Lagrange Multipliers And Economics Worksheet (Page 2 of 2) in pdf

Lagrange Multipliers and Economics

March 4, 1997

a) Find the optimality condition using Lagrange multipliers. (Note: r and w are assumed

constant.)

b) What is the interpretation of λ? Answer this two ways:

i) First, assume that the units of P (K, L) is something concrete. If you are an auto

manufacturer, the units would be cars. In order to interpret λ, start by determining

the units of λ. (cars per something? something per cars?).

ii) Now assume that production is measured in dollars (this is usually what economists

do, so they can combine diﬀerent sorts of production). Now what are the units of λ?

The interpretation of λ is now a certain sort of (pure) multiplier. Explain.

5. Sometimes companies have contracts for a ﬁxed amount of product. Then instead of maximizing

production subject to ﬁxed costs, you want to minimize cost for the given production, say, P

Assume you are producing cars. Use the expressions P (K, L) and rK + wL as before, with r, w

again constants.

a) Find the optimality condition.

b) Interpret λ, when production is in cars. A λ with this sort of interpretation is called a

“shadow price”. If another supplier oﬀered to sell you some similar cars (so you wouldn’t

have to produce them yourself), at a price lower than λ, what would you do?

6. A motor company makes one type of car and one type of truck. Its revenue is R(c, t), where c is

the number of cars it produces (per year) and t is the number of trucks. Suppose production is

constrained by the amount of steel available, where each car needs s

units of steel, each truck

needs s

units of steel, and S units of steel are available.

a) Write the constraint as an equation.

b) Use Lagrange multipliers to determine an optimality equality. What is the interpretation

of λ at the optimum?

7. Another motor company makes cars, trucks and vans. Its revenue is R(c, t, v), where c, t, v

are the number of cars, trucks, vans (repsectively) it produces per year. Suppose production is

constrained by the amount of steel available, and by the amount of aluminum available. Assume

each car, truck, van needs s

, s

units of steel, respectively, and a

, a

units of aluminum,

respectively. Suppose S units of steel and A of aluminum are available.

a) Write the constraints as equations.

b) Use Lagrange multipliers to determine optimality conditions. What are the interpretations

of the two λ’s? Hint: they are both shadow prices.

Lagrange Multipliers And Economics Worksheet Page 2