Mathematical Treatment of Consumer Equilibrium – Explained!

We begin with the simple model of a single commodity ‘X’. The utility function of the consumer may be written as

U = f(q„)

Where utility is measured in monetary units. If the consumer purchases q_x units of the commodity, his expenditure is p_x q_x for p_x price of this commodity.

Since the consumer seeks to maximise the difference between his utility and expenditure (U – p_x q_x), so the necessary condition for equilibrium requires to make partial derivative of the function with respect to q_x equal to zero. We have

Thus, the consumer is in equilibrium, when the marginal utility of commodity ‘X’ is equal to the price of this commodity. To derive equilibrium condition for a consumer consuming ‘n’ commodities, we first obtain the result in case of two goods X, and X₂ only. Let U = f(x₁, x₂) be the utility function. Further, the consumer has a fixed income or budget ‘M’. He spends the entire budget and purchases x, and x₂ units of the commodity. The budget equation is

^M = P₁ ^x₁ ⁺ P₂ ^X₂

Since utility (U) is to be maximised, subject to budget constraint (M), the net utility of the consumer will be

N = Utility as given in the utility function – Loss of utility on account of money spent to purchase the two goods

= Utility – (Marginal Utility of Money x Expenditure)

= U – λ M

= f(x₁, x₂) – X (p₁ x₁ + p₂ x₂) … (4.3)

The first condition for utility maximisation requires

The above two equations can also be obtained by equating first partial derivative of the Langrangian expression

The second order (sufficiency) condition is given by the law of diminishing marginal utility. The second order condition is satisfied, so long as the marginal utility curves for both goods are downward sloping. Symbolically,

Taken across all goods, this gives the general version of the law of equimarginal utility. This law states that the consumer equalizes the ratio of marginal utilities with the ratio of corresponding prices for each pair of goods consumed. Thus, we have

‘λ’ represents the common ratio of marginal utility of money or the marginal utility of income or the gain in total utility from the expenditure of a marginal unit of money. When a consumer consumes many commodities, expenditure on a single commodity represents only a small part of his total expenditure. Any ordinary change in its price will result in only a small change in the purchasing power.