The analysis will be first for a single household and the extension to the entire market is merely a matter of aggregating over all households in the market.

Let the utility function of a household be U(x,Y) where x is the level of consumption of the good under consideration and Y is the vector of the levels of consumption of all other goods and services. When a household maximizes utility subject to an income constraint p_xx + P_Y·Y = M they consume where

∂U/∂x = λp_x
 

The Lagrangian multiplier λ is such that dU/dM=λ. (The first order conditions for the other goods and services are not relevant here.)

For an increase in consumption of dx the increase in utility is given by:

dU = (∂U/∂x)dx = λp_xdx
 

But dU=(dU/dM)dM = λdM so the amount of money income necessary to achieve the same increase in utility satisfaction as an increase in consumption of dx is given by:

dU = λp_xdx = λdM
and thus
dM = p_xdx
 

The increase in income ΔM necessary to achieve the same increase in satisfaction as an increase in consumption from X to X+ΔX is given by the integral of the above relation over the range of x from X to X+ΔX; i.e.,

ΔM = ∫_X^S+ΔXp_xdx.
 

The demand function for a household is the relationship between the quantity x consumed as a function of its price with all other factors held fixed; i.e,,

X = f(p_x)
and the inverse demand function is
p_x = f^-1(X)
 

Thus the previous integral expression is just the area under the demand function with price being expressed as a function of consumption.

The sum of the equivalent money incomes for all consumers as a result of the sum of their increases in consumption is then the area under the aggregate market demand curve.