## Quick Reference

A variable introduced to solve a problem involving constrained optimization. Suppose that the function *f*(*x, y*) has to be maximized by choice of *x* and *y* subject to the constraint that *g*(*x, y*) ≤ *k*. The solution can be found by constructing the Lagrangean function

*L*(*x, y, λ*) = *f*(*x, y*) + *λ*[*k* − *g*(*x, y*)]

where *λ* is the Lagrange multiplier. Let *f _{x}* denote

*∂f*(*x, y*)/*∂x*,

etc. The optimal values of *x, y*, and *λ* solve the necessary conditions

*L*_{x} ≡ *f*_{x} − *λg*_{x} = 0, *L _{y}* ≡

*f*−

_{y}*λg*

_{y}= 0,

*λ*[

*k*−

*g*(

*x, y*)] = 0, and

*λ*≥ 0.

If the constraint is not binding, *λ* = 0 and the maximum occurs where *f _{x}* =

*f*= 0. If the constraint is binding,

_{y}*λ*> 0 and the optimum is found by solving the three equations

*L*= 0,

_{x}*L*= 0, and

_{y}*g*(

*x, y*) −

*k*= 0. The Lagrange multiplier,

*λ*, measures the increase in the objective function (

*f*(

*x, y*) that is obtained through a marginal relaxation in the constraint (an increase in

*k*). For this reason, the Lagrange multiplier is often termed a shadow price. For example, if

*f*(

*x, y*) is a utility function, which is maximized subject to the constraint that total spending on

*x*and

*y*is less than or equal to income,

*k*, then

*λ*measures the marginal utility of income—the additional utility provided by one more unit of income.

*Subjects:*
Science and Mathematics — Economics.

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