6.1.2 Exercises on interpretation of Lagrange multiplier

  1. 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.
    1. How much of input 2 does it use?
    2. What is the most that it is willing to bribe an inspector to allow it to use another unit of input 1?

  2. 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.
    1. How much of input y does it use?
    2. 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.)

[Solutions]