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\textbf{\Large Homework 5}
\textbf{due 4/6/2017}
\end{center}
\bigskip\noindent\textbf{Problem 1} Suppose that \textit{low-ability} workers have productivity of $D$, while \textit{high-ability} workers have productivity of $A$, where $A>D$. Firms cannot tell low-ability workers from high-ability workers \textit{ex ante}, but can observe a worker's education level $e$. Firms know that half of all workers are low-ability, and half are high-ability.
Any worker can acquire as much education as she wishes, but getting $e$ units of education costs a low-ability worker $B*e$, where $B>1$, and costs a high-ability worker $e$. Assume the labor market is competitive, so that a worker earns her expected productivity.
\medskip\noindent\textbf{a. } Suppose $A=15$, $B=4$, and $D=1$. Does there exist a pooling equilibrium in which both high- and low-ability workers get $1$ unit of education? If so, draw or describe a wage function that supports this equilibrium outcome, and determine whether or not the equilibrium satisfies the intuitive criterion. If not, explain why not.
\medskip\noindent\textbf{b. } Suppose $A=15$, $B=4$, and $D=1$. Does there exist a pooling equilibrium in which both high- and low-ability workers get $3$ units of education? If so, draw or describe a wage function that supports this equilibrium outcome, and determine whether or not the equilibrium satisfies the intuitive criterion. If not, explain why not.
\medskip\noindent\textbf{c. } Suppose $A=15$, $B=4$, and $D=1$. Solve for a separating equilibrium which \textit{does not} satisfy the intuitive criterion. Draw or describe a wage function that supports this outcome in an equilibrium. Clearly explain, using your graph and/or a verbal description, why this equilibrium fails the intuitive criterion.
\medskip\noindent\textbf{d. } For general $A$, $B$, and $D$, solve for the unique equilibrium which \textit{does} satisfy the intuitive criterion as a function of $A$, $B$ and $D$. How does the level of education obtained by the high types vary in $D$ in this equilibrium? What is the intuition?
\bigskip\noindent\textbf{Problem 2} This problem asks you to consider an extension of the basic Spence model to one in which education is productive and the cost of education is convex.
Suppose that high types with
education $e$ have productivity $y(H,e)=10+2e$, while low types have
productivity $y(L,e)=2+e$. Firms cannot observe whether a worker is
a high type or a low type, but know that half of all workers are of
each type. A competitive labor market ensures each type of worker
is paid her expected productivity. A high type can acquire $e$
units of education at cost $c_H(e)=\frac1{10}e^2$, while education
costs a low type $c_L(e)=\frac14(e+2)^2-1$.
\medskip\noindent\textbf{a. } Suppose $e_L=0$ and $e_H=12$.
What wage function would support this outcome as a separating equilibrium? Draw a picture and/or describe using an equation.
\medskip\noindent\textbf{b. } Does the equilibrium you
described in part a satisfy the intuitive criterion? Why or why
not?
\medskip\noindent\textbf{c. } Draw the set of all points which give the high type utility of 5. What is the slope of the indifference curve you drew, as a function of $e$? Determine the point of
tangency between the high type's indifference curve and the function
$y(H,e)$ (note that the high type may get more or less than 5 utility at the point of tangency). Do the same for the low type's indifference curve and the function $y(L,e)$.
\medskip\noindent\textbf{d. } Suppose that both types choose
their education level so that their indifference curve is tangent to
their productivity function. Describe, using a picture and/or an equation, a wage function that would
support this outcome as a perfect Bayesian equilibrium.
\medskip\noindent\textbf{e. } Does the equilibrium you described
in part d satisfy the intuitive criterion? Why or why not?
\bigskip\noindent\textbf{Problem 3} Workers choose their education
levels and then apply for jobs. $\frac14$ of workers are
\textit{high-ability}, while the remainder are \textit{low-ability}. Firms
cannot determine a given worker's ability level until after the worker is hired.
Regardless of education level, high ability workers increase firm profits by
\$100, while low ability workers increase firm profits by \$36. The cost of
education is $c_L(e)=e^2$ for low-ability workers, while it is $c_H(e)=2e$ for
high ability workers. Assume that workers are paid their expected productivity.
For outcomes 1-4 below, determine if each outcome is an equilibrium of the
game. If so, determine whether or not it satisfies the intuitive criterion, and
prove your answer.
\medskip\noindent\textbf{a. } Outcome 1:
\begin{align}
&e_H=12\nonumber\\
&e_L=2\nonumber\\
&w(e)=\left\{
\begin{array}{cc}
\$100 & \mbox{if } e=12 \\
\$36 & \mbox{ if } e\neq 12
\end{array}\right.
\end{align}
\medskip\noindent\textbf{b. } Outcome 2:
\begin{align}
&e_H=1\nonumber\\
&e_L=1\nonumber\\
&w(e)=\left\{
\begin{array}{cc}
\$52 & \mbox{if } e<25 \\
\$100 & \mbox{ if } e\geq 25
\end{array}\right.
\end{align}
\medskip\noindent\textbf{c. } Outcome 3:
\begin{align}
&e_H=5\nonumber\\
&e_L=5\nonumber\\
&w(e)=\left\{
\begin{array}{cc}
\$36 & \mbox{if } e<5\\
\$52 & \mbox{if } e= 5 \\
\$(32+4e) & \mbox{if } e\in(5,17)\\
\$100 & \mbox{if } e\geq17
\end{array}\right.
\end{align}
\medskip\noindent\textbf{d. } Outcome 4. For this part, your answer should
depend on the variables $X$ and $Y$.
\begin{align}
&e_H=X\nonumber\\
&e_L=Y \nonumber\\
&w(e)=\left\{
\begin{array}{cc}
\$36 & \mbox{if } e