6.1.2 Exercises on optimization with an equality constraint: interpretation of Lagrange multipliers
- A firm that uses two inputs to produce output has the production function 3x1/3y1/3, where x is the amount of input 1 and y is the amount of input 2. The price of output is 1 and the prices of the inputs are wx and wy. The firm is constrained by the government to use exactly 1000
units of input 1.
- How much of input 2 does it use?
- What is the most that it is willing to bribe an inspector to allow it to use another unit of input 1?
- Solving the first-order conditions we obtain y* = (10/wy)3/2.
- The maximal bribe is given by the value of the Lagrange multiplier, λ* = (1/100)·(10/wy)1/2 − wx. (If this is negative, then the firm isn't willing to pay any bribe to increase the amount of input 1 that it uses—it is instead willing to pay a bribe of λ* to decrease the amount of that input. You may verify that if wx and wy are such that in the absence of a constraint the firm chooses to use 1000 units of input 1 then λ* = 0 in the constrained problem.)
- A firm that uses two inputs to produce output has the production function 4x1/4y3/4. The price of output is 1 and the price of each input is 1. The firm is constrained to use exactly 1000 units of input x.
- How much of input y does it use?
- What is approximately the maximum amount the firm is willing to pay to be allowed to use ε more units of input x, for ε small? (Do not try to calculate your answer as a decimal number.)
- The firm's profit is 4x1/4y3/4 − x − y. Thus its problem is
maxx,y4x1/4y3/4 − x − y subject to x = 1000.The first-order conditions for this problem areThus (x, y) = (1000, 81000). Also we have λ = (81)3/4 − 1 = 26.
x−3/4y3/4 − 1 − λ = 0 3x1/4y−1/4 − 1 = 0 - If the firm is allowed to use ε more units of input x then its profit increases by approximately λε, or 26ε.