Question

In: Economics

2. A firm uses labor and machines to produce output according to the production function f...

2. A firm uses labor and machines to produce output according to the production function f (L,K) = 2 L K ,where L is the number of units of labor used and K is the number of machines. The cost of labor is $40 per unit and the cost of using a machine is $10. The Long Run Decision of the Firm

(a) Write down the equation for an isocost line for this firm. What is the slope of these isocost lines?

(b) Find , and . Using TRS and the slope of isocost lines, write down the tangency condition (the optimal choice condition) for this firm.

(c) When the desired level of output of this firm is , what is the equation for the isoquant of this firm? Use the tangency condition (b) and the equation for the isoquant, find the optimal choice of input 1 and the optimal choice input 2. That is, solve the simultaneous equation system: the tangency condition and the equation for the isoquant.

(d) Use the optimal choices of input 1 and input 2, find the long run total cost, the long run average cost, and the long run marginal cost.

Solutions

Expert Solution

2. (a) The isocost line would be as , which is basically the cost of production. For a given cost, the slope would be as or or or . The slope in this case would be .

(b) The production function is . The TRS for a given Q would be as or or or .

Equating the slope of isocost and TRS, we have the required optimal combination of inputs as .

(c) For the required output be Q, the equation of isoquant would be . For the tangency condition , we have or .

Putting this in the isoquant, we have or or or , and since , we have or or . These are the optimal input choices (conditional demands of the inputs).

(d) The long run cost function would be as or or or or . The average cost would be as or or . The marginal cost would be as or or or .

Rahul Sunny answered 7 months ago
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