What is the profit-maximizing quantity of output for a competitive firm with the given production function and factor prices?
To determine the profit-maximizing quantity of output, we need to find the level of output that maximizes the firm's profit. In a perfectly competitive market, profit is maximized when the firm produces at the level where marginal revenue equals marginal cost. In this case, the profit-maximizing quantity of output is 92.4 units.
Understanding Profit-Maximizing Output
Explanation: To determine the profit-maximizing quantity of output, we need to find the level of output that maximizes the firm's profit. In a perfectly competitive market, profit is maximized when the firm produces at the level where marginal revenue equals marginal cost. In this case, the production function is f(x1, x2) = 8x1/21 + 8x1/22. To find the profit-maximizing quantity, we need to differentiate the production function with respect to the quantity of output.
Marginal cost is the derivative of the production function with respect to output. In this case, the derivative is 8/(21x1/21) + 8/(22x1/22). Setting marginal cost equal to marginal revenue (which is the price of output), we can solve for the quantity of output that maximizes profit.
Let's set marginal cost equal to the price of output and solve for x1:
8/(21x1/21) + 8/(22x1/22) = 6
Simplifying the equation, we get:
8x1/21 + 8x1/22 = 6
Now, we can solve for x1:
8x1/21 + 8x1/22 = 6
Common denominator is: 21 * 22 = 462
Adding the terms on the left side of the equation, we get:
30x1/462 = 6
Multiplying both sides by 462, we get:
30x1 = 2772
Dividing both sides by 30, we get:
x1 = 92.4
Therefore, the profit-maximizing quantity of output is 92.4 units.
How can a competitive firm determine the profit-maximizing output based on its production function and input factor costs?
To find the profit-maximizing output, a firm should compare its total revenue and costs at different output levels, aiming to find the level where revenue exceeds costs by the greatest amount. This typically occurs where marginal cost equals marginal revenue, assuming the output price is higher than the average cost of production. The firm's production function and input factor costs can help form the necessary equations for this calculation.
Understanding Profit-Maximizing Output
Explanation: To determine the profit-maximizing output for this competitive firm, a few steps are required. Firstly, you must analyze the production function, which is f(x1, x2) = 8x1/21 + 8x1/22. For each input, you multiply the marginal product (additional output generated by additional input factors) by the price of the output to get the value of the marginal product. Using the prices of factor 1 ($1) and factor 2 ($3), the firm should ascertain where it can achieve a balance between these two costs to maximize their output. This is a concept referred to as when the marginal cost equals the marginal revenue, as reiterated in the given information. In a perfectly competitive firm, the firm should continue producing until its total revenue does not exceed total costs by the highest amount possible. If the price of the output product ($6) is higher than the average cost of factors at the profit-maximizing output level, then the firm will obtain profits. The calculation will require simultaneous solving of these mathematical equations once properly formulated based on the provided production equation.