Budget Constraint

A budget constraint is a line or plane that limits possible consumer choices.

Bivariate

A budget contraint in terms of two goods, x and y, is characterized as a line. The line is defined by the equality xP_x + yP_y = m; wherein

• i is the quantity of good i itself

• P_i is the price of good i

• m is the available income that can be spent

The intercepts of this line are m/P_x and m/P_y (on the x- and y-axes respectively).

The slope of this line is -P_x/P_y. This is generally referred to as the marginal rate of transformation (MRT).

Multivariate

A budget constraint is defined by the equality of some income constant m to the dot product of the price and quantity vectors p and q. (Recall that the dot product of two vectors like [x y] and [P_x P_y] is xP_x + yP_y.)

Also, the budget constraint is orthogonal to the price vector.

Non-linear Budget Sets

The above forms of budget constraints are linear. In some cases, the constraint is not constructed in this shape.

• Under autarky, the budget constraint conforms to the PPF.

• Rationing creates a piecewise budget constraint; it operates as normal up to some threshold where it becomes constant.
• Progressive (and regressive) taxing create complex piecewise budget constraints.

Usage

In neoclassical consumer choice theory, the optimal strategy is to pick the bundle where the budget constraint is tangent to the indifference curve:

m_budget = m_indif

MRT_xy = -P_x/P_y = -MU_x/MU_y

P_x/P_y = MU_x/MU_y

MU_x/P_x = MU_y/P_y

The intuitive explanation is that the optimal choice exists where, for all goods, the marginal utility given price is equal, so there is no incentive to substitute.

CategoryRicottone