A firm uses two inputs, labor services whose quantity is denoted by L and capital services whose quantity is denoted by K. The production function is given by Q = 100LK, and the price of labor w is $2 per unit and the price of capital r is $1 per unit. For this production function, the marginal products of labor and capital, respectively, are MP_L = 100K and MP_K = 100L.

a) Using the method of Lagrange, find the firm’s cost minimizing input combination (L, K) when it seeks to produce 5,000 units per year. What is the minimized  level of total cost TC when output equals 5,000?

b) Find the numerical value of the Lagrange multiplier, λ, which measures the firm’s marginal cost (the rate of change ΔTC/ΔQ) when output is 5,000.

c) Find the firm's cost-minimizing input (L, K), minimized total cost, and the value of λ if output is 5,001.

d) Verify that the increase in the firm's total cost when output increases from 5,000 to 5,001 is close to the values of λ you found in parts (b) and (c).