A consumer’s ordinary demand function sometimes also called a Marshallian demand function gives the quantity of a commodity that he will buy as a function of commodity prices and his income. They can be derived from the analysis of utility maximisation.

Let the utility function is \(U=\; x_{1}x_{2}\) and the budget constraint as \(y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2}\; =\; 0\). Form the expression

\(V\; =\; x_{1}x_{2}\; +\; \lambda \left( \; y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2} \right)\)

And set its partial derivatives equal to zero :

\(\frac{\partial V}{\partial x_{1}}=\; x_{2}\; -\; p_{1}\lambda \; =\; 0\)

\(\frac{\partial V}{\partial x_{2}}=\; x_{1}\; -\; p_{2}\lambda \; =\; 0\)

\(\frac{\partial V}{\partial \lambda}=\; y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2} = 0\)

Solving for *x** _{1}* and

*x*

*gives the demand functions :*

_{2}\(x_{1}\; =\; \frac{y}{2p_{_{1}}}\; and\; x_{2}\; =\; \frac{y}{2p_{2}}\)

The demand functions derived in this fashion are contingent on continued optimising behaviour by the consumer. Given the consumer’s income and prices of commodities, the quantities demanded by him can be determined from his demand functions. Of course these functions are the same as those obtained directly from the utility function.

Two important properties of demand functions can be deducted:

- The demand for any commodity is a single-valued function of prices and income,
- Demand functions are homogeneous of degree zero in prices and income, i.e., if all prices and income changes in the same proportion, the quantities demanded remains unchanged.

The first property follows from the strict quasi-concavity of the utility function; a single maximum, and therefore a single commodity combination, corresponds to a given set of prices and income.

To check the degree of homogeneity we can change all the variables i.e., prices and income in the same proportion. The budget constraint becomes

\(ay-ap_{1}q_{1}-ap_{2}q_{2}\; =\; 0\)

Where a is the factor of proportionality. The above equation “V” can be written as

\(V\; =\; f\left( x_{1},x_{2} \right)\; +\; \lambda \left( ay\; -\; ap_{1}q_{1}\; -\; ap_{2}q_{2} \right)\)

And the first order conditions are :

\(f_{1}\; -\; \lambda ap_{1}\; =\; 0\)

\(f_{2}\; -\; \lambda ap_{2}\; =\; 0\)

\(ay\; -\; ap_{1}q_{1}\; -\; ap_{2}q_{2}\; =\; 0\)

The last equation is the partial derivative with respect to Langrange multiplier and can be written as

\(a\left( y\; -\; p_{1}q_{1}\; -\; p_{2}q_{2} \right)\; =\; 0\)

Since *a *\(\neq \) 0 ,

\(y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2}\; =\; 0\)

By eliminating *a* from the first two equations from the above equations and dividing the two will give

\(\frac{f_{1}}{f_{2}}=\; \frac{p_{1}}{p_{2}}\)

Therefore the demand function for the price-income set \((ay\; -\; ap_{1}q_{1}\; -\; ap_{2}q_{2}\;)\) is derived from the same equations as for the price-income set \((y\; -\; p_{1}x_{1}\; -\; p_{2}x_{2}\;)\).

This proves that the demand functions are homogeneous of degree zero in prices and income. If all prices and the consumer’s income are increased in the same proportion, the quantities demanded by the consumer does not change.

It also implies some restrictions i.e. he will not behave weather the consumer becomes rich or poor in terms of real income if his income and prices rise in the same proportion. A rise in money income is desirable for the consumer but its benefits are illusionary if price changes proportionately.