Consider the following short-run production function (where L = variable input, Q =output): Q = 10L - 0.5L2 Suppose that output can be sold for $10 per unit.
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First, let's define the terms used in the short-run production function:
Q: Output
L: Variable input (in this case, labor)
The equation for the short-run production function is:
Q = 10L - 0.5L^2
Next, let's calculate the maximum output (Q) that can be produced. To find this, we need to find the level of L that maximizes Q. We can do this by taking the derivative of the production function with respect to L and setting it equal to zero:
dQ/dL = 10 - L = 0
Solving for L:
L = 10
Now, substitute this value of L back into the production function to find the maximum output (Q):
Q = 10(10) - 0.5(10)^2 = 100 - 0.5(100) = 100 - 50 = 50
Therefore, the maximum output (Q) that can be produced is 50 units.
Finally, since output can be sold for $10 per unit, we can calculate the total revenue (TR) from selling all units of output by multiplying the price per unit ($10) by the total output:
TR = $10 × 50 = $500
Therefore, the total revenue from selling all units of output is $500.
To find the profit-maximizing level of labor (L) for this production function, we need to determine where the marginal product of labor (MPL) is equal to the marginal cost of labor (MCL).
To begin, let's compute the MPL. The MPL is the additional output produced when one unit of labor is added:
MPL = change in Q / change in L
Taking the derivative of the production function with respect to L gives us:
dQ/dL = 10 - L
Now, let's compute the MCL. The MCL represents the additional cost incurred when one unit of labor is added. In this case, the MCL is the wage rate, which we'll assume is given as $10.
To find the profit-maximizing level of L, we set MPL equal to MCL and solve for L:
10 - L = 10
Simplifying the equation, we have:
L = 0
Therefore, for this production function, the profit-maximizing level of labor (L) is 0. This means that in the short run, it is optimal not to employ any labor because the marginal product of labor (MPL) is equal to zero at the point of zero labor input.
Please note that this result assumes that labor is the only variable input and that the wage rate is fixed at $10 per unit of labor. Additionally, we assume that there are no other costs associated with production, such as capital or materials.