Lagrange Multipliers

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Lagrange Multipliers

Lagrange Multipliers is a mathematical device introduced to find local extrema of a function subject to constraints. Problems of this general type are ubiquitous in the social sciences. For instance, they arise as optimization problems for quantities that depend on variables that satisfy some additional relations. An example in neoclassical economics is the optimization problem of the utility function with restrictions imposed by a fixed budget.

A typical problem of this type involves a function f(x₁, … x _n), say, dependent on n variables. One seeks values for the variables x_i so that f has a local extremum; that is, f has a local maximum or minimum. This requirement is equivalent to the statement that all partial derivatives of f vanish:

δflδx_i = 0 for all i = 1, …, n.

Quite often the variables x_i are not independent of each other but must satisfy some additional conditions. In such cases one cannot simply solve the above condition on the partial derivatives of f since the variables x_i are not independent of each other. The Lagrange multiplier technique can be employed in cases where the constraints on the variables can be expressed as equations:

g_α (x₁, …, x_n ) = 0, for α = 1, …, k.

In such cases the above extremization problem can be expressed in a different form. Consider a new function F defined through the relation:

This definition introduces k new variables λ_α—one for each constraint g_α. The sum in the above expression is over α. The new variables λ_αare called Lagrange multipliers.

In terms of F it is possible to express the previous problem of constrained extremization as a simple extremization problem. Since F depends on n + k variables, finding local extrema for F can be split up into two sets of equations:

1, …, n.

as well as:

δF/δλ_α = g_α = 0 for all α = 1, …, k.

The second set of equations imposes the constraints (g_α = 0), while the first set imposes a new condition. This apparently new condition actually reduces to the condition that the partial derivatives of f vanish once the constraints g_α = 0 are taken into account.

As an example from neoclassical economics, consider the problem of optimizing utility for a consumer who can buy two commodities, a and b, with price per unit p_a and p_b, respectively The consumer is constrained to spend within his or her budget w. Let the number of commodities of each kind, a and b, purchased by the consumer be given by x_a and x_b, respectively. The budget of the consumer imposes the constraint:

p_ax_a + p_bx_b ≤ w.

This states simply that the total price paid for the two commodities should be less than the available budget. In many neoclassical models the inequality is transformed into an equality. This is justified by hypothesizing one of a number of characteristics of consumer behavior. One hypothesis that justifies this modification is that the consumer is “insatiable” that is, the consumer will buy to the maximum of his or her buying ability. In any case, these hypotheses lead to the modified constraint:

p_a x_a + p_b x_b = w.

The utility function u is a measure of the happiness of the consumer. The utility is a function of x_a and x_b : u = u (x_a, x_b ). Consumers in neoclassical economics base their behavior on maximizing their utility function while satisfying the budgetary constraint. This problem is of the general type discussed above with f = u (x_a, x_b), and, g(x_a,x_b) = p_ax_a + p_bx_b – w. To illustrate the method, consider a utility function given by u(x_a,x_b) = x_ax_b.

To apply the Lagrange multiplier method one constructs a new function:

Setting the partial derivatives of F to zero, one finds:

These equations can be solved to yield:

x_a = w/2p_a

x_b = w/2p_b.

These give the proportions of each of the commodities that optimize the utility function. The Lagrange multiplier

λ = –w/2 p_a p_b

has an economic interpretation. The Lagrange multiplier can be expressed as a derivative of F:

λ = δF/δw.

It tells one how F changes when w is varied. Since on the extremum of F, the constraint is satisfied F coincides with u, the utility. The Lagrange multiplier is thus a measure of how the utility u changes when the budget w is varied, with this interpretation the Lagrange multiplier λ is called the shadow price.

SEE ALSO Constrained Choice; Maximization; Minimization; Shadow Prices; Utility Function