# Indirect Utility Function

In the cardinal utility approach, the utility function was direct in the sense that the total utility was related to the quantities of different commodities and services consumed by a consumer at a given time. The direct utility function can be expressed as :

$$U\; =\; f\left( q_{1},\; \; q_{2},\; \; q_{3},\; ……..q_{n} \right)$$,

The ordinary demand equations for the goods can be obtained through constrained maximisation of such utility functions. The demand function related to prices and quantities of different commodities and fixed level of income $$Y$$  can be expressed as below :

$$q_{i}\; =\; f_{i}\left( p_{1},p_{2},p_{3}………p_{n},\; y \right)$$

The indirect utility function shows that the maximum attainable utility is a function of price and income. It is possible to express the maximum attainable utility $$(U^{\star })$$  as below :

$$U^{\star }=\; f\; \left[ \phi _{1}\left( p_{1},p_{2}……..p_{n},y \right),\; \phi _{2}\left( p_{1},p_{2},……p_{n},y \right),\; ………….\phi _{n}\left( p_{1},p_{2}…..p_{n},y \right) \right]$$

In a simplified form it can be expressed as,

$$U^{\star }=\; f^{\star }\left( p_{1},p_{2}……….p_{n},y \right)$$

It can also be expressed in normalised form. The constrained has to be altered slightly for this purpose. It is assumed that $$y\; =\; p_{1}q_{1}\; +\; p_{2}q_{2}\; +\; …….p_{n}q_{n}$$ is the budget constraint.

Now dividing both sides by $$y$$

$$\frac{y}{y}\; =\; \frac{p_{1}q_{1}}{y}+\frac{p_{2}q_{2}}{y}+………………\frac{p_{n}q_{n}}{y}$$,

$$1\; =\; \frac{p_{1}q_{1}}{y}+\frac{p_{2}q_{2}}{y}+………………\frac{p_{n}q_{n}}{y}$$,

Now let $$v_{i}\; =\; \frac{p_{i}}{y}$$ , so the budget constraint can be written as :

$$1\; =\; q_{1}v_{1}\; +\; q_{2}v_{2}\; +\; …………..q_{n}v_{n}$$,

$$1\; =\; \sum_{i=1}^{n}{q_{i}v_{i}}$$,

Let the utility function $$U\; =\; f\left( q_{1},q_{2}…….q_{n} \right)$$ along with the equation above gives the following first order conditions for maximisation

$$f_{i}\; -\; \lambda v_{i}\; =\; 0$$ where i = 1 ,2, ……n,

$$1\; -\; \sum_{i=1}^{n}{q_{i}v_{i}} = 0$$,

Ordinary demand functions can be obtained by solving the above equations

$$q_{i}\; =\; D_{i}\; \left( v_{1}………v_{n} \right)$$,

The indirect utility function $$g\left( v_{1}………v_{n} \right)$$ is defined by

$$U\; =\; f\left[ D_{1}\left( v_{1},v_{2}……v_{n} \right),…….D_{n}\left( v_{1}………v_{n} \right) \right]\; =\; g\left( v_{1},\; ……..v_{n} \right)$$,

It gives maximum utility as a function of normalised prices. The indirect utility function reflects a degree of optimization and market prices.

The indirect utility function can provide the indirect indifference curves similar to those generated by the direct utility function. From geometrical angle, some differences between the two appear. Firstly, in case of indirect utility function v1and vcan be shown on the two axis. Secondly, the indirect indifference curve farther away from the origin will represent lower utility and the one nearer to the origin will represent the higher utility level. Thirdly, the optimum position related utility will coincide with the point of origin.