Question

Explain in technical details why the Lagrange multiplier is crucially used in analysing the theory of the consumer .Further as a managerial economics student ,show how you can apply this technique to advise your company to make sales without making loses and affecting your customers negatively

Answer 1

ECONOMIC APPLICATIONS OF LAGRANGE MULTIPLIERS

Maximization of a function with a constraint is common in economic situations. The first section consid- ers the problem in consumer theory of maximization of the utility function with a fixed amount of wealth to spend on the commodities. We consider three levels of generality in this treatment.

The second section presents an interpretation of a Lagrange multiplier in terms of the rate of change   of the value of extrema with respect to the change of the constraint constant. The first subsection gives a general presentation, the second subsection illustrates this formula for a particular situation, and then the third subsection derives the formula in general.

In the third section, we calculate a rate of change of the minimal cost of the output with respect to the change of price of one of the inputs.

1. Maximize Utility with Wealth Constraint

1.1. Three commodities. Assume there are three commodities with amounts x₁, x₂, and x₃, and prices p₁, p₂, and p₃. Assume the total value is fixed, p₁x₁ + p₂x₂ + p₃x₃ = w₀, where w₀> 0 is a fixed positive constant. Assume the utility is given by U = x₁x₂x₃. The maximum of U on the commodity bundles given by the wealth constraint satisfy the equations

  x2x₃ =  λp1

x1x₃ = λp2   

x₁x₂ = λp₃

w₀ = p₁x₁ + p₂x₂ + p₃x₃.

If we multiply the first equation by x₁, the second equation by x₂, and the third equation by x₃, then they are all equal:

x₁x₂x₃ = λp₁x₁ = λp₂x₂ = λp₃x₃.

One solution is λ = 0, but this forces one of the variables to equal zero and so the utility is zero. If λ = 0, then

p₁x₁ = p₂x₂ = p₃x₃

w₀ = 3p₁x₁

w_0/3=p₁x₁ = p₂x₂ = p₃x₃

Thus,one third of the wealth isspent on each commodity.This gives the maximization of U.

1.2. Maximize of a Weighted Utility. We make the same assumptions on the commodities as the last example, but assume the utility is given by U = x^a1 x^a2 x^a3 with a₁> 0, a₂> 0, and a₃> 0. This utility function gives a weight to the preference of the commodities as the solution to the maximization problem shows. The maximum of U on the commodity bundles given by the wealth constraint satisfy the equations

a1xa1−1xa2 xa3 = λp1

a2xa1 xa2−1xa3 = λp2

a3xa1 xa2 xa3−1 = λp3

p₁x₁ + p₂x₂ + p₃x₃ = w₀. If we multiply the first equation by x1/a1, the second equation by x2/a2, and the third equation by x3/a3, then they are all equal:

xa1 xa2 xa3 = λ p₁x_1/a1 = λp₂x_2/ a2= λp₃x3/a3

One solution is λ = 0, but this forces one of the variables to equal zero and so the utility is zero.

Thus, the wealth is distributed among the commodities in a way that uses the exponents as weights. Again, these choices give a maximization of U .

1.3. Maximize of a General Utility Function. Now we assume there are n commodities with amounts x_i for 1   i   n, and a utility function U (x₁, . . . , x_n) that depends on the amount of commodities but we do not give a specific formula. We assume the wealth w₀ = p₁x₁ +   + p_nx_n is fixed. The equations given by the Lagrange multiplier method are

∂U/∂x_i   = λp

for 1 ≤ i ≤ n

w₀ = p₁x₁ + · · · + p_nx_n.

Thus, the ratio of the marginal utility to price is the same for each commodity.

Interpretation of a Lagrange Multiplier

Let x = (x₁, . . . , x_n) be the variables. Consider the problem of finding the maximum of f (x) subject to the constraint g(x) = w. We discuss the problem in the case when f is the profit function of the inputs and w denotes the value of these inputs. For each choice of the constant w, let x^∗(w) of x that maximizes f , so f (x^∗(w)) is the maximal profit for fixed value of the inputs w. The derivative

d f (x^∗(w)), d

represents the rate of change in the optimal output from the change of the constant w.

Corresponding to x^∗(w) there is a value λ = λ^∗(w) such that they are a solution to the Lagrange multi- plier problem, i.e.,

∇f(x∗(w)) = λ∗(w)∇g(x∗(w)) w = g(x^∗(w)).

λ∗(w) =   d  f (x∗(w)).

Therefore, the Lagrange multiplier also equals this rate of the change in the optimal output resulting from the change of the constant w.

If f is the profit function of the inputs, and w denotes the value of these inputs, then the derivative is the rate of change of the profit from the change in the value of the inputs, i.e., the Lagrange multiplier is the “marginal profit of money”. For the example of the next subsection where the function f is the production function, the Lagrange multiplier is the “marginal product of money”. In Section 19.1 of the reference [1], the function f is a production function, there are several constraints and so several Lagrange multipliers, and the Lagrange multipliers are interpreted as the imputed value or shadow prices of inputs for production.

2.1. Change in budget constraint. In this subsection, we illustrate the validity of (1) by considering the maximization of the production function f (x, y) = x^2/3y^1/3, which depends on two inputs x and y, subject to the budget constraint

w = g(x, y) = p₁x + p₂y

where w is the fixed wealth, and the prices p₁ and p₂ are fixed. The equations for the Lagrange multiplier problem are

2. Change in inputs. In this subsection, we give a general derivation of the claim for two variables. The general case in n variables is the same, just replacing the sum of two terms by the sum of n terms. The details of the calculation are not important, but notice that is just uses the chain rule and the equations from Lagrange multipliers.

By the Chain Rule,

d  f (x^∗(w)) = ∂f  (x^∗(w)) dx∗1 (w) + ∂f  (x^∗(w)) dx∗2 (w).

dw                    ∂x₁                  dw            ∂x₂                 dw

3.Rate of Change of Minimal Cost of Production

Let Q = F (L, K) be the production function of a single output in terms of two inputs, labor L and capital K. Let w be the price of labor (wages) and r the price of capital (interest rate). Thus the cost function is C = wL + rK. Assume the output Q = Q₀ is fixed and the cost is minimized. Let L = L˜(w, r) be the amount of labor and K = K˜ (w, r) be the amount of capital which realizes this minimum. Define

C˜(w, r) = w L˜(w, r) + r K˜ (w, r)

be the minimal cost at these values. Shephard’s Lemma says that

∂C˜ (w , r ) = L˜(w , r )       and ∂C˜ (w , r ) = K˜ (w , r ).

This says that the rate of change of cost with respect to change in wages is equal to the size of the labor force, and does not depend on the change of the size of the labor force or amount of capital.