find the unemployment rate associated with such a policy.
 
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\begin_inset Formula $\vphantom{}$
\end_inset


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\begin_layout Subsubsection*
Problem 4 (Long Run) 
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\begin_layout Standard
Jimmy produces milk using milk-making machines (read: cows) 
\begin_inset Formula $(K)$
\end_inset

 and labor (milkmen) 
\begin_inset Formula $(L)$
\end_inset

.
 He has access to the technology given by
\end_layout

\begin_layout Standard
\align center
\begin_inset Formula $y=F(K,L)=K^{(1/3)}L^{(1/3)}$
\end_inset


\end_layout

\begin_layout Standard
The price of a gallon of milk is 
\begin_inset Formula $p=$
\end_inset

$1; the price of one machine (cow) is 
\begin_inset Formula $w_{k}=$
\end_inset

$2 and the (milkmen) wage rate is 
\begin_inset Formula $w_{L}=$
\end_inset

$1 
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\begin_layout Standard
a) Does this function exhibit increasing, constant or decreasing returns
 to scale? 
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\begin_layout Standard
b) Write down the profit function (a function that depends on 
\begin_inset Formula $K$
\end_inset

 and 
\begin_inset Formula $L$
\end_inset

)
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c) Find the optimal input levels, the output level and the maximal profit.
 (Hint: follow the steps shown in class - slides L14) 
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d) Argue that the optimal input choices also minimize the production cost
 of the optimal production level (from (c)) (show that the cost minimization
 condition holds.)
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