Question

4. A firm uses two inputs to produce a single product. If its production function is Q=x1/4y1/4 and if it sells its output for a dollar a unit and buys each input for $4 dollars a unit, find its profit-maximizing input bundle. (Check the second order conditions.)

Answer 1

The profit-maximizing input bundle for a firm using the production function Q=x^(1/4)y^(1/4), we need to calculate the ratio of marginal product to the price ratio. In this case, the quantity of both inputs x and y should be equal in the profit-maximizing input bundle.

Step-by-step explanation:

To find the profit-maximizing input bundle, we can use the marginal product of each input to calculate the ratio of their prices. The marginal product of an input is the additional output produced by adding one more unit of that input.

In this case, the production function is Q=x1/4y1/4. The marginal product of x is 1/4x-3/4y1/4, and the marginal product of y is 1/4x1/4y-3/4.

Setting the ratio of the marginal product to the price ratio, we get (1/4x-3/4y1/4) / (1/4x1/4y-3/4) = $4/$4, which simplifies to y = x.

Therefore, the profit-maximizing input bundle is when the quantities of inputs x and y are equal.

Answer 2

To find the profit-maximizing input bundle for a firm with a production function of Q=x^1/4y^1/4, we can use the marginal product of each input and the ratio of the input prices. The profit-maximizing input bundle occurs when the firm uses an equal quantity of both inputs.

Step-by-step explanation:

To find the profit-maximizing input bundle, we need to determine the quantity of each input that will maximize the firm's profits. In this case, the firm's production function is Q=x1/4y1/4. To find the profit-maximizing input bundle, we can use the marginal product of each input and the ratio of the input prices.

The marginal product of x is 1/4 * x-3/4 * y1/4. The marginal product of y is 1/4 * x1/4 * y-3/4.

We can set the ratio of the marginal products equal to the ratio of the input prices: (1/4 * x-3/4 * y1/4) / (1/4 * x1/4 * y-3/4) = $4 / $4. Simplifying, we get: x / y = 1.

Therefore, the profit-maximizing input bundle is when the firm uses an equal quantity of both inputs.