Question

A competitive firm's production function is f(x1, x2) = 6x1/21 + 8x1/22. The price of factor 1 is $1 and the price of factor 2 is $4. The price of output is $8. What is the profit-maximizing quantity of output?

Answer 1

Without complete information on costs, we cannot definitively calculate the exact profit-maximizing quantity of output for the firm. The firm will produce up to the point where marginal cost equals marginal revenue, which is the market price ($8) of the output.

Step-by-step explanation:

To find the profit-maximizing quantity of output for a competitive firm, we need to look at where marginal revenue (MR) equals marginal cost (MC). In a perfectly competitive market, MR is the same as the price of the output, which we are told is $8. The production function given is f(x1, x2) = 6x1/21 + 8x1/22. However, the information provided lacks cost functions for inputs or total cost, which are necessary to calculate marginal costs and, subsequently, the profit-maximizing output. Without these additional details, it is not possible to calculate the exact quantity of output that maximizes profit. However, the general principle is that the firm will continue to increase output until the cost of producing one more unit (marginal cost) is equal to the revenue from selling that unit (marginal revenue).

In the scenario where the market price of the output is higher than the average cost at the profit-maximizing output level, the firm will earn a profit. Conversely, if the market price is lower than the average cost at this output level, the firm will experience a loss. The given firm will achieve its profit-maximizing output at the quantity where this condition is met.

Answer 2

The profit-maximizing quantity of output is 41.6.

Set up the profit equation:

Profit = Total Revenue - Total Cost

`\begin{aligned}& \text { Profit }=\left(8 \cdot f\left(x_1, x_2\right)\right)-\left(1 \cdot x_1+4 \cdot x_2\right) \\& \text { Profit }=8 \cdot\left(6 x_1^(1 / 2)+8 x_2^(1 / 2)\right)-x_1-4 x_2\end{aligned}`

Find the marginal product of each factor:

`\begin{aligned}& M P_1=(\partial f\left(x_1, x_2\right))/(\partial x_1)=(3)/(x_1^(1 / 2)) \\& M P_2=(\partial f\left(x_1, x_2\right))/(\partial x_2)=(4)/(x_2^(1 / 2))\end{aligned}`

Set the marginal product of each factor equal to its factor price ratio:

MP1 / w1 = MP2 / w2

`(3)/(x_1^(1 / 2))=(4)/(x_2^(1 / 2))`

Simplifying, we get: x2 = 4x1

Substitute x2 in the production function:

`\begin{aligned}& f\left(x_1, x_2\right)=6 x_1^(1 / 2)+8\left(4 x_1\right)^(1 / 2) \\& f\left(x_1, x_2\right)=6 x_1^(1 / 2)+16 x_1^(1 / 2) \\& f\left(x_1, x_2\right)=22 x_1^(1 / 2)\end{aligned}`

Set the marginal product of the composite factor (x1, 4x1) equal to the output price:

`\mathrm{MP}=(\partial f\left(x_1, 4 x_1\right))/(\partial x_1)=(11)/(√(x_1))`

`(11)/(x_1^(1 / 2))=8`

Solving for x1, we get: x1 = 1.714

Find x2 using x2 = 4x1:

`x_2 = 4 * 1.714 = 6.856`

Find the profit-maximizing quantity of output:

`f\left((121)/(64), (242)/(9)\right)=22\left((121)/(64)\right)^(1 / 2) \approx 41.6`