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\multicolumn{2}{c} {\bfseries Problem Set 3 } \\
& \\
ECON 772001 - Math for Economists & Peter Ireland \\
Boston College, Department of Economics & Fall 2024 \\
& \\
\multicolumn{2}{c} {For Review and Practice - Not Collected or Graded}
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Many famous results from microeconomic theory are now understood to be special cases of the envelope theorem. This problem set will ask you to invoke the envelope theorem repeatedly to ``prove'' some of these results.
{\bfseries 1. Hotelling's Lemma}
Consider a firm that produces output $y$ with capital $k$ and labor $l$ according to the technology described by
$$
f(k,l) \geq y.
$$
The firm sells each unit of output at the price $p$, rents each unit of capital at the rate $r$, and hires each unit of labor at the wage $w$. Hence it chooses $y$, $k$, and $l$ to maximize profits
$$
py - rk - wl
$$
subject to the technological constraint just shown above.
\begin{enumerate}[label=\alph*.]
\item Set up the Lagrangian for this problem, letting $\lambda$ denote the multiplier on the constraint.
\item Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values $y^{*}$, $k^{*}$, and $l^{*}$ that solve the firm's problem, together with the associated value $\lambda^{*}$ for the multiplier.
\item Assume that the output and input prices $p$, $r$, and $w$ and the production function $f$ are such that it is possible to solve uniquely for the values of $y^{*}$, $k^{*}$, $l^{*}$, and $\lambda^{*}$ in terms of the parameters $p$, $r$, and $w$. Then the function $y^{*}(p,r,w)$ describing the optimal level of output represents the firm's supply function, and functions $k^{*}(p,r,w)$ and $l^{*}(p,r,w)$ describing the optimal inputs are the firm's factor demand curves. Along with these functions, define the firm's profit function as
$$
\pi(p,r,w) = \max_{y,k,l} py - rk - wl \text{ subject to } f(k,l) \geq y.
$$
Hotelling's lemma says that the derivative of this profit function with respect to the output price coincides with the supply function and that the derivatives of the profit function with respect to each input price, multiplied by $-1$, coincide with the associated factor demand curves. Use the definitions of all these functions to show that Hotelling's lemma is really just a special case of the envelope theorem.
\end{enumerate}
\pagebreak
{\bfseries 2. Shephard's Lemma}
Closely related to the profit maximization problem from above is the corresponding cost minimization problem in which the same firm chooses capital $k$ and labor $l$ inputs to minimize the cost of producing $y$ units of output:
$$
\min_{k,l} rk + wl \text{ subject to } f(k,l) \geq y.
$$
Note that in this version of the firm's problem, $y$ is a parameter instead of a choice variable.
\begin{enumerate}[label=\alph*.]
\item Set up the Lagrangian for this problem, letting $\mu$ denote the multiplier on the constraint.
\item Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values $k^{*}$ and $l^{*}$ that solve the firm's problem, together with the associated value $\mu^{*}$ for the multiplier.
\item Assume that the output requirement $y$, the input prices $r$ and $w$, and the production function $f$ are such that it is possible to solve uniquely for the values of $k^{*}$, $l^{*}$, and $\mu^{*}$ in terms of the parameters $r$, $w$, and $y$. Then the functions $k^{*}(r,w,y)$ and $l^{*}(r,w,y)$ describing the optimal inputs can be called the firm's conditional factor demand curves. Along with these functions, define the firm's cost function as
$$
c(r,w,y) = \min_{k,l} rk + wl \text{ subject to } f(k,l) \geq y.
$$
Shephard's lemma says that the derivatives of this cost function with respect to each input price coincides with the associated conditional factor demand curve. Use the definitions of all these functions to show that Shephard's lemma is really just a special case of the envelope theorem.
\end{enumerate}
{\bfseries 3. Roy's Identity}
Now consider a utility-maximizing consumer who solves
$$
\max_{c_{1},c_{2}} U(c_{1},c_{2}) \text{ subject to } I \geq p_{1}c_{1}+p_{2}c_{2},
$$
where $c_{1}$ and $c_{2}$ denote consumption of two goods, $U$ is the utility function, $I$ is income, and $p_{1}$ and $p_{2}$ are the prices of the two goods.
\begin{enumerate}[label=\alph*.]
\item Set up the Lagrangian for this problem, letting $\lambda$ denote the multiplier on the constraint.
\item Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values $c_{1}^{*}$ and $c_{2}^{*}$ that solve the consumer's problem, together with the associated value $\lambda^{*}$ for the multiplier.
\item Assume that income $I$, and goods prices $p_{1}$ and $p_{2}$, and the utility function $U$ are such that it is possible to solve uniquely for the values of $c_{1}^{*}$, $c_{2}^{*}$, and $\lambda^{*}$ in terms of the parameters $I$, $p_{1}$, and $p_{2}$. Recall that the function $c_{1}^{*}(p_{1},p_{2},I)$ and $c_{1}^{*}(p_{1},p_{2},I)$ describing these optimal choices are the Marshallian demand curves for the two goods. Along with these functions, define the indirect utility function
$$
v(p_{1},p_{2},I) = \max_{c_{1},c_{2}} U(c_{1},c_{2}) \text{ subject to } I \geq p_{1}c_{1}+p_{2}c_{2}.
$$
Roy's identity says that the Marshallian demand curves are related to the derivatives of the indirect utility function via
$$
c_{i}^{*}(p_{1},p_{2},I) = \frac {-v_{i}(p_{1},p_{2},I)}{v_{3}(p_{1},p_{2},I)}
$$
for each $i=1,2$. Use the definitions of all these functions to show that Roy's identity follows directly from the envelope theorem.
\end{enumerate}
{\bfseries 4. The Slutsky Equation}
Closely related to the utility maximization problem from above is the corresponding expenditure minimization problem in which the same consumer chooses consumptions $c_{1}$ and $c_{2}$ to minimize the cost of achieving a given utility level $\bar{U}$:
$$
\min_{c_{1},c_{2}} p_{1}c_{1}+p_{2}c_{2} \text{ subject to } U(c_{1},c_{2}) \geq \bar{U}.
$$
Note that in this version of the problem, $\bar{U}$ replaces $I$ as one of the parameters.
\begin{enumerate}[label=\alph*.]
\item Set up the Lagrangian for this problem, letting $\mu$ denote the multiplier on the constraint.
\item Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values $c_{1}^{*}$ and $c_{2}^{*}$ that solve this problem, together with the associated value $\mu^{*}$ for the multiplier.
\item Assume that the utility level $\bar{U}$, goods prices $p_{1}$ and $p_{2}$, and the utility function $U$ are such that it is possible to solve uniquely for the values of $c_{1}^{*}$, $c_{2}^{*}$, and $\mu^{*}$ in terms of the parameters $\bar{U}$, $p_{1}$, and $p_{2}$. Now the functions $c_{1}^{*}=h_{1}^{*}(p_{1},p_{2},\bar{U})$ and $c_{2}^{*}=h_{2}^{*}(p_{1},p_{2},\bar{U})$ describing the optimal choices are the Hicksian demand curves for the two goods (note that $h$'s instead of $c$'s are being used here to help distinguish these Hicksian demand curves from the Marshallian demand curves defined by the solution to the utility maximization problem). Along with these functions, define the expenditure function
$$
e(p_{1},p_{2},\bar{U}) = \min_{c_{1},c_{2}} p_{1}c_{1}+p_{2}c_{2} \text{ subject to } U(c_{1},c_{2}) \geq \bar{U}.
$$
Under most circumstances, it will be the case that the Marshallian and Hicksian demand curves coincide at the point where $I=e(p_{1},p_{2},\bar{U})$, so that the income $I$ from the utility maximization problem equals the expenditure required to attain the utility level $v(p_{1},p_{2},I)=\bar{U}$. This result can be summarized by stating that
$$
h_{i}^{*}(p_{1},p_{2},\bar{U}) = c_{i}^{*}(p_{1},p_{2},e(p_{1},p_{2},\bar{U}))
$$
for all values of $p_{1}$, $p_{2}$, and $\bar{U}$ and each $i=1,2$.
Use this last relationship, together with the definitions of all of the functions and the envelope theorem, to show that under these same circumstances, the Marshallian and Hicksian demand curves must also satisfy the Slutsky equation
$$
\frac{\partial c_{i}^{*}(p_{1},p_{2},I)}{\partial p_{i}} = \frac{\partial h_{i}^{*}(p_{1},p_{2},v(p_{1},p_{2},I))}{\partial p_{i}} - \frac{\partial c_{i}^{*}(p_{1},p_{2},I)}{\partial I} \, c_{i}^{*}(p_{1},p_{2},I)
$$
for each $i=1,2$.
\end{enumerate}
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