# Which utility function yields a constant price elasticity of demand function?

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?

• What is the alternative here? Is the consumer trying to solve a wealth allocation problem between two periods, two goods etc? – ramazan Sep 26 '15 at 0:52
• 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. – Thomas V. Sep 27 '15 at 10:13

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.

• 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. – Thomas V. Sep 27 '15 at 10:35
• 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? – Thomas V. Sep 27 '15 at 10:35
• @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. – Alecos Papadopoulos Sep 27 '15 at 11:38

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 ;

$$x\left(p\right)=Ap^{a}$$
$$elasticity=\frac{\frac{\partial x\left(p\right)}{\partial p}}{\frac{x\left(p\right)}{p}}$$
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.