Compensated Demand Function

Suppose there is a situation in which government imposes taxes or subsidies a consumer in such a way so as to leave his utility unchanged after a price change. Lets assume this is done by lump sum payment that will give the consumer the minimum income necessary to achieve his initial utility level.

The compensated demand function give the quantities of the commodities that he will buy as function of commodity prices under these conditions. They are obtained by minimising the consumer’s expenditure subject to the constraint that his utility is at the fixed level.

Suppose the utility function is $$U\; =\; x_{1}x_{2}$$. Form the expression

$$Z\; =\; p_{1}x_{1}\; +\; p_{2}x_{2}\; +\; \mu \; \left( U\; -\; x_{1}x_{2} \right)$$

And set its partial derivatives to zero :

$$\frac{\partial z}{\partial x_{1}}=\; p_{1}\; -\; \mu x_{2}\; =\; 0$$,

$$\frac{\partial z}{\partial x_{2}}=\; p_{2}\; -\; \mu x_{1}\; =\; 0$$,

$$\frac{\partial z}{\partial \mu }=\; U\; -\; x_{1}x_{2}\; =\; 0$$,

Solving the above equations for x1 and x2 gives the compensated demand functions :

$$x_{1}\; =\; \sqrt{\frac{U\; p_{2}}{p_{1}}}$$ and $$x_{2}\; =\; \sqrt{\frac{U\; p_{1}}{p_{2}}}$$

These are also homogeneous functions of degree zero.