ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank - Set 2
Liberty University
Question 1
Question
Suppose a production function is given by Q= 2LK, where Qis the total
output, Lis the amount of labor, and Kis the amount of capital. If the price
of labor is wand the price of capital is r, determine the equations for the total
cost of labor (T CL) and total cost of capital (T CK) in terms of w,r,L, and K.
Solution
Step 1: To find the total cost of labor (T CL), we need to multiply the amount
of labor Lby the price of labor w.
T CL=wL
Step 2: To find the total cost of capital (T CK), we need to multiply the
amount of capital Kby the price of capital r.
T CK=rK
Question 2
Question
Consider a production function given by Q=L0.5K0.5, where Qrepresents the
level of output, Lis the quantity of labor, and Kis the quantity of capital.
If the wage rate is w= 10 and the rental rate of capital is r= 20, find the
minimum cost combination of labor and capital that can produce 100 units of
output.
Solution
Step 1: The cost of production is given by C=wL +rK.
Step 2: We want to minimize the cost Csubject to the production constraint
Q= 100. Substituting the production function into the cost function, we have:
C= 10L+ 20K
Step 3: Using the production function Q=L0.5K0.5and the given produc-
tion level Q= 100, we can rewrite the production function as 100 = L0.5K0.5.
Step 4: From the production constraint, we have K= (100/L)2.
Step 5: Substituting K= (100/L)2into the cost function, we get:
C= 10L+ 20 100
L2
Step 6: To find the minimum cost combination of labor and capital, we
differentiate Cwith respect to Land set it equal to 0 to find the minimum:
dC
dL = 10 −40 100
L3= 0
Step 7: Solving for L, we get L= 10.
Step 8: Substituting L= 10 back into the production constraint Q= 100,
we find K= (100/10)2= 100.
Step 9: Therefore, the minimum cost combination of labor and capital that
can produce 100 units of output is L= 10 units and K= 100 units.
Question 3
Question
Let Q= 4LK represent the production function where Qrepresents the total
output, Lrepresents the amount of labor input, and Krepresents the amount
of capital input. If the price of labor is wand the price of capital is r, find the
cost minimizing combination of labor and capital inputs required to produce
100 units of output given that w= 3 and r= 2.
Solution
Step 1: The cost minimizing combination of inputs can be found by minimizing
the total cost function. The total cost function (C) can be defined as C=
wL +rK.
Step 2: Substituting Q= 100 into the production function, we have:
100 = 4LK
2
Step 3: Solving for Kin terms of Lfrom the production function:
K=100
4L= 25L−1
Step 4: Substitute the expression for Kinto the cost function:
C= 3L+ 2(25L−1)=3L+ 50L−1
Step 5: To minimize the total cost function, we need to find the critical
points by taking the derivative of Cwith respect to Land setting it equal to
zero: dC
dL = 3 −50L−2= 0
Step 6: Solving for L:
3 = 50L−2
1
L2=3
50
L=r50
3=5√6
3
Step 7: Substitute the value of Lback into the expression for K:
K= 25 5√6
3!−1
= 25 3
5√6=15
√6=15√6
6=5√6
2
Step 8: Therefore, the cost minimizing combination of labor and capital
inputs required to produce 100 units of output is L=5√6
3and K=5√6
2.
Question 4
Question
Consider a firm that produces output (Q) using two inputs, labor (L) and
capital (K), according to the production function Q= 2L0.5K0.5. If the wage
rate (w) is
$
10 per unit of labor and the rental rate (r) is
$
20 per unit of capital,
determine the optimal combination of labor and capital that minimizes the cost
of producing 100 units of output.
Solution
Step 1: Write the cost function. The cost of producing 100 units of output using
labor (L) and capital (K) is given by the cost function:
C=w·L+r·K
3
Step 2: Determine the firm’s objective function. The firm seeks to mini-
mize the cost of production, i.e., minimize the cost function Csubject to the
production function Q= 2L0.5K0.5and the output level Q= 100.
Step 3: Use the production function to substitute for Q. Since Q= 100
units, we substitute Qin the production function to get:
100 = 2L0.5K0.5
Step 4: Rewrite the above equation to solve for one of the variables. From
the production function, we can express Kin terms of Las:
K=100
2L0.52
=10000
4L= 2500L−1
Step 5: Substitute the cost function and the rewritten production function
to get the total cost function in terms of one variable. Substitute K= 2500L−1
into the cost function:
C= 10L+ 20 10000
4L= 10L+ 5000L−1
Step 6: Find the derivative of the cost function with respect to Land set it
equal to 0 to find the minimum cost.
dC
dL = 10 −5000L−2= 0
5000L−2= 10
L2= 500
L=√500 = 10√5
Step 7: Calculate the corresponding value of K. Plug L= 10√5 into the
production function to find K:
K= 2500(10√5)−1=2500
10√5=250
√5= 50√5
Therefore, the optimal combination of labor and capital that minimizes the
cost of producing 100 units of output is L= 10√5 and K= 50√5.
Question 5
Question
Let Q=K0.3L0.7represent a production function where Qis the quantity of
output, Kis the quantity of capital, and Lis the quantity of labor. Determine
the marginal product of labor (MPL) at the point (K, L) = (100,49).
4
Solution
Step 1: Calculate the partial derivative of the production function with respect
to labor L.∂Q
∂L = 0.7K0.3L−0.3
Step 2: Evaluate the MPL at the point (K, L) = (100,49) by substituting
the values of Kand Linto the partial derivative.
∂Q
∂L = 0.7(100)0.3(49)−0.3
∂Q
∂L = 0.7(4.6416)(0.7925)
∂Q
∂L = 2.6146
Therefore, the marginal product of labor (MPL) at the point (K, L) =
(100,49) is 2.6146.
Question 6
Question
Let Q= 2LK be a production function where Qis the quantity produced, Lis
the quantity of labor input, and Kis the quantity of capital input. Determine
the marginal product of labor and the marginal product of capital.
Solution
We are given the production function Q= 2LK, where Qis the quantity pro-
duced, Lis the quantity of labor input, and Kis the quantity of capital input.
Step 1: Marginal Product of Labor The marginal product of labor
(MPL) is given by the partial derivative of the production function with respect
to labor:
MPL=∂Q
∂L
Taking the partial derivative of Q= 2LK with respect to L, we get:
MPL= 2K
Step 2: Marginal Product of Capital Similarly, the marginal product
of capital (MPK) is given by the partial derivative of the production function
with respect to capital:
MPK=∂Q
∂K
Taking the partial derivative of Q= 2LK with respect to K, we get:
MPK= 2L
5
Therefore, the marginal product of labor is 2Kand the marginal product of
capital is 2L.
Question 7
Question
Consider a firm with the production function Q=L3
4K1
4, where Qis the level
of output, Lis the amount of labor employed, and Kis the amount of capital
employed. If the firm currently employs 64 units of labor and 16 units of capital,
what is the marginal product of labor at this level of input?
Solution
Step 1: Calculate the marginal product of labor.
MPL=∂Q
∂L
=3
4L
−1
4K1
4
=3
4
K1
4
L1
4
=3
4
161
4
641
4
=3
4
2
4
=3
8
Therefore, the marginal product of labor when 64 units of labor and 16 units
of capital are employed is 3
8.
Question 8
Question
Consider a production function given by Q= 6L0.5K0.5, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. If the wage
rate is w= 4 and the rental rate for capital is r= 9, find the values of Land
Kthat minimize the cost of producing 36 units of output.
Solution
Step 1: Write the cost function. Given that the cost of production is given by
C=wL +rK, we can substitute Q= 36 into the production function to find
the cost function.
6
C= 4L+ 9K
Step 2: Minimize the cost function subject to the production function. To
minimize C= 4L+ 9Ksubject to Q= 36 and Q= 6L0.5K0.5, we set up the
Lagrange function:
L= 4L+ 9K+λ(36 −6L0.5K0.5)
Step 3: Find the first-order conditions. Taking the partial derivatives of L
with respect to L,K, and λand setting them equal to zero:
∂L
∂L = 4 −3λL−0.5K0.5= 0
∂L
∂K = 9 −3λL0.5K−0.5= 0
∂L
∂λ = 36 −6L0.5K0.5= 0
Solving these equations simultaneously will give us the values of Land K
that minimize the cost of producing 36 units of output.
Question 9
Question
Consider a production function given by f(x, y) = 4x0.5y0.5. Find the total
output produced when the inputs xand yare both increased by 10
Solution
1. To find the total output produced when the inputs xand yare both increased
by 10Let the original values be x0and y0, and the new values be x1and y1.
Given that x1=x0+0.1x0and y1=y0+0.1y0, we can simplify these expressions
as x1= 1.1x0and y1= 1.1y0.
2. Now, we will find the total output produced when the inputs are increased.
The new total output, denoted by f(x1, y1), is calculated as:
f(x1, y1) = 4(1.1x0)0.5(1.1y0)0.5
3. Simplifying the above expression, we get:
f(x1, y1)=4·1.10.5x0.5
0·1.10.5y0.5
0
4. Further simplifying, we have:
f(x1, y1) = 4 ·1.0487x0.5
0y0.5
0
5. Therefore, the total output produced when the inputs xand yare both
increased by 10
7
Question 10
Question
Consider a production function given by Q= 10L0.5K0.5, where Qis the quan-
tity of output, Lis the quantity of labor, and Kis the quantity of capital. The
wage rate is w= 4 and the rental rate of capital is r= 9. Given these condi-
tions, determine the equations for the isocost and isoquant curves, and find the
optimal combination of labor and capital that minimizes the cost of production
while producing 100 units of output.
Solution
Step 1: The isocost curve equation is determined by the total cost (T C) equa-
tion, which is given by:
T C =wL +rK
Substitute w= 4 and r= 9:
T C = 4L+ 9K
Therefore, the equation for the isocost curve is:
4L+ 9K=C
where Cis the total cost.
Step 2: The isoquant curve equation can be found by setting the production
function equal to the desired output level of 100:
10L0.5K0.5= 100
Simplify to:
L0.5K0.5= 10
Square both sides to get:
LK = 100
Therefore, the equation for the isoquant curve is:
LK =Q
Step 3: To find the optimal combination of labor and capital that mini-
mizes the cost of production while producing 100 units of output, we set up the
following optimization problem:
(Minimize 4L+ 9K
Subject to LK = 100
We can solve this by substituting the isoquant equation into the cost function
and applying the Lagrange multiplier method. After solving the system of
equations, we find the optimal values of Land K.
8
Question 11
Question
Consider a firm with the production function given by: Q=K0.3L0.7, where
Qis the quantity of output, Kis the quantity of capital, and Lis the quantity
of labor. If the firm’s total cost function is given by T C = 1000 + 10K+ 4L,
find the total cost of producing 1000 units of output at the most cost-efficient
combinations of capital and labor.
Solution
Step 1: To find the most cost-efficient combinations of capital and labor, we
need to minimize the total cost function subject to the production constraint.
Step 2: We can rewrite the total cost function in terms of Q,K, and Lusing
the production function. Substituting Q= 1000 (given in the question):
T C = 1000 + 10K+ 4L
Step 3: Substituting the production function Q=K0.3L0.7into the total
cost function, we get:
T C = 1000 + 10K+ 4L
Step 4: Now, we substitute Q= 1000 into the production function to get
the most cost-efficient combinations of capital and labor:
1000 = K0.3L0.7
Step 5: Since the question asks for the total cost of producing 1000 units
of output at the most cost-efficient combinations of capital and labor, we have
narrowed down the variables.
Step 6: To find the cost-minimizing values of Kand L, we can use the La-
grange multiplier method or solve the constrained optimization problem directly
by substituting Q= 1000 into the production function and then into the total
cost function.
Step 7: After solving for the optimal values of Kand L, we can substitute
these values into the total cost function to find the total cost of producing 1000
units of output.
Question 12
Question
Consider a production function f(K, L) = KαLβ, where α, β > 0. Show that
this production function exhibits constant returns to scale.
9
Solution
To show that the production function f(K, L) = KαLβexhibits constant returns
to scale, we need to prove that doubling the inputs Kand Lwill result in a
doubling of the output f.
Step 1: Calculate the production function with inputs doubled Let
us consider the production function with inputs that are doubled:
f(2K, 2L) = (2K)α(2L)β
Step 2: Simplify the expression
f(2K, 2L) = 2αKα·2βLβ= 2α+βKαLβ
Step 3: Compare the scaled output to the original output Now, we
compare f(K, L) with f(2K, 2L):
f(2K, 2L)=2α+βKαLβ= 2α+βf(K, L)
Step 4: Conclusion Since 2α+βis a constant multiplier that does not
depend on Kor L, we can see that doubling the inputs results in a doubling
of the output. Therefore, the production function f(K, L) = KαLβexhibits
constant returns to scale.
Question 13
Question
Consider a production function given by Q=AKαLβ, where Qrepresents the
total output, Krepresents the amount of capital input, Lrepresents the amount
of labor input, Ais a positive constant, and αand βare positive constants.
Given that the wage rate is wand the rental rate for capital is r, derive the
expressions for the marginal product of labor (MPL) and the marginal product
of capital (MPK).
Solution
Step 1: To find the marginal product of labor (M PL), we first need to differen-
tiate the production function Qwith respect to L, holding Kconstant.
∂Q
∂L =AαKαLβ−1
Step 2: Simplify the expression to obtain MPL:
MPL=∂Q
∂L =AαKαLβ−1
10
Step 3: Next, to find the marginal product of capital (M PK), we differentiate
the production function Qwith respect to K, holding Lconstant.
∂Q
∂K =AβKα−1Lβ
Step 4: Simplify the expression to obtain MPK:
MPK=∂Q
∂K =AβKα−1Lβ
Therefore, the expressions for the marginal product of labor and capital are:
MPL=AαKαLβ−1
MPK=AβKα−1Lβ
Question 14
Question
Consider a production function f(x, y) = 3x2y−2xy2. Compute the marginal
product of ywhen x= 2 and y= 3.
Solution
To find the marginal product of y, we need to find the partial derivative of the
production function f(x, y) with respect to y.
Step 1: Find the partial derivative of f(x, y) with respect to y.
∂f
∂y = 3x2−4xy
Step 2: Evaluate ∂f
∂y at x= 2 and y= 3.
∂f
∂y (2,3)
= 3(2)2−4(2)(3) = 12 −24 = −12
Therefore, the marginal product of ywhen x= 2 and y= 3 is −12.
Question 15
Question
Consider a production function Q=L0.3K0.7, where Qis the quantity of output,
Lis the quantity of labor input, and Kis the quantity of capital input. If
the wage rate is 10perunitoflaborandtherentalrateofcapitalis20 per unit of
capital, what is the minimum cost of producing 100 units of output?
11
Solution
Step 1: The cost of production is given by the cost of labor and the cost of
capital. Since we are trying to minimize the cost, we need to find the quantities
of labor and capital that will minimize the cost while producing 100 units of
output.
Step 2: The cost function can be defined as T C =wL+rK, where T C is the
total cost, wis the wage rate, ris the rental rate of capital, Lis the quantity
of labor input, and Kis the quantity of capital input.
Step 3: We are given that the production function is Q=L0.3K0.7and
we need to produce 100 units of output. Substituting this into the production
function gives us:
100 = L0.3K0.7
Step 4: To minimize the cost, we need to differentiate the cost function with
respect to both labor and capital inputs and set the resulting equations equal
to zero. This is because the minimum point of a cost function occurs where the
marginal cost of the inputs is equal.
Step 5: Differentiating the cost function T C =wL +rK with respect to
labor gives us:
dT C
dL =w−∂Q/∂L
∂Q/∂K r= 0
Step 6: Similarly, differentiating the cost function with respect to capital
gives us:
dT C
dK =r−∂Q/∂K
∂Q/∂L w= 0
Step 7: Solving the system of equations from Step 5 and Step 6 will give us
the optimal quantities of labor and capital inputs.
Step 8: Once we have the optimal quantities of labor and capital, we can
calculate the total cost by substituting these values into the cost function T C =
wL +rK. This will give us the minimum cost of producing 100 units of output.
Question 16
Question
Consider a production function given by Q= 10L1/2K1/3, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. Suppose the
price of labor is w= 4 and the price of capital is r= 9. Find the cost minimizing
combination of labor and capital that will produce 100 units of output.
Solution
Step 1: Calculate the marginal products of labor and capital. The marginal
product of labor (MP L) is given by the partial derivative of the production
12
function with respect to labor:
MP L =∂Q
∂L =1
2·10L−1/2K1/3= 5rK
L
The marginal product of capital (MP K) is given by the partial derivative
of the production function with respect to capital:
MP K =∂Q
∂K =1
3·10L1/2K−2/3=10
33
√K
Step 2: Calculate the ratio of the marginal products to their respective
prices.
MP L
w=5qK
L
4=5
4√L
√K
MP K
r=10
27 3
√K
Step 3: Set up the cost minimizing condition. At the cost minimizing combi-
nation, the ratios of the marginal products to prices for labor and capital should
be equal: MP L
w=MP K
r
5
4√L
√K=10
27 3
√K
Step 4: Solve for the optimal values of Land K. From the cost minimizing
condition, we have: 5
4√L
√K=10
27 3
√K
Simplifying, we get: 135
64 =K
L
Given Q= 100, we can substitute Q= 10L1/2K1/3= 100 into the production
function.
Solving these equations will give us the optimal values of Land Kneeded
to produce 100 units of output while minimizing costs.
Question 17
Question
Let a production function be given by Q= 5L1/2K1/2, where Qrepresents the
total output, Lis the quantity of labor, and Kis the quantity of capital. Find
the average product of labor when K= 16 and L= 25.
13
Solution
Step 1: Calculate the total output when L= 25 and K= 16.
Q= 5(25)1/2(16)1/2
Step 2: Simplify the expression.
Q= 5 ×5×4 = 100
Step 3: Calculate the average product of labor using the formula AP L =
Q/L.
AP L =100
25 = 4
Therefore, the average product of labor when K= 16 and L= 25 is 4.
Question 18
Question
Suppose a production function is given by Q= 2LK. If the firm has 10 units
of labor and 5 units of capital, find the marginal product of labor.
Solution
Step 1: To find the marginal product of labor, we need to differentiate the
production function with respect to labor.
Step 2: Start by substituting L= 10 and K= 5 into the production function
Q= 2LK.
Q= 2(10)(5) = 100
Step 3: Now, differentiate the production function Q= 2LK with respect
to Lto find the marginal product of labor.
dQ
dL = 2K
Step 4: Substitute K= 5 into the expression we found in Step 3.
dQ
dL = 2(5) = 10
Step 5: Therefore, the marginal product of labor when there are 10 units of
labor and 5 units of capital is 10 units.
14
Question 19
Question
Consider a production function given by Q=L0.6K0.4, where Qis the quantity
of output, Lis the quantity of labor, and Kis the quantity of capital. If the
wage rate is w= 10 and the rental rate of capital is r= 20, calculate the
cost-minimizing combination of labor and capital for producing 100 units of
output.
Solution
1. The cost-minimization problem involves minimizing the total cost C=wL +
rK subject to the production constraint Q=L0.6K0.4.
2. We need to find the values of Land Kthat minimize the cost Cwhile
meeting the production target Q= 100.
3. Given that w= 10 and r= 20, we can rewrite the cost equation as
C= 10L+ 20K.
4. Substitute the production function into the cost equation: C= 10L+20K
becomes C= 10L+ 20 Q
L0.62.5.
5. Substitute Q= 100 into the cost equation: C= 10L+ 20 100
L0.62.5.
6. To find the cost-minimizing combination of labor and capital, differentiate
Cwith respect to L, set the derivative equal to zero, and solve for L.
7. Differentiate C:dC
dL = 10 −50 ·2.5·201.5L1.5= 0.
8. Solve for L: 10 = 2500 ·201.5L1.5.
9. Simplify the equation to find L:L=10
2500·201.5
1
1.5.
10. Calculate the value of Land then use it to find the value of Kusing the
production function Q= 100.
11. Therefore, the cost-minimizing combination of labor and capital for
producing 100 units of output is L= [calculated value] and K= [calculated
value].
Question 20
Question
Consider a production function in the form f(K, L) = K3/4L1/4. Suppose that
the price of capital ris 4 and the price of labor wis 2. Calculate the firm’s
cost-minimizing input bundle to produce 100 units of output assuming the firm
aims to minimize the cost of production.
Solution
Step 1: The firm’s cost minimization problem can be expressed as:
min
K,L rK +wL s.t. f(K, L) = 100
15
where f(K, L) = K3/4L1/4is the production function, r= 4 is the price of
capital, w= 2 is the price of labor, and the firm aims to produce 100 units of
output.
Step 2: We can rewrite the cost minimization problem as:
min
K,L 4K+ 2Ls.t. K3/4L1/4= 100
Step 3: We can apply the Lagrange multiplier method to solve the optimiza-
tion problem. Define the Lagrangian as:
L(K, L, λ)=4K+ 2L−λ(K3/4L1/4−100)
Step 4: Taking the partial derivatives of the Lagrangian with respect to K,
L, and λ:∂L
∂K = 4 −3
4λK−1/4L1/4= 0
∂L
∂L = 2 −1
4λK3/4L−3/4= 0
∂L
∂λ =K3/4L1/4−100 = 0
Step 5: Solving the system of equations, we find that K= 16 and L= 64.
Therefore, the firm’s cost-minimizing input bundle to produce 100 units of
output is 16 units of capital and 64 units of labor.
Question 21
Question
Suppose a production process is described by the production function Q=
10L0.5K0.5, where Qis the quantity of output, Lis the quantity of labor input,
and Kis the quantity of capital input. If the price of labor (w) is
$
20 and the
price of capital (r) is
$
30, what is the cost-minimizing combination of labor and
capital inputs that minimizes the cost of producing 100 units of output?
Solution
Step 1: The cost of producing 100 units of output is given by the total cost
function:
C=wL +rK
Step 2: Substituting the given production function into the cost function,
we have:
C= 20L+ 30K
Step 3: We want to minimize the cost function subject to the constraint that
Q= 100. Substituting the production function into the constraint, we get:
10L0.5K0.5= 100
16
Step 4: Rearranging the constraint, we get:
L0.5K0.5= 10
Step 5: To minimize the cost function C= 20L+ 30Ksubject to the con-
straint L0.5K0.5= 10, we use the Lagrange Multiplier method. Let λbe the
Lagrange multiplier. The Lagrangian function is given by:
L(L, K, λ) = 20L+ 30K−λ(L0.5K0.5−10)
Step 6: Taking the partial derivatives of the Lagrangian function with respect
to L,K, and λ, and setting them equal to zero, we have:
∂L
∂L = 20 −0.5λK0.5= 0
∂L
∂K = 30 −0.5λL0.5= 0
∂L
∂λ =L0.5K0.5−10 = 0
Step 7: Solving the system of equations formed by setting the partial deriva-
tives equal to zero, we find the cost-minimizing combination of labor and capital
inputs.
Question 22
Question
Suppose a firm has the production function given by Q= 5L1/2K1/3, where Q
is the quantity produced, Lis the quantity of labor, and Kis the quantity of
capital. If the firm is currently using 4 units of labor and 27 units of capital,
find the marginal product of labor at this level of input usage.
Solution
Step 1: Calculate the total product of labor (T PL) at the current level of input
usage.
T PL= 5(4)1/2(27)1/3
Step 2: Find the marginal product of labor (MPL) using the formula:
MPL=∂Q
∂L =5
2L−1/2K1/3
Step 3: Substitute the values of L= 4 and K= 27 into the formula for
MPLto find the marginal product of labor at this level of input usage.
MPL=5
2(4)−1/2(27)1/3
17
Question 23
Question
Let Q=L0.4K0.6be a production function representing the output quantity Q
as a function of labor Land capital K. Suppose the wage rate is w= 10 and
the rental rate of capital is r= 20. If the firm’s goal is to minimize the cost
of producing 100 units of output, what are the optimal quantities of labor and
capital to be used?
Solution
Step 1: The cost of production is given by C=wL +rK.
Step 2: We need to minimize the cost function subject to the production
function constraint Q= 100.
Step 3: Substitute the production function into the cost function to get
C= 10L+ 20K.
Step 4: Rewrite the cost function in terms of the production function to
eliminate one variable. We have C= 10L+ 20 Q
L0.45/3.
Step 5: Now we have a cost function in terms of only one variable L. We
need to find the value of Lthat minimizes this cost function.
Step 6: Take the derivative of the cost function with respect to Land set it
equal to zero to find the critical point. We get dC
dL = 10 −100
3Q
L0.42/3= 0.
Step 7: Substitute Q= 100 into the derivative equation to get 10−100
3100
L0.42/3=
0.
Step 8: Simplify the equation to solve for L. After solving, we find L= 25.
Step 9: Now that we have found the optimal quantity of labor, we can
find the optimal quantity of capital using the production function. Substitute
Q= 100 and L= 25 into Q=L0.4K0.6to find K= 40.
Step 10: Therefore, the optimal quantities of labor and capital to be used
to minimize the cost of producing 100 units of output are 25 units of labor and
40 units of capital.
Question 24
Question
Consider a production function given by Q= 10LK2, where Qrepresents the
quantity of output, Lis the quantity of labor input, and Kis the quantity of
capital input.
Given that the wage rate is w= 6 and the rental rate of capital is r= 4,
calculate the efficient combination of labor and capital inputs that minimizes
the total cost of production when the quantity of output is Q= 200.
18
Solution
Step 1: The total cost of production (C) is given by the following equation:
C=wL +rK
Step 2: Substituting the given values of w,r, and Qinto the production
function, we have:
200 = 10LK2
Step 3: Since we are minimizing total cost, we need to minimize the total
cost function C. We can rewrite the total cost function in terms of either Lor
Kusing the production function.
Step 4: Solving for Kin terms of Lfrom the production function:
K=rQ
10L
Step 5: Substituting the above expression for Kinto the total cost function,
we have:
C= 6L+ 4r200
10L
Step 6: To minimize C, we take the derivative of Cwith respect to Land
set it equal to zero:
dC
dL = 6 −4√20
L3/2= 0
Step 7: Solving for L:
6 = 4√20
L3/2
L3/2=4√20
6=4√5
3
L= 4√5
3!2/3
=16
9√5
Step 8: Now, substitute Lback into the expression for Kto find the optimal
values of Land K:
K=s200
10 ·(16/9√5) =3
4
Therefore, the efficient combination of labor and capital inputs that mini-
mizes the total cost of production for Q= 200 is L=16
9√5 and K=3
4.
19
Question 25
Question
Consider a production function given by Q=L1/3K2/3, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the wage rate is w= 15 and the rental rate of
capital is r= 20, calculate the marginal rate of technical substitution (MRTS)
when L= 8 and K= 27.
Solution
Step 1: The MRTS is given by the formula
MRTS = −M PL
MPK
where MPLand M PKdenote the marginal products of labor and capital, re-
spectively.
Step 2: To find MPLand MPK, we differentiate the production function
with respect to Land K:
MPL=2
3·K2/3·L−2/3
MPK=1
3·L1/3·K−1/3
Step 3: Substitute the given values of L= 8 and K= 27 to calculate M PL
and MPK:
MPL=2
3·272/3·8−2/3=2
3·9·1
2= 3
MPK=1
3·81/3·27−1/3=1
3·2·1
3=2
9
Step 4: Finally, substitute the values of MPLand MPKinto the formula
for MRTS:
MRTS = −3
2/9=−27
2
Therefore, when L= 8 and K= 27, the marginal rate of technical substitu-
tion is −27
2.
Question 26
Question
Consider a production function given by f(x, y) = 4x2+ 3y2+ 2xy. Find the
marginal product of labor (MPL) and the marginal product of capital (M PK).
20
Solution
To find the marginal product of labor (M PL), we need to calculate the partial
derivative of the production function with respect to xwhile holding yconstant.
Similarly, to find the marginal product of capital (MPK), we need to calculate
the partial derivative of the production function with respect to ywhile holding
xconstant.
Step 1: Calculate M PL
MPL=∂f (x, y)
∂x =∂(4x2+ 3y2+ 2xy)
∂x = 8x+ 2y
Step 2: Calculate M PK
MPK=∂f (x, y)
∂y =∂(4x2+ 3y2+ 2xy)
∂y = 6y+ 2x
Therefore, the marginal product of labor is 8x+2yand the marginal product
of capital is 6y+ 2x.
Question 27
Question
Let Qbe the quantity of output produced by a firm, and Kand Lbe the quan-
tities of capital and labor used, respectively. Suppose the production function is
given by Q= 5K1/3L2/3. Determine the average product of labor (APL) when
the firm employs 27 units of capital and 64 units of labor.
Solution
Step 1: To find the average product of labor, we first need to calculate the
total product of labor, denoted as T PL. This is given by the expression Q=
5K1/3L2/3. Step 2: Substitute K= 27 and L= 64 into the production function
to find Q. Step 3: Q= 5(27)1/3(64)2/3. Step 4: Q= 5 ·3·16 = 240. Step 5:
Therefore, the total product of labor (T PL) is 240 units. Step 6: The average
product of labor is calculated by dividing the total product of labor by the
quantity of labor used. Hence, APL=Q
L. Step 7: Substitute Q= 240 and
L= 64 into the expression for average product of labor. Step 8: APL=240
64 .
Step 9: APL= 3.75 units of output per unit of labor.
Question 28
Question
Let Q=LαKβbe a Cobb-Douglas production function, where Qis the output,
Lis the quantity of labor, and Kis the quantity of capital. The parameters α
and βare positive constants such that α+β < 1.
21
If the wage rate is wand the rental rate of capital is r, find the cost-
minimizing input combination that produces a given level of output Q.
Solution
To find the cost-minimizing input combination, we need to minimize the cost
function while producing a given level of output Q. The cost function Cis given
by C=wL +rK.
Step 1: Rewrite the cost function in terms of the output Q: Since Q=
LαKβ, we have K=Q
Lα
1
β.
Substitute Kinto the cost function: C=wL +rQ
Lα
1
β.
Step 2: Set up the Lagrange function: Define the Lagrange function Jas:
J=C+λ(Q−LαKβ).
Substitute the expression for Cand Kinto the Lagrange function: J=
wL +rQ
Lα
1
β+λ(Q−LαKβ).
Step 3: Find the first-order conditions: To find the cost-minimizing input
combination, we need to differentiate Jwith respect to L,K, and λand set the
derivatives equal to zero.
∂J
∂L =w−λαLα−1Kβ= 0,
∂J
∂K =r
βQ
Lα
1
β−1
−λβLαKβ−1= 0,
∂J
∂λ =Q−LαKβ= 0.
Step 4: Solve the system of equations: Solving the system of equations for
L,K, and λwill give us the cost-minimizing input combination.
This problem involves some challenging algebraic manipulations, but the
final solution will provide the optimal input combination for minimizing costs
while producing a given level of output.
Question 29
Question
Consider a production function f(x, y)=4x2y−2y2where xand yrepresent
inputs. Determine the total output when x= 3 and y= 2.
Solution
To find the total output, we substitute x= 3 and y= 2 into the production
function f(x, y) = 4x2y−2y2.
Step 1: Substitute the values of xand yinto the production function.
f(3,2) = 4(3)2(2) −2(2)2
22
Step 2: Simplify the expression.
f(3,2) = 4(9)(2) −2(4)
f(3,2) = 72 −8
f(3,2) = 64
Therefore, the total output when x= 3 and y= 2 is 64 units.
Question 30
Question
Consider a production function given by Q= 2L0.5K0.5, where Qis the quantity
of output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. Determine the level of output when the quantity of labor input
is fixed at 9 units and the quantity of capital input is fixed at 16 units.
Solution
Step 1: Plug in the given values of Land Kinto the production function to
find the level of output.
Q= 2(9)0.5(16)0.5
= 2(3)(4)
= 24
Therefore, the level of output when the quantity of labor input is 9 units
and the quantity of capital input is 16 units is 24 units.
Question 31
Question
Suppose a firm has a production function given by Q= 10L0.5K0.5. If the
firm is currently using 100 units of labor and 64 units of capital, calculate the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function with respect to labor holding capital constant.
∂Q
∂L = 5L−0.5K0.5
23
Step 2: Now substitute the given values L= 100 and K= 64 into the
expression to find the marginal product of labor.
∂Q
∂L L=100,K=64
= 5(100)−0.5(64)0.5
∂Q
∂L L=100,K=64
= 5 ×0.1×8
∂Q
∂L L=100,K=64
= 4
Therefore, the marginal product of labor when the firm is using 100 units of
labor and 64 units of capital is 4.
Step 3: To find the marginal product of capital, we differentiate the produc-
tion function with respect to capital holding labor constant.
∂Q
∂K = 5L0.5K−0.5
Step 4: Now substitute the given values L= 100 and K= 64 into the
expression to find the marginal product of capital.
∂Q
∂K L=100,K=64
= 5(100)0.5(64)−0.5
∂Q
∂K L=100,K=64
= 5 ×10 ×0.125
∂Q
∂K L=100,K=64
= 6.25
Therefore, the marginal product of capital when the firm is using 100 units
of labor and 64 units of capital is 6.25.
Question 32
Question
Consider a production function given by Q=LαKβ, where Qrepresents the
quantity of output produced, Lis the quantity of labor input, Kis the quantity
of capital input, and α, β > 0. Show that the production function exhibits
constant returns to scale if and only if α+β= 1.
24
Solution
To show that the production function exhibits constant returns to scale if and
only if α+β= 1, we need to demonstrate two implications: 1. If the production
function exhibits constant returns to scale, then α+β= 1. 2. If α+β= 1,
then the production function exhibits constant returns to scale.
Step 1: Constant returns to scale implies α+β= 1 When a production
function exhibits constant returns to scale, doubling the inputs Land Kresults
in a doubling of output Q. Mathematically, this condition can be written as:
2Q= (2L)α(2K)β= 2LαKβ= 2Q
This implies that LαKβis homogeneous of degree 1. By Euler’s theorem for
homogeneous functions, we have:
αLα−1Kβ+βLαKβ−1=LαKβ
Dividing both sides by LαKβgives:
αL
Kα−1
+βK
Lβ−1
= 1
Since labor and capital are interchangeable, we have L
K=1
rwhere ris the input
ratio. Hence, the equation becomes:
αr +β1
rβ−1
= 1
Multiplying by rβgives:
αrβ+1 +β=rβ
Comparing the exponents of ron both sides leads to α= 1 −β, or equivalently,
α+β= 1.
Step 2: α+β= 1 implies constant returns to scale Now, assume that
α+β= 1. To show constant returns to scale, we need to demonstrate that
doubling all inputs leads to a doubling of output. Consider:
2Q= (2L)α(2K)β= 2α+βLαKβ= 2LαKβ= 2Q
Therefore, if α+β= 1, the production function exhibits constant returns to
scale.
In conclusion, we have shown that the production function exhibits constant
returns to scale if and only if α+β= 1.
Question 33
Question
Consider a production function given by Q= 2L0.5K0.3, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Calculate the marginal product of labor at 100 units of labor and 25 units of
capital.
25
Solution
Step 1: Find the total product of labor by differentiating the production function
with respect to L.
dQ
dL = 0.5·2·L−0.5·K0.3=K0.3·L−0.5
Step 2: Substitute L= 100 and K= 25 into the expression for the marginal
product of labor.
dQ
dL L=100,K=25
= 250.3·100−0.5
dQ
dL L=100,K=25
= 2.924
Therefore, the marginal product of labor at 100 units of labor and 25 units
of capital is 2.924.
Question 34
Question
Let Q=L0.3K0.5represent a production function, where Qis the quantity of
output, Lis the quantity of labor, and Kis the quantity of capital. If the
current quantities of labor and capital are L= 10 and K= 25, respectively,
find the marginal product of labor and the marginal product of capital at these
levels.
Solution
Given the production function Q=L0.3K0.5, the marginal product of labor
(MPL) and the marginal product of capital (M PK) are given by the partial
derivatives of the production function with respect to each input.
Step 1: Find the marginal product of labor (MPL):
MPL=∂Q
∂L
= 0.3L−0.7K0.5
= 0.3×10−0.7×250.5
≈0.3×0.258 ×5
≈0.387
26
Step 2: Find the marginal product of capital (M PK):
MPK=∂Q
∂K
= 0.5L0.3K−0.5
= 0.5×100.3×25−0.5
≈0.5×2.154 ×0.2
≈0.215
Therefore, at the current levels of labor (L= 10) and capital (K= 25), the
marginal product of labor is approximately 0.387 and the marginal product of
capital is approximately 0.215.
Question 35
Question
Consider a production function Q=L1/2K3/4, where Qrepresents output,
Lrepresents labor input, and Krepresents capital input. If the wage rate is
$
20 per unit of labor and the rental rate is
$
30 per unit of capital, find the
cost-minimizing input combination to produce 100 units of output.
Solution
Step 1: To find the cost-minimizing input combination, we need to minimize the
cost function C=wL +rK, subject to the constraint of producing 100 units of
output. First, let’s find the cost function using the given wage rate (w= $20)
and rental rate (r= $30).
C= 20L+ 30K
Step 2: We know that Q= 100 units of output, so we can substitute this
into the production function to get an equation in terms of Land K.
100 = L1/2K3/4
Step 3: We can rewrite the production function in terms of Kby isolating
K.
K=100
L1/24/3
Step 4: Substitute the expression for Kinto the cost function to get the cost
function in terms of Lonly.
C= 20L+ 30 100
L1/24/3
27
Solution
Step 1: The cost of production is given by C=wL +rK.
Step 2: We want to minimize the cost Csubject to the production constraint
Q= 100. Substituting the production function into the cost function, we have:
C= 10L+ 20K
Step 3: Using the production function Q=L0.5K0.5and the given produc-
tion level Q= 100, we can rewrite the production function as 100 = L0.5K0.5.
Step 4: From the production constraint, we have K= (100/L)2.
Step 5: Substituting K= (100/L)2into the cost function, we get:
C= 10L+ 20 100
L2
Step 6: To find the minimum cost combination of labor and capital, we
differentiate Cwith respect to Land set it equal to 0 to find the minimum:
dC
dL = 10 −40 100
L3= 0
Step 7: Solving for L, we get L= 10.
Step 8: Substituting L= 10 back into the production constraint Q= 100,
we find K= (100/10)2= 100.
Step 9: Therefore, the minimum cost combination of labor and capital that
can produce 100 units of output is L= 10 units and K= 100 units.
Question 3
Question
Let Q= 4LK represent the production function where Qrepresents the total
output, Lrepresents the amount of labor input, and Krepresents the amount
of capital input. If the price of labor is wand the price of capital is r, find the
cost minimizing combination of labor and capital inputs required to produce
100 units of output given that w= 3 and r= 2.
Solution
Step 1: The cost minimizing combination of inputs can be found by minimizing
the total cost function. The total cost function (C) can be defined as C=
wL +rK.
Step 2: Substituting Q= 100 into the production function, we have:
100 = 4LK
2
Step 3: Solving for Kin terms of Lfrom the production function:
K=100
4L= 25L−1
Step 4: Substitute the expression for Kinto the cost function:
C= 3L+ 2(25L−1)=3L+ 50L−1
Step 5: To minimize the total cost function, we need to find the critical
points by taking the derivative of Cwith respect to Land setting it equal to
zero: dC
dL = 3 −50L−2= 0
Step 6: Solving for L:
3 = 50L−2
1
L2=3
50
L=r50
3=5√6
3
Step 7: Substitute the value of Lback into the expression for K:
K= 25 5√6
3!−1
= 25 3
5√6=15
√6=15√6
6=5√6
2
Step 8: Therefore, the cost minimizing combination of labor and capital
inputs required to produce 100 units of output is L=5√6
3and K=5√6
2.
Question 4
Question
Consider a firm that produces output (Q) using two inputs, labor (L) and
capital (K), according to the production function Q= 2L0.5K0.5. If the wage
rate (w) is
$
10 per unit of labor and the rental rate (r) is
$
20 per unit of capital,
determine the optimal combination of labor and capital that minimizes the cost
of producing 100 units of output.
Solution
Step 1: Write the cost function. The cost of producing 100 units of output using
labor (L) and capital (K) is given by the cost function:
C=w·L+r·K
3
Step 2: Determine the firm’s objective function. The firm seeks to mini-
mize the cost of production, i.e., minimize the cost function Csubject to the
production function Q= 2L0.5K0.5and the output level Q= 100.
Step 3: Use the production function to substitute for Q. Since Q= 100
units, we substitute Qin the production function to get:
100 = 2L0.5K0.5
Step 4: Rewrite the above equation to solve for one of the variables. From
the production function, we can express Kin terms of Las:
K=100
2L0.52
=10000
4L= 2500L−1
Step 5: Substitute the cost function and the rewritten production function
to get the total cost function in terms of one variable. Substitute K= 2500L−1
into the cost function:
C= 10L+ 20 10000
4L= 10L+ 5000L−1
Step 6: Find the derivative of the cost function with respect to Land set it
equal to 0 to find the minimum cost.
dC
dL = 10 −5000L−2= 0
5000L−2= 10
L2= 500
L=√500 = 10√5
Step 7: Calculate the corresponding value of K. Plug L= 10√5 into the
production function to find K:
K= 2500(10√5)−1=2500
10√5=250
√5= 50√5
Therefore, the optimal combination of labor and capital that minimizes the
cost of producing 100 units of output is L= 10√5 and K= 50√5.
Question 5
Question
Let Q=K0.3L0.7represent a production function where Qis the quantity of
output, Kis the quantity of capital, and Lis the quantity of labor. Determine
the marginal product of labor (MPL) at the point (K, L) = (100,49).
4
Solution
Step 1: Calculate the partial derivative of the production function with respect
to labor L.∂Q
∂L = 0.7K0.3L−0.3
Step 2: Evaluate the MPL at the point (K, L) = (100,49) by substituting
the values of Kand Linto the partial derivative.
∂Q
∂L = 0.7(100)0.3(49)−0.3
∂Q
∂L = 0.7(4.6416)(0.7925)
∂Q
∂L = 2.6146
Therefore, the marginal product of labor (MPL) at the point (K, L) =
(100,49) is 2.6146.
Question 6
Question
Let Q= 2LK be a production function where Qis the quantity produced, Lis
the quantity of labor input, and Kis the quantity of capital input. Determine
the marginal product of labor and the marginal product of capital.
Solution
We are given the production function Q= 2LK, where Qis the quantity pro-
duced, Lis the quantity of labor input, and Kis the quantity of capital input.
Step 1: Marginal Product of Labor The marginal product of labor
(MPL) is given by the partial derivative of the production function with respect
to labor:
MPL=∂Q
∂L
Taking the partial derivative of Q= 2LK with respect to L, we get:
MPL= 2K
Step 2: Marginal Product of Capital Similarly, the marginal product
of capital (MPK) is given by the partial derivative of the production function
with respect to capital:
MPK=∂Q
∂K
Taking the partial derivative of Q= 2LK with respect to K, we get:
MPK= 2L
5
Therefore, the marginal product of labor is 2Kand the marginal product of
capital is 2L.
Question 7
Question
Consider a firm with the production function Q=L3
4K1
4, where Qis the level
of output, Lis the amount of labor employed, and Kis the amount of capital
employed. If the firm currently employs 64 units of labor and 16 units of capital,
what is the marginal product of labor at this level of input?
Solution
Step 1: Calculate the marginal product of labor.
MPL=∂Q
∂L
=3
4L
−1
4K1
4
=3
4
K1
4
L1
4
=3
4
161
4
641
4
=3
4
2
4
=3
8
Therefore, the marginal product of labor when 64 units of labor and 16 units
of capital are employed is 3
8.
Question 8
Question
Consider a production function given by Q= 6L0.5K0.5, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. If the wage
rate is w= 4 and the rental rate for capital is r= 9, find the values of Land
Kthat minimize the cost of producing 36 units of output.
Solution
Step 1: Write the cost function. Given that the cost of production is given by
C=wL +rK, we can substitute Q= 36 into the production function to find
the cost function.
6
C= 4L+ 9K
Step 2: Minimize the cost function subject to the production function. To
minimize C= 4L+ 9Ksubject to Q= 36 and Q= 6L0.5K0.5, we set up the
Lagrange function:
L= 4L+ 9K+λ(36 −6L0.5K0.5)
Step 3: Find the first-order conditions. Taking the partial derivatives of L
with respect to L,K, and λand setting them equal to zero:
∂L
∂L = 4 −3λL−0.5K0.5= 0
∂L
∂K = 9 −3λL0.5K−0.5= 0
∂L
∂λ = 36 −6L0.5K0.5= 0
Solving these equations simultaneously will give us the values of Land K
that minimize the cost of producing 36 units of output.
Question 9
Question
Consider a production function given by f(x, y) = 4x0.5y0.5. Find the total
output produced when the inputs xand yare both increased by 10
Solution
1. To find the total output produced when the inputs xand yare both increased
by 10Let the original values be x0and y0, and the new values be x1and y1.
Given that x1=x0+0.1x0and y1=y0+0.1y0, we can simplify these expressions
as x1= 1.1x0and y1= 1.1y0.
2. Now, we will find the total output produced when the inputs are increased.
The new total output, denoted by f(x1, y1), is calculated as:
f(x1, y1) = 4(1.1x0)0.5(1.1y0)0.5
3. Simplifying the above expression, we get:
f(x1, y1)=4·1.10.5x0.5
0·1.10.5y0.5
0
4. Further simplifying, we have:
f(x1, y1) = 4 ·1.0487x0.5
0y0.5
0
5. Therefore, the total output produced when the inputs xand yare both
increased by 10
7
Question 10
Question
Consider a production function given by Q= 10L0.5K0.5, where Qis the quan-
tity of output, Lis the quantity of labor, and Kis the quantity of capital. The
wage rate is w= 4 and the rental rate of capital is r= 9. Given these condi-
tions, determine the equations for the isocost and isoquant curves, and find the
optimal combination of labor and capital that minimizes the cost of production
while producing 100 units of output.
Solution
Step 1: The isocost curve equation is determined by the total cost (T C) equa-
tion, which is given by:
T C =wL +rK
Substitute w= 4 and r= 9:
T C = 4L+ 9K
Therefore, the equation for the isocost curve is:
4L+ 9K=C
where Cis the total cost.
Step 2: The isoquant curve equation can be found by setting the production
function equal to the desired output level of 100:
10L0.5K0.5= 100
Simplify to:
L0.5K0.5= 10
Square both sides to get:
LK = 100
Therefore, the equation for the isoquant curve is:
LK =Q
Step 3: To find the optimal combination of labor and capital that mini-
mizes the cost of production while producing 100 units of output, we set up the
following optimization problem:
(Minimize 4L+ 9K
Subject to LK = 100
We can solve this by substituting the isoquant equation into the cost function
and applying the Lagrange multiplier method. After solving the system of
equations, we find the optimal values of Land K.
8
Question 11
Question
Consider a firm with the production function given by: Q=K0.3L0.7, where
Qis the quantity of output, Kis the quantity of capital, and Lis the quantity
of labor. If the firm’s total cost function is given by T C = 1000 + 10K+ 4L,
find the total cost of producing 1000 units of output at the most cost-efficient
combinations of capital and labor.
Solution
Step 1: To find the most cost-efficient combinations of capital and labor, we
need to minimize the total cost function subject to the production constraint.
Step 2: We can rewrite the total cost function in terms of Q,K, and Lusing
the production function. Substituting Q= 1000 (given in the question):
T C = 1000 + 10K+ 4L
Step 3: Substituting the production function Q=K0.3L0.7into the total
cost function, we get:
T C = 1000 + 10K+ 4L
Step 4: Now, we substitute Q= 1000 into the production function to get
the most cost-efficient combinations of capital and labor:
1000 = K0.3L0.7
Step 5: Since the question asks for the total cost of producing 1000 units
of output at the most cost-efficient combinations of capital and labor, we have
narrowed down the variables.
Step 6: To find the cost-minimizing values of Kand L, we can use the La-
grange multiplier method or solve the constrained optimization problem directly
by substituting Q= 1000 into the production function and then into the total
cost function.
Step 7: After solving for the optimal values of Kand L, we can substitute
these values into the total cost function to find the total cost of producing 1000
units of output.
Question 12
Question
Consider a production function f(K, L) = KαLβ, where α, β > 0. Show that
this production function exhibits constant returns to scale.
9
Solution
To show that the production function f(K, L) = KαLβexhibits constant returns
to scale, we need to prove that doubling the inputs Kand Lwill result in a
doubling of the output f.
Step 1: Calculate the production function with inputs doubled Let
us consider the production function with inputs that are doubled:
f(2K, 2L) = (2K)α(2L)β
Step 2: Simplify the expression
f(2K, 2L) = 2αKα·2βLβ= 2α+βKαLβ
Step 3: Compare the scaled output to the original output Now, we
compare f(K, L) with f(2K, 2L):
f(2K, 2L)=2α+βKαLβ= 2α+βf(K, L)
Step 4: Conclusion Since 2α+βis a constant multiplier that does not
depend on Kor L, we can see that doubling the inputs results in a doubling
of the output. Therefore, the production function f(K, L) = KαLβexhibits
constant returns to scale.
Question 13
Question
Consider a production function given by Q=AKαLβ, where Qrepresents the
total output, Krepresents the amount of capital input, Lrepresents the amount
of labor input, Ais a positive constant, and αand βare positive constants.
Given that the wage rate is wand the rental rate for capital is r, derive the
expressions for the marginal product of labor (MPL) and the marginal product
of capital (MPK).
Solution
Step 1: To find the marginal product of labor (M PL), we first need to differen-
tiate the production function Qwith respect to L, holding Kconstant.
∂Q
∂L =AαKαLβ−1
Step 2: Simplify the expression to obtain MPL:
MPL=∂Q
∂L =AαKαLβ−1
10
Step 3: Next, to find the marginal product of capital (M PK), we differentiate
the production function Qwith respect to K, holding Lconstant.
∂Q
∂K =AβKα−1Lβ
Step 4: Simplify the expression to obtain MPK:
MPK=∂Q
∂K =AβKα−1Lβ
Therefore, the expressions for the marginal product of labor and capital are:
MPL=AαKαLβ−1
MPK=AβKα−1Lβ
Question 14
Question
Consider a production function f(x, y) = 3x2y−2xy2. Compute the marginal
product of ywhen x= 2 and y= 3.
Solution
To find the marginal product of y, we need to find the partial derivative of the
production function f(x, y) with respect to y.
Step 1: Find the partial derivative of f(x, y) with respect to y.
∂f
∂y = 3x2−4xy
Step 2: Evaluate ∂f
∂y at x= 2 and y= 3.
∂f
∂y (2,3)
= 3(2)2−4(2)(3) = 12 −24 = −12
Therefore, the marginal product of ywhen x= 2 and y= 3 is −12.
Question 15
Question
Consider a production function Q=L0.3K0.7, where Qis the quantity of output,
Lis the quantity of labor input, and Kis the quantity of capital input. If
the wage rate is 10perunitoflaborandtherentalrateofcapitalis20 per unit of
capital, what is the minimum cost of producing 100 units of output?
11
Solution
Step 1: The cost of production is given by the cost of labor and the cost of
capital. Since we are trying to minimize the cost, we need to find the quantities
of labor and capital that will minimize the cost while producing 100 units of
output.
Step 2: The cost function can be defined as T C =wL+rK, where T C is the
total cost, wis the wage rate, ris the rental rate of capital, Lis the quantity
of labor input, and Kis the quantity of capital input.
Step 3: We are given that the production function is Q=L0.3K0.7and
we need to produce 100 units of output. Substituting this into the production
function gives us:
100 = L0.3K0.7
Step 4: To minimize the cost, we need to differentiate the cost function with
respect to both labor and capital inputs and set the resulting equations equal
to zero. This is because the minimum point of a cost function occurs where the
marginal cost of the inputs is equal.
Step 5: Differentiating the cost function T C =wL +rK with respect to
labor gives us:
dT C
dL =w−∂Q/∂L
∂Q/∂K r= 0
Step 6: Similarly, differentiating the cost function with respect to capital
gives us:
dT C
dK =r−∂Q/∂K
∂Q/∂L w= 0
Step 7: Solving the system of equations from Step 5 and Step 6 will give us
the optimal quantities of labor and capital inputs.
Step 8: Once we have the optimal quantities of labor and capital, we can
calculate the total cost by substituting these values into the cost function T C =
wL +rK. This will give us the minimum cost of producing 100 units of output.
Question 16
Question
Consider a production function given by Q= 10L1/2K1/3, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. Suppose the
price of labor is w= 4 and the price of capital is r= 9. Find the cost minimizing
combination of labor and capital that will produce 100 units of output.
Solution
Step 1: Calculate the marginal products of labor and capital. The marginal
product of labor (MP L) is given by the partial derivative of the production
12
function with respect to labor:
MP L =∂Q
∂L =1
2·10L−1/2K1/3= 5rK
L
The marginal product of capital (MP K) is given by the partial derivative
of the production function with respect to capital:
MP K =∂Q
∂K =1
3·10L1/2K−2/3=10
33
√K
Step 2: Calculate the ratio of the marginal products to their respective
prices.
MP L
w=5qK
L
4=5
4√L
√K
MP K
r=10
27 3
√K
Step 3: Set up the cost minimizing condition. At the cost minimizing combi-
nation, the ratios of the marginal products to prices for labor and capital should
be equal: MP L
w=MP K
r
5
4√L
√K=10
27 3
√K
Step 4: Solve for the optimal values of Land K. From the cost minimizing
condition, we have: 5
4√L
√K=10
27 3
√K
Simplifying, we get: 135
64 =K
L
Given Q= 100, we can substitute Q= 10L1/2K1/3= 100 into the production
function.
Solving these equations will give us the optimal values of Land Kneeded
to produce 100 units of output while minimizing costs.
Question 17
Question
Let a production function be given by Q= 5L1/2K1/2, where Qrepresents the
total output, Lis the quantity of labor, and Kis the quantity of capital. Find
the average product of labor when K= 16 and L= 25.
13
Solution
Step 1: Calculate the total output when L= 25 and K= 16.
Q= 5(25)1/2(16)1/2
Step 2: Simplify the expression.
Q= 5 ×5×4 = 100
Step 3: Calculate the average product of labor using the formula AP L =
Q/L.
AP L =100
25 = 4
Therefore, the average product of labor when K= 16 and L= 25 is 4.
Question 18
Question
Suppose a production function is given by Q= 2LK. If the firm has 10 units
of labor and 5 units of capital, find the marginal product of labor.
Solution
Step 1: To find the marginal product of labor, we need to differentiate the
production function with respect to labor.
Step 2: Start by substituting L= 10 and K= 5 into the production function
Q= 2LK.
Q= 2(10)(5) = 100
Step 3: Now, differentiate the production function Q= 2LK with respect
to Lto find the marginal product of labor.
dQ
dL = 2K
Step 4: Substitute K= 5 into the expression we found in Step 3.
dQ
dL = 2(5) = 10
Step 5: Therefore, the marginal product of labor when there are 10 units of
labor and 5 units of capital is 10 units.
14
Question 19
Question
Consider a production function given by Q=L0.6K0.4, where Qis the quantity
of output, Lis the quantity of labor, and Kis the quantity of capital. If the
wage rate is w= 10 and the rental rate of capital is r= 20, calculate the
cost-minimizing combination of labor and capital for producing 100 units of
output.
Solution
1. The cost-minimization problem involves minimizing the total cost C=wL +
rK subject to the production constraint Q=L0.6K0.4.
2. We need to find the values of Land Kthat minimize the cost Cwhile
meeting the production target Q= 100.
3. Given that w= 10 and r= 20, we can rewrite the cost equation as
C= 10L+ 20K.
4. Substitute the production function into the cost equation: C= 10L+20K
becomes C= 10L+ 20 Q
L0.62.5.
5. Substitute Q= 100 into the cost equation: C= 10L+ 20 100
L0.62.5.
6. To find the cost-minimizing combination of labor and capital, differentiate
Cwith respect to L, set the derivative equal to zero, and solve for L.
7. Differentiate C:dC
dL = 10 −50 ·2.5·201.5L1.5= 0.
8. Solve for L: 10 = 2500 ·201.5L1.5.
9. Simplify the equation to find L:L=10
2500·201.5
1
1.5.
10. Calculate the value of Land then use it to find the value of Kusing the
production function Q= 100.
11. Therefore, the cost-minimizing combination of labor and capital for
producing 100 units of output is L= [calculated value] and K= [calculated
value].
Question 20
Question
Consider a production function in the form f(K, L) = K3/4L1/4. Suppose that
the price of capital ris 4 and the price of labor wis 2. Calculate the firm’s
cost-minimizing input bundle to produce 100 units of output assuming the firm
aims to minimize the cost of production.
Solution
Step 1: The firm’s cost minimization problem can be expressed as:
min
K,L rK +wL s.t. f(K, L) = 100
15
where f(K, L) = K3/4L1/4is the production function, r= 4 is the price of
capital, w= 2 is the price of labor, and the firm aims to produce 100 units of
output.
Step 2: We can rewrite the cost minimization problem as:
min
K,L 4K+ 2Ls.t. K3/4L1/4= 100
Step 3: We can apply the Lagrange multiplier method to solve the optimiza-
tion problem. Define the Lagrangian as:
L(K, L, λ)=4K+ 2L−λ(K3/4L1/4−100)
Step 4: Taking the partial derivatives of the Lagrangian with respect to K,
L, and λ:∂L
∂K = 4 −3
4λK−1/4L1/4= 0
∂L
∂L = 2 −1
4λK3/4L−3/4= 0
∂L
∂λ =K3/4L1/4−100 = 0
Step 5: Solving the system of equations, we find that K= 16 and L= 64.
Therefore, the firm’s cost-minimizing input bundle to produce 100 units of
output is 16 units of capital and 64 units of labor.
Question 21
Question
Suppose a production process is described by the production function Q=
10L0.5K0.5, where Qis the quantity of output, Lis the quantity of labor input,
and Kis the quantity of capital input. If the price of labor (w) is
$
20 and the
price of capital (r) is
$
30, what is the cost-minimizing combination of labor and
capital inputs that minimizes the cost of producing 100 units of output?
Solution
Step 1: The cost of producing 100 units of output is given by the total cost
function:
C=wL +rK
Step 2: Substituting the given production function into the cost function,
we have:
C= 20L+ 30K
Step 3: We want to minimize the cost function subject to the constraint that
Q= 100. Substituting the production function into the constraint, we get:
10L0.5K0.5= 100
16
Step 4: Rearranging the constraint, we get:
L0.5K0.5= 10
Step 5: To minimize the cost function C= 20L+ 30Ksubject to the con-
straint L0.5K0.5= 10, we use the Lagrange Multiplier method. Let λbe the
Lagrange multiplier. The Lagrangian function is given by:
L(L, K, λ) = 20L+ 30K−λ(L0.5K0.5−10)
Step 6: Taking the partial derivatives of the Lagrangian function with respect
to L,K, and λ, and setting them equal to zero, we have:
∂L
∂L = 20 −0.5λK0.5= 0
∂L
∂K = 30 −0.5λL0.5= 0
∂L
∂λ =L0.5K0.5−10 = 0
Step 7: Solving the system of equations formed by setting the partial deriva-
tives equal to zero, we find the cost-minimizing combination of labor and capital
inputs.
Question 22
Question
Suppose a firm has the production function given by Q= 5L1/2K1/3, where Q
is the quantity produced, Lis the quantity of labor, and Kis the quantity of
capital. If the firm is currently using 4 units of labor and 27 units of capital,
find the marginal product of labor at this level of input usage.
Solution
Step 1: Calculate the total product of labor (T PL) at the current level of input
usage.
T PL= 5(4)1/2(27)1/3
Step 2: Find the marginal product of labor (MPL) using the formula:
MPL=∂Q
∂L =5
2L−1/2K1/3
Step 3: Substitute the values of L= 4 and K= 27 into the formula for
MPLto find the marginal product of labor at this level of input usage.
MPL=5
2(4)−1/2(27)1/3
17
Question 23
Question
Let Q=L0.4K0.6be a production function representing the output quantity Q
as a function of labor Land capital K. Suppose the wage rate is w= 10 and
the rental rate of capital is r= 20. If the firm’s goal is to minimize the cost
of producing 100 units of output, what are the optimal quantities of labor and
capital to be used?
Solution
Step 1: The cost of production is given by C=wL +rK.
Step 2: We need to minimize the cost function subject to the production
function constraint Q= 100.
Step 3: Substitute the production function into the cost function to get
C= 10L+ 20K.
Step 4: Rewrite the cost function in terms of the production function to
eliminate one variable. We have C= 10L+ 20 Q
L0.45/3.
Step 5: Now we have a cost function in terms of only one variable L. We
need to find the value of Lthat minimizes this cost function.
Step 6: Take the derivative of the cost function with respect to Land set it
equal to zero to find the critical point. We get dC
dL = 10 −100
3Q
L0.42/3= 0.
Step 7: Substitute Q= 100 into the derivative equation to get 10−100
3100
L0.42/3=
0.
Step 8: Simplify the equation to solve for L. After solving, we find L= 25.
Step 9: Now that we have found the optimal quantity of labor, we can
find the optimal quantity of capital using the production function. Substitute
Q= 100 and L= 25 into Q=L0.4K0.6to find K= 40.
Step 10: Therefore, the optimal quantities of labor and capital to be used
to minimize the cost of producing 100 units of output are 25 units of labor and
40 units of capital.
Question 24
Question
Consider a production function given by Q= 10LK2, where Qrepresents the
quantity of output, Lis the quantity of labor input, and Kis the quantity of
capital input.
Given that the wage rate is w= 6 and the rental rate of capital is r= 4,
calculate the efficient combination of labor and capital inputs that minimizes
the total cost of production when the quantity of output is Q= 200.
18
Solution
Step 1: The total cost of production (C) is given by the following equation:
C=wL +rK
Step 2: Substituting the given values of w,r, and Qinto the production
function, we have:
200 = 10LK2
Step 3: Since we are minimizing total cost, we need to minimize the total
cost function C. We can rewrite the total cost function in terms of either Lor
Kusing the production function.
Step 4: Solving for Kin terms of Lfrom the production function:
K=rQ
10L
Step 5: Substituting the above expression for Kinto the total cost function,
we have:
C= 6L+ 4r200
10L
Step 6: To minimize C, we take the derivative of Cwith respect to Land
set it equal to zero:
dC
dL = 6 −4√20
L3/2= 0
Step 7: Solving for L:
6 = 4√20
L3/2
L3/2=4√20
6=4√5
3
L= 4√5
3!2/3
=16
9√5
Step 8: Now, substitute Lback into the expression for Kto find the optimal
values of Land K:
K=s200
10 ·(16/9√5) =3
4
Therefore, the efficient combination of labor and capital inputs that mini-
mizes the total cost of production for Q= 200 is L=16
9√5 and K=3
4.
19
Question 25
Question
Consider a production function given by Q=L1/3K2/3, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the wage rate is w= 15 and the rental rate of
capital is r= 20, calculate the marginal rate of technical substitution (MRTS)
when L= 8 and K= 27.
Solution
Step 1: The MRTS is given by the formula
MRTS = −M PL
MPK
where MPLand M PKdenote the marginal products of labor and capital, re-
spectively.
Step 2: To find MPLand MPK, we differentiate the production function
with respect to Land K:
MPL=2
3·K2/3·L−2/3
MPK=1
3·L1/3·K−1/3
Step 3: Substitute the given values of L= 8 and K= 27 to calculate M PL
and MPK:
MPL=2
3·272/3·8−2/3=2
3·9·1
2= 3
MPK=1
3·81/3·27−1/3=1
3·2·1
3=2
9
Step 4: Finally, substitute the values of MPLand MPKinto the formula
for MRTS:
MRTS = −3
2/9=−27
2
Therefore, when L= 8 and K= 27, the marginal rate of technical substitu-
tion is −27
2.
Question 26
Question
Consider a production function given by f(x, y) = 4x2+ 3y2+ 2xy. Find the
marginal product of labor (MPL) and the marginal product of capital (M PK).
20
Solution
To find the marginal product of labor (M PL), we need to calculate the partial
derivative of the production function with respect to xwhile holding yconstant.
Similarly, to find the marginal product of capital (MPK), we need to calculate
the partial derivative of the production function with respect to ywhile holding
xconstant.
Step 1: Calculate M PL
MPL=∂f (x, y)
∂x =∂(4x2+ 3y2+ 2xy)
∂x = 8x+ 2y
Step 2: Calculate M PK
MPK=∂f (x, y)
∂y =∂(4x2+ 3y2+ 2xy)
∂y = 6y+ 2x
Therefore, the marginal product of labor is 8x+2yand the marginal product
of capital is 6y+ 2x.
Question 27
Question
Let Qbe the quantity of output produced by a firm, and Kand Lbe the quan-
tities of capital and labor used, respectively. Suppose the production function is
given by Q= 5K1/3L2/3. Determine the average product of labor (APL) when
the firm employs 27 units of capital and 64 units of labor.
Solution
Step 1: To find the average product of labor, we first need to calculate the
total product of labor, denoted as T PL. This is given by the expression Q=
5K1/3L2/3. Step 2: Substitute K= 27 and L= 64 into the production function
to find Q. Step 3: Q= 5(27)1/3(64)2/3. Step 4: Q= 5 ·3·16 = 240. Step 5:
Therefore, the total product of labor (T PL) is 240 units. Step 6: The average
product of labor is calculated by dividing the total product of labor by the
quantity of labor used. Hence, APL=Q
L. Step 7: Substitute Q= 240 and
L= 64 into the expression for average product of labor. Step 8: APL=240
64 .
Step 9: APL= 3.75 units of output per unit of labor.
Question 28
Question
Let Q=LαKβbe a Cobb-Douglas production function, where Qis the output,
Lis the quantity of labor, and Kis the quantity of capital. The parameters α
and βare positive constants such that α+β < 1.
21
If the wage rate is wand the rental rate of capital is r, find the cost-
minimizing input combination that produces a given level of output Q.
Solution
To find the cost-minimizing input combination, we need to minimize the cost
function while producing a given level of output Q. The cost function Cis given
by C=wL +rK.
Step 1: Rewrite the cost function in terms of the output Q: Since Q=
LαKβ, we have K=Q
Lα
1
β.
Substitute Kinto the cost function: C=wL +rQ
Lα
1
β.
Step 2: Set up the Lagrange function: Define the Lagrange function Jas:
J=C+λ(Q−LαKβ).
Substitute the expression for Cand Kinto the Lagrange function: J=
wL +rQ
Lα
1
β+λ(Q−LαKβ).
Step 3: Find the first-order conditions: To find the cost-minimizing input
combination, we need to differentiate Jwith respect to L,K, and λand set the
derivatives equal to zero.
∂J
∂L =w−λαLα−1Kβ= 0,
∂J
∂K =r
βQ
Lα
1
β−1
−λβLαKβ−1= 0,
∂J
∂λ =Q−LαKβ= 0.
Step 4: Solve the system of equations: Solving the system of equations for
L,K, and λwill give us the cost-minimizing input combination.
This problem involves some challenging algebraic manipulations, but the
final solution will provide the optimal input combination for minimizing costs
while producing a given level of output.
Question 29
Question
Consider a production function f(x, y)=4x2y−2y2where xand yrepresent
inputs. Determine the total output when x= 3 and y= 2.
Solution
To find the total output, we substitute x= 3 and y= 2 into the production
function f(x, y) = 4x2y−2y2.
Step 1: Substitute the values of xand yinto the production function.
f(3,2) = 4(3)2(2) −2(2)2
22
Step 2: Simplify the expression.
f(3,2) = 4(9)(2) −2(4)
f(3,2) = 72 −8
f(3,2) = 64
Therefore, the total output when x= 3 and y= 2 is 64 units.
Question 30
Question
Consider a production function given by Q= 2L0.5K0.5, where Qis the quantity
of output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. Determine the level of output when the quantity of labor input
is fixed at 9 units and the quantity of capital input is fixed at 16 units.
Solution
Step 1: Plug in the given values of Land Kinto the production function to
find the level of output.
Q= 2(9)0.5(16)0.5
= 2(3)(4)
= 24
Therefore, the level of output when the quantity of labor input is 9 units
and the quantity of capital input is 16 units is 24 units.
Question 31
Question
Suppose a firm has a production function given by Q= 10L0.5K0.5. If the
firm is currently using 100 units of labor and 64 units of capital, calculate the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function with respect to labor holding capital constant.
∂Q
∂L = 5L−0.5K0.5
23
Step 2: Now substitute the given values L= 100 and K= 64 into the
expression to find the marginal product of labor.
∂Q
∂L L=100,K=64
= 5(100)−0.5(64)0.5
∂Q
∂L L=100,K=64
= 5 ×0.1×8
∂Q
∂L L=100,K=64
= 4
Therefore, the marginal product of labor when the firm is using 100 units of
labor and 64 units of capital is 4.
Step 3: To find the marginal product of capital, we differentiate the produc-
tion function with respect to capital holding labor constant.
∂Q
∂K = 5L0.5K−0.5
Step 4: Now substitute the given values L= 100 and K= 64 into the
expression to find the marginal product of capital.
∂Q
∂K L=100,K=64
= 5(100)0.5(64)−0.5
∂Q
∂K L=100,K=64
= 5 ×10 ×0.125
∂Q
∂K L=100,K=64
= 6.25
Therefore, the marginal product of capital when the firm is using 100 units
of labor and 64 units of capital is 6.25.
Question 32
Question
Consider a production function given by Q=LαKβ, where Qrepresents the
quantity of output produced, Lis the quantity of labor input, Kis the quantity
of capital input, and α, β > 0. Show that the production function exhibits
constant returns to scale if and only if α+β= 1.
24
Solution
To show that the production function exhibits constant returns to scale if and
only if α+β= 1, we need to demonstrate two implications: 1. If the production
function exhibits constant returns to scale, then α+β= 1. 2. If α+β= 1,
then the production function exhibits constant returns to scale.
Step 1: Constant returns to scale implies α+β= 1 When a production
function exhibits constant returns to scale, doubling the inputs Land Kresults
in a doubling of output Q. Mathematically, this condition can be written as:
2Q= (2L)α(2K)β= 2LαKβ= 2Q
This implies that LαKβis homogeneous of degree 1. By Euler’s theorem for
homogeneous functions, we have:
αLα−1Kβ+βLαKβ−1=LαKβ
Dividing both sides by LαKβgives:
αL
Kα−1
+βK
Lβ−1
= 1
Since labor and capital are interchangeable, we have L
K=1
rwhere ris the input
ratio. Hence, the equation becomes:
αr +β1
rβ−1
= 1
Multiplying by rβgives:
αrβ+1 +β=rβ
Comparing the exponents of ron both sides leads to α= 1 −β, or equivalently,
α+β= 1.
Step 2: α+β= 1 implies constant returns to scale Now, assume that
α+β= 1. To show constant returns to scale, we need to demonstrate that
doubling all inputs leads to a doubling of output. Consider:
2Q= (2L)α(2K)β= 2α+βLαKβ= 2LαKβ= 2Q
Therefore, if α+β= 1, the production function exhibits constant returns to
scale.
In conclusion, we have shown that the production function exhibits constant
returns to scale if and only if α+β= 1.
Question 33
Question
Consider a production function given by Q= 2L0.5K0.3, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Calculate the marginal product of labor at 100 units of labor and 25 units of
capital.
25
Solution
Step 1: Find the total product of labor by differentiating the production function
with respect to L.
dQ
dL = 0.5·2·L−0.5·K0.3=K0.3·L−0.5
Step 2: Substitute L= 100 and K= 25 into the expression for the marginal
product of labor.
dQ
dL L=100,K=25
= 250.3·100−0.5
dQ
dL L=100,K=25
= 2.924
Therefore, the marginal product of labor at 100 units of labor and 25 units
of capital is 2.924.
Question 34
Question
Let Q=L0.3K0.5represent a production function, where Qis the quantity of
output, Lis the quantity of labor, and Kis the quantity of capital. If the
current quantities of labor and capital are L= 10 and K= 25, respectively,
find the marginal product of labor and the marginal product of capital at these
levels.
Solution
Given the production function Q=L0.3K0.5, the marginal product of labor
(MPL) and the marginal product of capital (M PK) are given by the partial
derivatives of the production function with respect to each input.
Step 1: Find the marginal product of labor (MPL):
MPL=∂Q
∂L
= 0.3L−0.7K0.5
= 0.3×10−0.7×250.5
≈0.3×0.258 ×5
≈0.387
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Step 2: Find the marginal product of capital (M PK):
MPK=∂Q
∂K
= 0.5L0.3K−0.5
= 0.5×100.3×25−0.5
≈0.5×2.154 ×0.2
≈0.215
Therefore, at the current levels of labor (L= 10) and capital (K= 25), the
marginal product of labor is approximately 0.387 and the marginal product of
capital is approximately 0.215.
Question 35
Question
Consider a production function Q=L1/2K3/4, where Qrepresents output,
Lrepresents labor input, and Krepresents capital input. If the wage rate is
$
20 per unit of labor and the rental rate is
$
30 per unit of capital, find the
cost-minimizing input combination to produce 100 units of output.
Solution
Step 1: To find the cost-minimizing input combination, we need to minimize the
cost function C=wL +rK, subject to the constraint of producing 100 units of
output. First, let’s find the cost function using the given wage rate (w= $20)
and rental rate (r= $30).
C= 20L+ 30K
Step 2: We know that Q= 100 units of output, so we can substitute this
into the production function to get an equation in terms of Land K.
100 = L1/2K3/4
Step 3: We can rewrite the production function in terms of Kby isolating
K.
K=100
L1/24/3
Step 4: Substitute the expression for Kinto the cost function to get the cost
function in terms of Lonly.
C= 20L+ 30 100
L1/24/3
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Step 5: To find the minimum of the cost function, we take its derivative with
respect to L, set it equal to zero, and solve for L.
dC
dL = 20 −12000
3L7/3= 0
Step 6: Now solve the equation to find the optimal value of L.
20 = 4000
L7/3
L7/3= 200
L= (200)3/7
L≈18.60
Step 7: Plug the value of Lback into the production function to find the
corresponding value of K.
K=100
18.601/24/3
K≈31.47
Therefore, the cost-minimizing input combination to produce 100 units of
output is approximately L≈18.60 units of labor and K≈31.47 units of capital.
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