I need my answers checked for this question, thanks.
A firm's production function:
Q = 100 K^0.5 L^0.5
During the last production period, the firm operated efficiently and used input rates of 100 and 25 for labor and capital respectively.
(a) What is the marginal product of capital and the marginal product of labor based on the input rates specified?
-marginal product of capital=100
-marginal product of labor=25
(b) If the price of capital was $20 per unit, what was the wage rate?
-wage rate=5
(c) For the next production period, the price per unit of capital is expected to increase to $25 while the wage rate and the labor input will remain unchanged. If the firm maintains efficient production, what input rate of capital will be used?
-i need help on this question.
Thanks.
A good problem. Unfortunately, my calculas skills are not what they once were. Fortunately, this particular problem has a particular feature which makes the solution almost obvious (once you see it).
Assume, this firm operates in a perfectly competitive market. So, the price of its output is given. Step 1, determine the price is sells its output. A firm will expand production until the marginal cost of producing an additional unit equals the marginal revenue (price) of that unit. You know that marginal product of labor is 25 unit and the cost of one unit of labor is $5. So, the marginal cost of an additional unit is 5/25 = $0.20. (Similarly, the mp of K is 100 and one K costs 20. So the MC of and additional unit is 20/100= $0.20 again.)
So, the price of the output is $0.20 per unit.
Now calculate total revenue and net profit under the initial conditions. Q=100K^.5L^.5 = 100*(25)^.5 * (100).5 = 5000. Total revenue is 0.2*5000=1000. Total cost is 100*$5 + 25*$20 = 1000. Profit = 0.
Under price of capital increase, the problem becomes:
max(profit) = .2*(100K^.5 L^.5) - 25K - 5L. (here you have a two-variable maximization calculas problem. I am on shaky ground here).
However, one can see the solution ahead of this step. By operating efficiently, given the initial input prices, the firms best solution was to earn zero profits. It barely covered its variable costs. With the increase in the cost of capital, the firm cannot even hope to break even. So, the optimal solution is to shut down.
Of course, this assumes the firm is operates in a competitive environment. If the firm is a monopolist or has some monopoly power, then, I believe, you need more information about the marginal revenue obtained by the firm.
Lotsa luck
To find the input rate of capital that will be used in the next production period, we need to understand the concept of profit maximization. In a perfectly competitive market, firms maximize their profits by equating their marginal costs to the market price.
In our case, we have already determined the market price to be $0.20 per unit of output. To find the optimal input rate of capital, we need to calculate the marginal cost of capital (MCk) and equate it to the market price.
The marginal cost of capital (MCk) is the change in total cost when one additional unit of capital is used. From our initial information, we know that the cost of one unit of capital is $20.
Now, we need to calculate the change in total cost when one additional unit of capital is used. We can do this by taking the derivative of the production function with respect to K and multiplying it by the cost of one unit of capital. The derivative of the production function with respect to K is 0.5 * 100 * K^(-0.5) * L^0.5.
So, the marginal cost of capital (MCk) is 0.5 * 100 * K^(-0.5) * L^0.5 * $20.
Next, we equate MCk to the market price ($0.20 per unit of output) to find the optimal input rate of capital.
0.5 * 100 * K^(-0.5) * L^0.5 * $20 = $0.20
Now, substitute the values of L, which is 25, to simplify the equation:
0.5 * 100 * K^(-0.5) * 25^0.5 * $20 = $0.20
Simplify further:
0.5 * 100 * K^(-0.5) * 5 * $20 = $0.20
Now, solve for K:
100 * K^(-0.5) = $0.20 / (0.5 * 5 * $20)
100 * K^(-0.5) = $0.20 / $10
K^(-0.5) = 0.02
Take the reciprocal of both sides:
K^(0.5) = 1 / 0.02
K^(0.5) = 50
Square both sides:
K = 50^2
K = 2500
Therefore, the input rate of capital that will be used in the next production period is 2500 units.