Question

In: Economics

2) Mary opens up a shop making flip-flops. Let: q(L,K) = sqrt(L)sqrt(2K) w = price of...

2) Mary opens up a shop making flip-flops. Let:

q(L,K) = sqrt(L)sqrt(2K)

w = price of labor per unit = $5

r = price of capital per unit = $15

p = price of flip-flops per unit= $5

a) Suppose Mary contracts on 50 units of flip-flop making machinery (kapital)… what is her optimal Labor demand in the short run i.e. what quantity of labor (L) should she hire if capital is fixed at K = 50?

b) Suppose wage increases to $10 per unit… how does her optimal short run labor demand change (i.e. what is the new optimal short run L)? Intuitively… what change resulted and why?

c) Mary is now planning for the long run… she must make 1000 flip flops – use the substitution method to write her problem as a single variable optimization (hint she wants to spend as little as possible subject to her production being at least q=1000). Let w=5 and r=15 still. DO NOT SOLVE…

Solutions

Expert Solution

2. The production function is given as . The cost of production would be or . The total revenue would be or .

(a) In the short run, K is fixed at 50. The production function would be or . The cost would be or . The profit would be or or . The profit would be maximum where or or or or or . The optimal labor demand would hence be 25 units.

(b) If the wage increases, then the cost function changes to be , and the profit would be in the short run. The profit would be maximum where or or or or . If the wage increases, the optimal labor demand would be 6.25 units. As the wage increases, the labor demand decreases, because increase in wages increases the cost, and decreases the profit. The relation between wage and marginal labor product is that . As price remains constant, an increase in wage require marginal product of labor to increase, which would require the labor input to decrease.

(c) In this case, both K and L are variable, and since output is given, the cost should be minimized with the output constraint. The problem would be as below.

Minimize

Subject to

The Lagrangian function in this case would be , and solving its FOCs would give us the required labor and capital demand.

Rahul Sunny answered 2 years ago

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