6. A firm, which is operating in the conditions of perfect competition, produces
ID: 1118310 • Letter: 6
Question
6. A firm, which is operating in the conditions of perfect competition, produces good Y. The production function of this firm is y(x) = 16 x1/2 , where y denotes the quantity of output Y and x denotes the quantity of input X used in the production process. Suppose that the the price per unit of input X is 4 euros and the price per unit of output Y is 5 euros. a) Find the profit-maximizing level of input X and the profit-maximizing level of output Y for this firm. (10points) b) Find the profit that the firm can obtain in case of its profit-maximizing output level. (5 points)
Explanation / Answer
y = 16x1/2
(a) Profit is maximized when (Output price x MPx1) = Input price
MPx1 = dy / dx1 = 16 x (1/2) / x1/2 = 8 / x1/2
Therefore,
5 x (8 / x1/2) = 4
40 / x1/2 = 4
x1/2 = 10
x = 100
y = 16 x 10 = 160
(b)
Profit = (Output price x y) - (Input price x Quantity of input used)
= (5 x 160) - (4 x 100) = 800 - 400
= 400
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