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How do I know which utility function I can use to find an isoelastic demand function, e.g., $x(p)=Ap^a$? And similarly, which cost function can I use to find an isoelastic supply function? Does it work through trial and error, or is there a particular method?

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  • $\begingroup$ What is the alternative here? Is the consumer trying to solve a wealth allocation problem between two periods, two goods etc? $\endgroup$ – ramazan Sep 26 '15 at 0:52
  • $\begingroup$ Not quite sure to understand what your question asks. Here is again some intuition: Say you have a Cobb-Douglas utility function, and you maximize utility s.t. a budget constraint to derive some Marshallian demand functions. Then my question is whether it is possible to find out that this Marshallian demand function was derived from a CD utility function. Or in my case, can I find the utility function that lead to the isoelastic demand function. $\endgroup$ – Thomas V. Sep 27 '15 at 10:13
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By Roy's Identity we have that Marshallian (uncompensated) demand for good $x_i$ is

$$x_i^M = \frac {\partial U^*/\partial p_i}{\partial U^*/\partial B} \tag{1}$$

where $U^*$ is optimized utility over goods vector $\mathbf x = (x_1,...,x_i,...x_n)$ and $B$, which is the available budget, and given the price vector $\mathbf p = (p_1,...,p_i,...,p_n)$. To obtain a constant elasticity demand function we require

$$\eta \equiv \frac {\partial x_i^M}{\partial p_i}\cdot \frac{p_i} {x_i^M}=const. \tag{2}$$

Using $(1)$ we have that

$$\frac {\partial x_i^M}{\partial p_i} = \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2} \tag{3}$$

Inserting $(3)$ and $(1)$ into $(2)$ we have

$$\eta \equiv \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{\big[\partial U^*/\partial B\big]^2}\cdot \frac{p_i} {\frac {\partial U^*/\partial p_i}{\partial U^*/\partial B}}$$

$$= \frac {(\partial^2 U^*/\partial p^2_i)\cdot(\partial U^*/\partial B) -(\partial^2 U^*/\partial B^2)\cdot (\partial U^*/\partial p_i)}{(\partial U^*/\partial B)\cdot (\partial U^*/\partial p_i)} \cdot p_i$$

$$\implies \eta = p_i\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) \tag{4}$$

For this expression to be constant over the whole range of $p_i$ we need that

$$\left(\frac {\partial^2 U^*/\partial p^2_i}{ \partial U^*/\partial p_i}- \frac{\partial^2 U^*/\partial B^2}{\partial U^*/\partial B}\right) = \frac {C}{p_i} \tag{5}$$

for some constant $C$, which becomes the constant value of the price elasticity of demand.

That's the general condition that must be satisfied.

You could check whether the generalized Cobb-Douglas standard Utility function specification

$$U(\mathbf x) = \prod_{i=1}^n x_i^{a_i}$$

satisfies the condition, perhaps under some restrictions.

THE CASE OF QUASI-LINEAR UTILITY FUNCTION

The OP ponders the case of a quasi-linear utility function, so let's solve this forward. We have

$$\max_{x,m} [cx^{\theta} +m],;\;\;\; s.t. \;\;\;p_xx + m = B,\;\;\; 0<\theta <1,\;\; c>0$$

where $m$ is residual income for all other goods. The Lagrangean is

$$\Lambda = cx^{\theta} +m + \lambda[B-p_xx - m]$$ and first order conditions are

$$c\theta x^{\theta-1}=\lambda p_x,\\\;\;\; \lambda =1$$

So

$$x^* = \left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)},\;\;\; m^* = B-p_xx^*$$

So the indirect utility function is

$$U^* = c\left(\frac {c\theta}{p_x}\right)^{\theta/(1-\theta)} + B-p_x\left(\frac {c\theta}{p_x}\right)^{1/(1-\theta)}$$

$$= \left[c(c\theta)^{\theta/(1-\theta)} - (c\theta)^{1/(1-\theta)}\right] \cdot \frac {1}{p_x^{\theta/(1-\theta)}} + B$$

One can easily verify that this indirect utility function satisfies the required condition $(5)$ for isoelastic demand, and also see how the preference parameters map to the demand parameters.

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  • $\begingroup$ Thank you. Your result is quite general. Shouldn't it be actually the indirect utility? If not, I am not quite sure how to test whether the generalized CD utility satisfies your condition, as the utility function is not a function of price. $\endgroup$ – Thomas V. Sep 27 '15 at 10:35
  • $\begingroup$ I thought also to go backwards through Roy's identity. However, I tried to do it as follows: Demand function is given as X=AP^α. In order to derive the indirect utility, knowing that X≡∂V/∂P/∂V/∂m=AP^α. I take the integral of the function w.r.t. p and m: ∫∂V/∂P dP=(1/(a+1))AP^(a+1); and ∫∂V/∂m dm=m Then the indirect utility function is V(P)=1/(α+1) AP^(α+1)+m Solving X(P) for P=(X/A)^(1/α) and substituting it in the indirect utility function I get the function U(X)=1/(α+1) A(X/A)^((1+α)/α)+m. Can you tell me if this is a possible approach? $\endgroup$ – Thomas V. Sep 27 '15 at 10:35
  • $\begingroup$ @ThomasV. Comment 1: I wrote the utility function $U$ prior to solving the optimization. So one should first solve the optimization problem to arrive at the indirect utility function $U^*$ and then check. $\endgroup$ – Alecos Papadopoulos Sep 27 '15 at 11:38
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For a two good economy:

$I$ is a the collection of goods such that $I=(i_1,...i_x)$ and $x$ is therefore the quantity of $i$ type goods.

In the same way, $K=(k_1,..., ky)$, and $y$ is the amount of $k$ type goods.

$p$ is the relative price of $i$ and $k$

$ p = x/y $

The demand for good $k$ is

$ x(p) = Ap^a $

That is,

$ x = A( $ $x \over y $ $)^a $

$ x = Ax^a( $ $1 \over y^a $ $) $

$x^{1-a}\over A$ = $1 \over y^a $

$A\over x^{1-a}$ =$y^a $

$y=$ $({A \over x^{1-a} } )$ $^a$

$({A \over x^{1-a} } )$ $^a -y = 0 $

Now, trade occurs when $U_k(x) \ge U_i(y)$. Demand can be understood as a function $D(x, y)$ which plots the points at which the marginal utility of the $x$th $k$ type good and the $y$th $i$ type good is equal:

$U_k(x)=U_i(y)$

$U_k(x) - U_i(y) = 0$

Now we can say that

$U_k(x) - U_k(y)$ = $({A \over x^{1-a} } )$ $^a -y$

$U_k(x)$ = $({A \over x^{1-a} } )$ $^a +U_i(y) -y$

Finally

z = $U_i(y) -y$

And the marginal utility curve is:

$U_k(x)$ = $({A \over x^{1-a} } )$ $^a +z$

$U_k(x)$ = $({Ax^{a-1}})^a +z$

Now, the value of z will change the plotting, but as you can see, not the shape of the curve. Now, if good $i$ is you numéraire you can apply this to any $n$-good economy.

Hope it helps.

EDIT: As user "optimal control" noted before me, $a$ is your elasticity. So you can just plug in the desired elasticity above.

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  • $\begingroup$ Thank you. This seems like an interesting approach. Can I then simply integrate the marginal utility function to get the utility function? $\endgroup$ – Thomas V. Sep 27 '15 at 10:20
  • $\begingroup$ That's the idea. However I just noticed a mistake in my answer. I will edit my response as soon as I have the time. What I got wrong is that z can not be considered conatant because I defined y as a function of x. $\endgroup$ – Pablo Derbez Sep 27 '15 at 15:04
  • $\begingroup$ My idea now is to use a Lagrangian function with the above equations. If I get any interesting result I will post it here. $\endgroup$ – Pablo Derbez Sep 27 '15 at 15:09
  • $\begingroup$ I haven't been able to take my mind of this problem for hours. I'm too tired now, but I think there are infinite solutions and the best you can hope for is finding the relationship of the derivatives of $Uk$ and $Ui$. Basically, for any utility function $Uk$ you can construct a function $Ui$ such that the resulting demand function is isoelastic, as long as their derivatives follow a certain proportion. As long as they follow that proportion, there is an infinite amount of derivatives that can satisfy your demand model. So it's pretty hopeless without any further information. $\endgroup$ – Pablo Derbez Sep 28 '15 at 2:45
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Simply, an iso-elastic demand function exhibits a constant elasticity. For example, CRRA type of utility function is an iso elastic function. The best way to see if the demand function is iso elastic or not, you could differenciate it as ;

Your demand function is ;

$$x\left(p\right)=Ap^{a}$$

To see that if elasticity is constant or not ;

$$elasticity=\frac{\frac{\partial x\left(p\right)}{\partial p}}{\frac{x\left(p\right)}{p}}$$

When you make the calculations, you will see that elasticity will be simply equal to $a$ which is a constant term. (of course in iso elastic demand function.) Same counts for the cost function.

Hope that it helps.

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  • $\begingroup$ Thank you. I actually understand why the isoelastic demand function has a constant elasticity. The question was rather which utility function I need to maximize s.t. budget constraint, to find the isoelastic demand. $\endgroup$ – Thomas V. Sep 27 '15 at 10:16

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