ECON 350 - CLASSICAL ECONOMICS - Production Functions Question Bank with Complete Detailed Solutions- Set 5

ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank with Complete Detailed
Solutions- Set 5
Liberty University




Question 1
Question
Consider a production function Q = 3L0.5K0.3, where Q represents the total output,
L is the amount of labor input, and K is the amount of capital input. If the wage
rate is 12 and the rental rate of capital is 20, find the minimum cost of producing
Q = 1000 units of output.


Solution
Step 1: Determine the cost minimization condition. Given the production function
Q = 3L0.5K0.3 and the cost of producing Q = 1000 units of output, we need to
minimize the cost function C = wL + rK subject to the production constraint Q =
1000.
Step 2: Express L and K in terms of Q. From the production function, we have:
Q = 3L0.5K0.3 Substitute Q = 1000 into the production function: 1000 = 3L0.5K0.3
Step 3: Solve for L in terms of K. Solving for L in terms of K, we get:

Step 4: Cost function in terms of K. Substitute the expression for L into the cost
function:
Step 5: Minimize the cost function. To find the minimum cost, take the
derivative of the cost function with respect to K and set it equal to zero: dKdC = 0
Step 6: Solve for the optimal value of K. Solving for the optimal value of K, we
can find the corresponding value of L using the production function.
Step 7: Calculate the minimum cost. Substitute the optimal values of L and K
into the cost function to find the minimum cost of producing Q = 1000 units of
output.

,Question 2
Question
Consider a production function given by Q = 2L0.5K0.5 where Q is the quantity
produced, L is the quantity of labor, and K is the quantity of capital. Determine the
marginal product of labor (MPL) and the marginal product of capital (MPK).

Solution
Step 1: To find the marginal product of labor (MPL), we need to take the partial
derivative of the production function with respect to labor L.



Step 2: Taking the partial derivative with respect to L, we get:



Step 3: Simplifying the expression, we find the marginal product of labor:



Step 4: Similarly, to find the marginal product of capital (MPK), we need to take
the partial derivative of the production function with respect to capital K.



Step 5: Taking the partial derivative with respect to K, we get:



Step 6: Simplifying the expression, we find the marginal product of capital:



Therefore, the marginal product of labor is and the marginal
product of capital is .


Question 3
Question
Consider a production function given by Q = 10L3/4K1/4, where Q is the quantity of
output, L is the quantity of labor input, and K is the quantity of capital input.
Suppose the price of labor is w = 4 and the price of capital is r = 2. Calculate the
minimum cost of producing 100 units of output.




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,Solution
Step 1: The cost of production (C) can be expressed as C = wL + rK, where w is the
price of labor and r is the price of capital.
Step 2: To minimize cost, we need to minimize the cost function C subject to
the constraint that the production function Q = 10L3/4K1/4 produces 100 units of
output. This gives us the optimization problem:

Minimize C = 4L + 2K

subject to the constraint
10L3/4K1/4 = 100
Step 3: We can rewrite the constraint as 10 = 100L−3/4K−1/4.
Step 4: Using the Lagrange multiplier method, the Lagrangian function is:

L(L,K,λ) = 4L + 2K − λ(100L−3/4K−1/4 − 10)

Step 5: Taking partial derivatives and setting them equal to zero, we get the
following three equations:




Step 6: Solve the system of equations to find the values of L and K. Once you
have found L and K, substitute these values back into the cost function
C = 4L + 2K to calculate the minimum cost of producing 100 units of output.


Question 4
Question
Consider a production function Q = 2L0.5K0.5, where Q is the quantity of output
produced, L is the quantity of labor input, and K is the quantity of capital input.
Calculate the marginal product of labor and the marginal product of capital given
that L = 16 and K = 25.


Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function Q with respect to L:



Step 2: Substitute L = 16 and K = 25 into the formula to find MPL:




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, MPL = 250.5 = 5
Therefore, the marginal product of labor is 5 units of output per additional
unit of labor.
Step 3: To find the marginal product of capital (MPK), we differentiate the
production function Q with respect to K:



Step 4: Substitute L = 16 and K = 25 into the formula to find MPK:

MPK = 160.5 = 4

Therefore, the marginal product of capital is 4 units of output per additional
unit of capital.


Question 5
Question
Let Q denote the total output of a firm and K and L represent capital and labor
inputs, respectively. The production function of the firm is given by Q = K0.4L0.6. If
the firm’s total capital input is fixed at 16 units, what is the equation for the firm’s
total variable cost function in terms of L? Simplify the derived function.

Solution
Step 1: Recall that total variable cost (TVC) is the cost of all inputs except fixed
inputs. In this case, the only non-fixed input is labor.
Step 2: The firm’s total variable cost (TVC) is given by the cost of labor, which
is the variable input. Since the cost of labor is the product of the wage rate (w) and
the amount of labor used (L), the TVC function can be expressed as TV C = wL.
Step 3: To determine the equation for the firm’s TVC in terms of L, we need to
find the expression for w first.
Step 4: The wage rate (w) can be found by taking the partial derivative of the
production function with respect to labor L, and dividing it by the marginal
product of labor ( .
Step 5: Calculate the partial derivative of Q with respect to L:




Step 6: Calculate the marginal product of labor (MPL) by taking the derivative
of the production function with respect to L:




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