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Corners 1. A consumer has utility function $u(x_1, x_2) = x_1 + \log(x_2)$ and solves $V(w, p) = \max_{x \ge 0} u(x)$ s.t. $px \le w$ a. Show that $Du(x_1, x_2) = (1, 1/x_2)$, that $\partial^2 u/\partial x_1 \partial x_1 = 0$, and that $\partial^2 u/\partial x_2 \partial x_2 < 0$. b. Find the set of $(x_1, x_2)$ such that $\partial u/\partial x_1 < \partial u/\partial x_2$. c. At prices $p = (p_1, p_2)$, find the set of $(x_1, x_2)$ such that the utils per dollar spent on good 1 are smaller than the utils per dollar spent on good 2. d. Give the isoquants of $u(\cdot, \cdot)$ and their slopes. e. Put the previous two pieces of the problem together to find the consumer's demad functions $x_1^(p, w)$ and $x_2^(p, w)$. f. Find the possible patterns of the income expansion paths, that is, graph $x^*(p, w) \in \mathbb{R}^2_+$ as $w$ increases.

          Corners 1. A consumer has utility function $u(x_1, x_2) = x_1 + \log(x_2)$ and solves
$V(w, p) = \max_{x \ge 0} u(x)$ s.t. $px \le w$
a. Show that $Du(x_1, x_2) = (1, 1/x_2)$, that $\partial^2 u/\partial x_1 \partial x_1 = 0$, and that $\partial^2 u/\partial x_2 \partial x_2 < 0$.
b. Find the set of $(x_1, x_2)$ such that $\partial u/\partial x_1 < \partial u/\partial x_2$.
c. At prices $p = (p_1, p_2)$, find the set of $(x_1, x_2)$ such that the utils per dollar
spent on good 1 are smaller than the utils per dollar spent on good 2.
d. Give the isoquants of $u(\cdot, \cdot)$ and their slopes.
e. Put the previous two pieces of the problem together to find the consumer's
demad functions $x_1^*(p, w)$ and $x_2^*(p, w)$.
f. Find the possible patterns of the income expansion paths, that is, graph
$x^*(p, w) \in \mathbb{R}^2_+$ as $w$ increases.

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Corners 1. A consumer has utility function u(x1, x2) = x1 + log(x2) and solves
V(w, p) = maxx ≥ 0 u(x) s.t. px ≤ w
a. Show that Du(x1, x2) = (1, 1/x2), that ∂^2 u/∂ x1 ∂ x1 = 0, and that ∂^2 u/∂ x2 ∂ x2 < 0.
b. Find the set of (x1, x2) such that ∂ u/∂ x1 < ∂ u/∂ x2.
c. At prices p = (p1, p2), find the set of (x1, x2) such that the utils per dollar
spent on good 1 are smaller than the utils per dollar spent on good 2.
d. Give the isoquants of u(·, ·) and their slopes.
e. Put the previous two pieces of the problem together to find the consumer's
demad functions x1^*(p, w) and x2^*(p, w).
f. Find the possible patterns of the income expansion paths, that is, graph
x^*(p, w) ∈ℝ^2+ as w increases.

Added by Kristen H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/14/2023

Step 1

The first-order partial derivatives of u(x₁, x₂) are: ∂u/∂x₁ = 1 ∂u/∂x₂ = 1/x₂ Show more…

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Corners 1. A consumer has utility function u(x₁, x₂) = x₁ + log(x₂) and solves V(w, p) = maxz>0 u(x) s.t. px w ≤ a. Show that Du(x₁, x₂) = (1, 1/x₂), that ∂²u/∂x₁∂x₁ = 0, and that ∂²u/∂x₂∂x₂ < 0. b. Find the set of (x₁, x₂) such that ∂u/∂x₁ < ∂u/∂x₂. c. At prices p = (P₁, P₂), find the set of (x₁, x₂) such that the utils per dollar spent on good 1 are smaller than the utils per dollar spent on good 2. d. Give the isoquants of u(·,·) and their slopes. e. Put the previous two pieces of the problem together to find the consumer's demand functions x₁(p, w) and x₂(p, w). f. Find the possible patterns of the income expansion paths, that is, graph x*(p, w) ∈ R² as w increases.

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00:01 Let's take a look at this problem.

00:03 So, consumer preferences over two goods x1 and x2 is given by a utility function u x1, x2 which is equal to x1 power 1 by 2 plus x2 power 1 by 2.

00:23 So, now i want to check for the monotonicity and the convexity for these preferences.

00:41 Okay, so let's briefly discuss the monotonicity first.

00:59 Okay, so monotonicity means that more of a good is always preferred to less of a good.

01:25 Now, we have the utility function as x1 1 by 2 plus x2 power 1 by 2.

01:33 Now, from this we can see that the consumer always prefer a bundle with more of both goods more of both goods to a bundle with less of both goods.

02:11 Okay, so let's look at our convexity.

02:25 The convexity refers to the property that if a consumer prefers one bundle of goods over another he will always prefer any bundle that is a weighted average of those two bundles.

02:39 So, if a consumer prefers one bundle of good over another then he always prefers any bundle, any good that is the weighted average of those two bundles.

03:33 Okay, so let's take our utility function here u x1 comma x2 is given by x1 power 1 by 2 plus x2 power 1 by 2.

03:47 So, let's check for the monotonicity.

03:50 Let's find my marginal utility.

04:01 Now, for good x1 the marginal utility mu x1 is equal to the partial derivative with respect to x1 dou u by dou x1 which is equal to 1 by 2 x1 power minus 1 by 2...

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