Consider the production function Q = 20K 1/2 L 1/2. The firm operates in the short run with 100 units of capital.
a. The firm’s short-run production function is Q = __________.
c. The average product of labor function is AP = __________.
d. The marginal product of labor function is MP = _________. Show that the marginal diminishes for all levels of labor usage.
e. Assuming that the firm in question 2 is in the long run where its fixed endowment is $1000, wage rate is $10, and interest payment for capital is $5, what is the combination of capital-labor ratio that will maximize output? Show the graphical combination. Assume that the wage rate falls to $5, what is the new combination on the same graph?
f. What kind of returns to scale is depicted for each of the Cobb-Douglas production functions below? Explain. Q = K 1/4 L 1/4; Q = K 1/2 L 2/3
What is your question??
a. The firm’s short-run production function is Q = __________.
c. The average product of labor function is AP = __________.
d. The marginal product of labor function is MP = _________. Show that the marginal diminishes for all levels of labor usage.
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a. The firm's short-run production function can be obtained by plugging in the given value of capital (100 units) into the production function equation.
So, the firm's short-run production function is:
Q = 20(100)^(1/2)L^(1/2)
b. The average product of labor (AP) is determined by dividing total product (Q) by the level of labor usage (L). In this case, the labor usage is the only variable input, so AP is the same as the production function.
Hence, the average product of labor function is:
AP = 20(100)^(1/2)L^(1/2)
c. The marginal product of labor (MP) is the derivative of the production function with respect to labor (L).
Taking the derivative of the production function Q = 20(100)^(1/2)L^(1/2) gives us:
MP = (1/2) * 20(100)^(1/2)L^(-1/2)
To show that the marginal product diminishes for all levels of labor usage, we can take the second derivative and confirm if it is negative:
MP' = (-1/4) * 20(100)^(1/2)L^(-3/2) < 0
Since the second derivative is negative, we can conclude that the marginal product diminishes for all levels of labor usage.
e. In the long run, the firm can adjust both capital (K) and labor (L) inputs, unlike the short run where only capital is fixed. To maximize output, the firm needs to choose the optimal combination of capital-labor ratio.
The optimal combination of capital-labor ratio can be determined by equating the Marginal Product of Capital (MPK) to the Marginal Product of Labor (MPL) and solving for K/L.
In this case, the Cobb-Douglas production function is Q = K^(1/2) * L^(1/2). Assuming that the firm's fixed endowment is $1000, wage rate is $10, and interest payment for capital is $5, we can set up the following equation:
MPK/PK = MPL/PL
Taking the partial derivatives of the production function with respect to K and L will give us MPK and MPL:
MPK = (1/2) * (K^(-1/2)) * L^(1/2)
MPL = (1/2) * K^(1/2) * L^(-1/2)
Substituting the given values for PK ($5) and PL ($10), we can solve for the optimal K/L ratio:
(1/2) * (K^(-1/2)) * L^(1/2) / $5 = (1/2) * K^(1/2) * L^(-1/2) / $10
Simplifying the equation, we get:
L/K = 2
Hence, the combination of capital-labor ratio that will maximize output is L/K = 2.
If the wage rate falls to $5, the new combination on the same graph can be obtained by substituting the new PL ($5) into the equation:
(1/2) * (K^(-1/2)) * L^(1/2) / $5 = (1/2) * K^(1/2) * L^(-1/2) / $5
Simplifying the equation, we still get:
L/K = 2
Therefore, even with a different wage rate, the optimal combination of capital-labor ratio will remain L/K = 2.
f. The type of returns to scale depicted by a Cobb-Douglas production function can be determined based on the sum of the exponents of K and L.
In the production function Q = K^(1/4) * L^(1/4), the sum of the exponents is 1/4 + 1/4 = 1/2. This indicates that the Cobb-Douglas production function exhibits Constant Returns to Scale (CRS). This means that if inputs are increased by a certain proportion, the output will increase by the same proportion.
In the production function Q = K^(1/2) * L^(2/3), the sum of the exponents is 1/2 + 2/3 = 5/6. This indicates that the Cobb-Douglas production function exhibits Decreasing Returns to Scale (DRS). This means that if inputs are increased by a certain proportion, the output will increase by a smaller proportion.