Suppose that your company is offered a contract to produce 1,000 units of a good X per day. Your company uses labor and capital to produce good X. The quantity of labor used,L, is expressed by hours of work, and the wage rate is $10 per hour. The quantity of capital utilized,K, is expressed in machine-hours, and the rental rate per machine hour is $40.
Thus total cost of production is TC = 10L + 40K. The production function is given by Q = 20L 0.5 K 0.5. In the contract, your company is required to produce 1,000 of good X.
a) Using the Lagrange function, determine the amount of labor and capital to use to minimize the cost of producing 1,000 units of good X.
b) Interpret lambda and what it means to your company.