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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

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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.

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    $x_1(p_1, p_2, I) = \frac{I}{p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}} + p_2}$ is the same thing posted in my question. I just fiddled around with the powers. Thats why my $x_1$ and $x_2$ look a bit different. I got the demand functions the same way you did, by forming the Lagrangian, so I understand that part. What I do not understand is how I compute, for example, the own price elasticity using $\frac{\partial x_i}{\partial p_i}\frac{p_i}{x_i}$2012-04-28
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    Do I simply take the derivative of the demand function of $x_1$, for instance, with respect to $p_1$ and then multiply it by $\frac{x_1}{p_1}$?2012-04-28
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    About the Engel's curve. In the assignment it is the following: _A locus of points (in $x_1$ ,$x_2$ space) of tangency between the budget line and the consumer’s indifference curve for given prices and different values of income $I$ is called an Engle’s curve or income expansion path. Describe the Engle’s curve for these preferences._2012-04-28
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    As I understand it(and I think it is correct) it is how the agents choice between good $x_1$ and good $x_2$ changes when income changes: $I=x_1({p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}} + p_2})=x_2({p_2(\frac{p_2}{p_1})^{\frac{1}{a-1}} + p_1})$2012-04-28
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    I'm not sure how you got from my demand functions to yours sorry, when I try to clean it up a bit I get $$ x_1(p_1, p_2, I) = \frac{I}{p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}} + p_2} \\ = \frac{I}{\frac{p_1^{\frac{a}{a-1}}}{p_2^{\frac{1}{a-1}}} + p_2} \\ = \frac{Ip_2^{\frac{1}{a-1}}}{p_1^{\frac{a}{a-1}} + p_2^{\frac{a}{a-1}}} $$2012-04-29
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    Yes you just take the partial derivative of say $x_1$ with respect to the price of good $j$ (using the chain rule) and multiply that by $\frac{p_1}{x_j(p_1, p_2, I)}$.2012-04-29
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    Going entirely off the description you give above for the Engel curve I suspect for a given set of prices it maps out the utility maximising bundles for varying levels of income? Currently trying to work out how to actually get that (note above you substituted $x_i$ for $p_i$ which can't be right).2012-04-29
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    Sorry, that's the offer curve that I describe above, I think the Engel curve might just be the cross section of each demand function for given prices like I said at the start.2012-04-29
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    About your demand function and mine being identical: When you subbed $ x_1 = x_2(\frac{p_1}{p_2})^{\frac{1}{a-1}}$ into the budget constraint you solved for $x_2$ not $x_1$. I corrected that in your answer. Now, lets talk about the expression in the denominator of the demand function for $x_2$: $$ p_1(\frac{p_1}{p_2})^{\frac{1}{a-1}}+p_2= $$ $$ p_1 p_1 ^{\frac{1}{a-1}} p_2 ^{\frac{-1}{a-1}}+p_2= $$ $$ p_2(p_1 ^{\frac{a}{a-1}} p_2 ^{\frac{-1}{a-1}}p_2 ^{-1} +1)= $$ $$ p_2(1+p_1 ^{\frac{a}{a-1}}p_2 ^{\frac{-a}{a-1}})= $$ $$ p_2(1+(\frac{p_2}{p_1})^{\frac{a}{1-a}})= $$2012-04-29
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    $$ =p_2(1+(\frac{p_1}{p_2})^{\frac{a}{a-1}}) $$ Which is identical to the denominator in the demand function for $x_2$ I posted in my question.2012-04-29
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    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