How to derive a utility function using indirect utility function?

Question

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.

Answer 1

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

Answer 2

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: $$ u(x_1, x_2) = (\sqrt{x_1} + \sqrt{x_2})^2. $$