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Microeconomic Analysis

Hal R. Varian

Chapter 2

Profit Maximization - all with Video Answers

Educators

Chapter Questions

03:13

Problem 1

Use the Kuhn-Tucker theorem to derive conditions for profit maximization and cost minimization that are valid even for boundary solutions, i.e., when some factor is not used.

Noah Musser
Noah Musser
Numerade Educator
02:17

Problem 2

Show that a profit-maximizing bundle will typically not exist for a technology that exhibits increasing returns to scale as long as there is some point that yields a positive profit.

Nick Johnson
Nick Johnson
Numerade Educator
00:00

Problem 3

Calculate explicitly the profit function for the technology $y=x^{a},$ for $0<a<1$ and verify that it is homogeneous and convex in $(p, w)$.

Oluwadamilola Ameobi
Oluwadamilola Ameobi
Numerade Educator
01:03

Problem 4

Let $f\left(x_{1}, x_{2}\right)$ be a production function with two factors and let $w_{1}$ and $w_{2}$ be their respective prices. Show that the elasticity of the factor share $\left(w_{2} x_{2} / w_{1} x_{1}\right)$ with respect to $\left(x_{1} / x_{2}\right)$ is given by $1 / \sigma-1$.

Tyler Moulton
Tyler Moulton
Numerade Educator
03:09

Problem 5

Show that the elasticity of the factor share with respect to $\left(w_{2} / w_{1}\right)$ is $1-\sigma$.

Chai Santi
Chai Santi
Numerade Educator
04:55

Problem 6

Let $\left(\mathbf{p}^{t}, \mathbf{y}^{t}\right)$ for $t=1, \ldots, T$ be a set of observed choices that satisfy WAPM, and let $Y I$ and $Y O$ be the inner and outer bounds to the true production set $Y$. Let $\pi^{+}(\mathbf{p})$ be the profit function associated with $Y O$ and $\pi^{-}(\mathbf{p})$ be the profit function associated with $Y I,$ and $\pi(\mathbf{p})$ be the profit function associated with $Y$. Show that for all $\mathbf{p}, \pi^{+}(\mathbf{p}) \geq \pi(\mathbf{p}) \geq \pi^{-}(\mathbf{p})$.

Matt Just
Matt Just
Numerade Educator
01:10

Problem 7

The production function is $f(x)=20 x-x^{2}$ and the price of output is normalized to $1 .$ Let $w$ be the price of the $x$ -input. We must have $x \geq 0$
(a) What is the first-order condition for profit maximization if $x>0 ?$
(b) For what values of $w$ will the optimal $x$ be zero?
(c) For what values of $w$ will the optimal $x$ be $10 ?$
(d) What is the factor demand function?
(e) What is the profit function?
(f) What is the derivative of the profit function with respect to $w ?$

Carson Merrill
Carson Merrill
Numerade Educator