# How to derive a utility function using indirect utility function?

I am working on this question to solve for utility function. The indirect utility function is given as follows:

$$V(p_1,p_2,w)=(\frac {1}{p_1} + \frac {1}{p_2})*w$$

w stands for income, $$p_1$$ and $$p_2$$ are the price of each goods

I tried the method of deriving hicksian demand function by finding the partial derivatives of income by each goods price, but I stuck at finding the price ratio since the differentiated product is in the form of $$X^c= {Vp_2-Vp_1p_2\over (p_1+p_2)^2}$$

Any help is appreciated. Thank you.

• You can't find Hicksian demands using the indirect utility function, you need to use the expenditure function for that. I think you're attempting to find marshallian demands in your approach by using Roy's identity but that will be of no use since we need Hicksian demands Jun 17 at 20:09

We need to solve for the Utility function given the indirect utility function (IUF).

The Indirect Utility Function is: $$V(p_1,p_2,w)=w\left(\frac{1}{p_1}+\frac{1}{p_2}\right)$$

From IUF we can write the expenditure function as: $$E(p_1,p_2,U)=\frac{Up_1p_2}{p_1+p_2}$$

Using Shepherd's lemma we can find the Hicksian demand functions. Let us denote the two goods by $$x$$ and $$y$$, respectively.

Thus, $$x^h(p_1,p_2,U)=\frac{\partial E}{\partial p_1}=\frac{Up_2^2}{(p_1+p_2)^2}$$ and $$y^h(p_1,p_2,U)=\frac{\partial E}{\partial p_2}=\frac{Up_1^2}{(p_1+p_2)^2}$$

Since we are trying to find a utility function of the form $$U(x,y)$$ we need to use the expression for $$x^h$$ and $$y^h$$ obtained above to solve for $$U(x,y)$$

let $$p=\frac{p_1}{p_2}$$ and rewrite $$x^h$$ and $$y^h$$ as: $$\begin{eqnarray} x=x^h=\frac{U}{\left(\frac{p_1}{p_2}+1\right)^2}=\frac{U}{(p+1)^2} \tag{1} \\ y=y^h=\frac{U}{\left(1+\frac{p_2}{p_1}\right)^2}=\frac{Up^2}{(p+1)^2} \tag{2} \end{eqnarray}$$

substituting $$(1)$$ in $$(2)$$ gives us the relation: $$p^2=\frac{y}{x} \implies p=\sqrt{\frac{y}{x}}\tag{3}$$

Lastly, substituting $$(3)$$ in $$(1)$$ gives us: $$x=\frac{Ux}{(\sqrt x +\sqrt y)^2}$$

Rewriting above we get: $$\boxed{U(x,y)=(\sqrt x+ \sqrt y)^2}$$

• To check whether my answer is correct or not, you can use this utility function and solve the UMP to see if you're getting the same IUF or not. Jun 17 at 20:07

An alternative is to use the dual of the utility maximisation problem: $$u(x_1, x_2) = \min_{p_1, p_2} v(p_1, p_2, 1) \text{ s.t. } p_1 x_1 + p_2 x_2 = 1.$$ In your example, this becomes: $$u(x_1, x_2) = \min_{p_1, p_2} (1/p_1 + 1/p_2) \text{ s.t. } p_1 x_1 + p_2 x_2 = 1.$$

The first order conditions give: \begin{align*} &-\frac{1}{(p_1)^2} = \lambda x_1,\\ &-\frac{1}{(p_2)^2} = \lambda x_2\\ &p_1 x_1 + p_2 x_2 = 1 \end{align*} Solving these for $$p_1$$ and $$p_2$$ gives: \begin{align*} &p_1 = \frac{1}{(\sqrt{x_1} + \sqrt{x_2})\sqrt{x_1}}\\ &p_1 = \frac{1}{(\sqrt{x_1} + \sqrt{x_2})\sqrt{x_2}} \end{align*}

Substituting into $$v(p_1, p_2, 1)$$ produces the following direct utility function: $$u(x_1, x_2) = (\sqrt{x_1} + \sqrt{x_2})^2.$$

• I see you arrived at the right answer, but can you explain why the direct utility is the minimum over prices of the indirect utility? Jul 29 at 14:34
• @Nicolas Torres: the answer would be a bit too long to put here, but for a reference, you can find it in the handbook of Jehle and Reny (Advanced Microeconomic Theory, Third edition) in section 2.1.3 (Indirect utility and consumer preferences) or in the excellent book of Cornes (Duality and modern economics), section 2.5 (duality between u(q) and v(p))
– tdm
Jul 30 at 7:43