0
$\begingroup$

A consumer has the utility function $u(x_1,x_2)=(x_1^a+x_2^a)^{1/a}$ where 0\neq a<1. Her expenditure must satisfy $p_1x_1+p_2x_2=I$, where $p_i$ is the price of a good i, and I is her income. Find the optimum consumption bundle. Describe Engel's curve for these preferences. Compute the own price elasticity and cross-price elasticity for both goods.

Attempt:

I found the optimum consumption bundle by forming the Lagrangian. The optimum consumption bundle is:

$x_1=\frac{I}{p_1(1+(\frac{p_1}{p_2})^{\frac{a}{1-a}})}\;\;\;\;\;\;\;\;\;x_2=\frac{I}{p_2(1+(\frac{p_2}{p_1})^{\frac{a}{1-a}})}$

Now I need to describe the Engel's curve for these preferences. What should I do?

I also have little idea how to compute the own price elasticity and cross-price elasticity of both goods.

1 Answers 1

1

When I did it I got $x_1(p_1, p_2, I) = \frac{I}{p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}} + p_2}$.

These are your (Marshallian) demand functions, the own (point-)price elasticity of demand for good $x_i$ is given by $\frac{\partial x_i}{\partial p_i}\frac{p_i}{x_i}$ and the cross (point-)price elasticity of demand for good $x_i$ (for a change in the price of good $x_j$) is given by $\frac{\partial x_i}{\partial p_j}\frac{p_j}{x_i}$.

I'm not sure about the Engel curve? Is it just the cross-section of the demand function for fixed prices?

Here's my working for the demand functions:

$ L(x_1, x_2, \lambda) = (x_1^a + x_2^a)^{\frac{1}{a}} + \lambda[I - p_1x_1 - p_2x_2] \\ \frac{\partial L}{\partial x_1} = x_1^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}} - \lambda p_1 = 0 \\ \lambda = \frac{x_1^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}}}{p_1} \\ \frac{\partial L}{\partial x_2} = x_2^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}} - \lambda p_2 = 0 \\ \lambda = \frac{x_2^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}}}{p_2} \\ \therefore \frac{x_1^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}}}{p_1} = \frac{x_2^{a-1}(x_1^a + x_2^a)^{\frac{1-a}{a}}}{p_2} \\ \therefore x_1^{a-1} = x_2^{a-1}\frac{p_1}{p_2} \\ \therefore x_1 = x_2(\frac{p_1}{p_2})^{\frac{1}{a-1}} \\ \text{Subbing into the budget constraint you have} \\ I = p_1(x_2(\frac{p_1}{p_2})^{\frac{1}{a-1}}) + p_2x_2 \\ \therefore x_2 = \frac{I}{p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}} + p_2} = x_2(p_1, p_2, I). $

Then you get $x_1(p_1, p_2, I)$ by the problem being symmetric.

  • 0
    Yeah$I$see I substituted $x_i$ for $p_1$. Thanks. About the Engel's curve: Yeah you are right. Graphically it will be a curve or a line that connects utility maximizing bundles for varying levels of income. And we know from the demand functions of $x_1$ and $x_2$ that $\;I=x_1(p_1(\frac{p_1}{p_2})^{\frac{1}{a−1}}+p_2)=x_2(p_1( \frac{p_1}{p_2})^{\frac{1}{a-1}}+p_2)$. Since Engle's curve tells how the agents choice between good $x_1$ and good $x_2$ changes when income changes its slope will be $\frac{\partial x_2}{\partial x_1}$.2012-04-29