Indirect utility function

The indirect utility function describes the maximum amount of utility a consumer can achieve, given a fixed amount of income and facing fixed prices. For two goods it is defined as

V(P_1,P_2,M) = U(X_1^*,X_2^*)

where $V$ denotes the indirect utility function, P_1 is the price of good 1, P_2 is the price of good 2, M is the consumers income, U is the utility function, X_1^* is the quantity of good 1 consumed in equilibrium and X_2^* is the quantity of good 2 consumed in equilibrium.

Cobb Douglas Utility function

Consider the scenario where a consumer has the utility function described by Cobb Douglas Preferences, such that

 U = X_1^\alpha X_2^{(1-\alpha)}

and has the following budget constraint:

 M = P_1 X_1 + P_2 X_2

It is important to note that X_1 and X_2 are just feasible options for good 1 and 2 and are not the equilibrium level of consumption X_1^* and X_2^*.

To calculate the indirect utility function, we need to first find the demand equations X_1^* and X_2^*. Using our equilibrium condition

\frac{MU \text{ of } X_1}{P_1}=\frac{MU\text{ of }X_2}{P_1}

And substituting in the marginal utility for both good X_1 and good X_2

MU \text{ of } X_1 = \alpha \frac{U}{X_1}

MU \text{ of } X_2 = (1-\alpha) \frac{U}{X_2}

gives us

\frac{\alpha U}{P_1 X_1}=\frac{(1-\alpha) U}{P_2 X_2}

which allows us to solve for X_1 in terms of X_2 such that:

X_1=X_2\frac{alpha P_2}{(1-\alpha) P_1 }

which we now substitute into the budget constraint, such that:

M = P_1X_2\frac{\alpha P_2}{(1-\alpha) P_1 } + P_2 X_2

(1-\alpha) M = \alpha P_2 X_2 + (1-\alpha) P_2 X_2

(1-\alpha) M = \alpha P_2 X_2 + P_2 X_2 - \alpha P_2 X_2

(1-\alpha) M = \alpha P_2 X_2 + P_2 X_2 - \alpha P_2 X_2

 (1-\alpha)M = P_2 X_2

Which can be re-arranged to give us the demand equation, which we denote as X_2^*

X_2^* = \frac{(1-\alpha)M}{P_2}

By symmetry, we should hopefully see that:

X_1^* = \frac{(\alpha)M}{P_1}

Substituting both these demand equations into our utility function gives the indirect utility function:

V(P_1,P_2,M) = U(X_1^*,X_2^*) = (\frac{(\alpha)M}{P_1})^\alpha (\frac{(\alpha)M}{P_2})^{1-\alpha}

V(P_1,P_2,M) = M(\frac{(\alpha)}{P_1})^\alpha (\frac{(\alpha)}{P_2})^{1-\alpha}

 

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