ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank - Set 5
Liberty University
Question 1
Question
Consider a production function Q= 3L0.5K0.3, where Qrepresents the total
output, Lis the amount of labor input, and Kis 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 func-
tion Q= 3L0.5K0.3and 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 Land Kin terms of Q. From the production function,
we have: Q= 3L0.5K0.3Substitute Q= 1000 into the production function:
1000 = 3L0.5K0.3
Step 3: Solve for Lin terms of K. Solving for Lin terms of K, we get:
L=1000
3K0.32
Step 4: Cost function in terms of K. Substitute the expression for Linto
the cost function: C= 12 1000
3K0.32+ 20K
Step 5: Minimize the cost function. To find the minimum cost, take the
derivative of the cost function with respect to Kand set it equal to zero: dC
dK = 0
Step 6: Solve for the optimal value of K. Solving for the optimal value of
K, we can find the corresponding value of Lusing the production function.
Step 7: Calculate the minimum cost. Substitute the optimal values of Land
Kinto 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= 2L0.5K0.5where Qis the quantity
produced, Lis the quantity of labor, and Kis the quantity of capital. Deter-
mine the marginal product of labor (M PL) 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.
∂Q
∂L =∂(2L0.5K0.5)
∂L
Step 2: Taking the partial derivative with respect to L, we get:
∂Q
∂L = 1L−0.5K0.5
Step 3: Simplifying the expression, we find the marginal product of labor:
MPL=K0.5
L0.5
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.∂Q
∂K =∂(2L0.5K0.5)
∂K
Step 5: Taking the partial derivative with respect to K, we get:
∂Q
∂K = 1L0.5K−0.5
Step 6: Simplifying the expression, we find the marginal product of capital:
MPK=L0.5
K0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=L0.5
K0.5.
Question 3
Question
Consider a production function given by Q= 10L3/4K1/4, where Qis the
quantity of output, Lis the quantity of labor input, and Kis 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.
2
Solution
Step 1: The cost of production (C) can be expressed as C=wL +rK, where
wis the price of labor and ris the price of capital.
Step 2: To minimize cost, we need to minimize the cost function Csubject
to the constraint that the production function Q= 10L3/4K1/4produces 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:
4 + 3
4λL−7/4K−1/4= 0
2 + 1
4λL−3/4K−5/4= 0
100L−3/4K−1/4−10 = 0
Step 6: Solve the system of equations to find the values of Land K. Once
you have found Land K, substitute these values back into the cost function
C= 4L+ 2Kto calculate the minimum cost of producing 100 units of output.
Question 4
Question
Consider a production function Q= 2L0.5K0.5, where Qis the quantity of
output produced, Lis the quantity of labor input, and Kis 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 Qwith respect to L:
MPL=∂Q
∂L = 0.5×2×L−0.5×K0.5=K0.5
Step 2: Substitute L= 16 and K= 25 into the formula to find MPL:
MPL= 250.5= 5
3
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 Qwith respect to K:
MPK=∂Q
∂K = 0.5×2×L0.5×K−0.5=L0.5
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 Qdenote the total output of a firm and Kand Lrepresent 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 T V C =wL.
Step 3: To determine the equation for the firm’s TVC in terms of L, we need
to find the expression for wfirst.
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 (MP L): w=∂Q/∂L
MP L .
Step 5: Calculate the partial derivative of Qwith respect to L:
∂Q
∂L = 0.6K0.4L−0.4
= 0.6K0.4
L0.4
4
Step 6: Calculate the marginal product of labor (M P L) by taking the deriva-
tive of the production function with respect to L:
MP L =∂Q
∂L = 0.6K0.4L−0.4
= 0.6K0.4
L0.4
Step 7: Now, we can find the wage rate (w):
w=
0.6K0.4
L0.4
0.6K0.4
L0.4= 1
Step 8: Substitute w= 1 back into the TVC function T V C =wL to get the
TVC function in terms of L:
T V C = 1 ×L=L
Step 9: Therefore, the equation for the firm’s total variable cost function in
terms of Lis T V C =L.
Question 6
Question
Suppose a firm uses the production function Q=L1/3K1/2to produce output,
where Qis the quantity of output, Lis the quantity of labor, and Kis the
quantity of capital. If the firm currently has 16 units of capital and is producing
64 units of output, determine:
1. The marginal product of labor.
2. The average product of labor.
3. The rate at which the firm can substitute labor for capital while keeping
output constant.
Solution
1. To find the marginal product of labor, we need to calculate the derivative of
the production function with respect to labor L:
dQ
dL =1
3L−2/3K1/2
Given that L= 643/2and K= 16, we can substitute the values into the
derivative to find the marginal product of labor.
5
2. To find the average product of labor, we need to divide the total product
by the quantity of labor. The average product of labor (AP L) can be calculated
as:
AP L =Q
L=L1/3K1/2
L
Substitute the given values of Land Kin to find AP L.
3. The rate at which the firm can substitute labor for capital while keeping
output constant can be found using the MRTS (Marginal Rate of Technical
Substitution) formula:
MRT S =−MPL
MPK
Calculate the marginal products of labor and capital individually, then substi-
tute them into the MRTS formula to find the rate at which labor and capital
can be substituted while keeping output constant.
Question 7
Question
Suppose a production function is given by Q= 4K0.5L0.5. If the wage rate is
w= 10 and the rental rate of capital is r= 5, find the minimum cost to produce
100 units of output.
Solution
Step 1: Given the production function Q= 4K0.5L0.5, we need to minimize the
cost to produce 100 units of output. The cost function is given by C=wL+rK.
Step 2: We know that Q= 100, so substitute this into the production
function to get: 100 = 4K0.5L0.5.
Step 3: Rewrite the production function in terms of K:
K=100
4L2
=625
L2
Step 4: Substitute the value of Kback into the cost function:
C= 10L+ 5 625
L2
Step 5: To minimize cost, take the derivative of the cost function with respect
to Land set it equal to zero:
dC
dL = 10 −10 625
L3= 0
Step 6: Solve for L:
10 = 6250
L3
6
L3= 625
L= 5
Step 7: Substitute the value of Lback into the equation for K:
K=625
52= 25
Step 8: Substitute the values of Kand Lback into the cost function to find
the minimum cost:
C= 10(5) + 5(25) = 50 + 125 = 175
Therefore, the minimum cost to produce 100 units of output is 175.
Question 8
Question
Consider the production function Q= 10LK −0.1L2K2, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK). Then, find the values of Land Kthat maximize output.
Solution
Step 1: To find the marginal product of labor (MPL), we need to differentiate
the production function with respect to labor L:
MP L =∂Q
∂L = 10K−0.2LK2
Step 2: Similarly, to find the marginal product of capital (MPK), we differ-
entiate the production function with respect to capital K:
MP K =∂Q
∂K = 10L−0.2L2K
Step 3: To maximize output, we need to find the values of Land Kthat
satisfy the first-order conditions:
∂Q
∂L = 0 and ∂Q
∂K = 0
Step 4: Equating the MPL to zero:
10K−0.2LK2= 0
10 −0.2LK = 0
7
L=10
0.2K
Step 5: Substituting the expression for Linto the equation for MPK and
solving for K:
MP K = 10 10
0.2K−0.210
0.2K2
= 0
100 −2·100 = 0
100 −200 = 0
−100 = 0
Step 6: Since the equation −100 = 0 has no solutions, there seems to be an
error in the calculations. Double-check the derivatives and the steps to identify
and correct the mistake.
Question 9
Question
Consider a production function Q=LαKβ, where Qrepresents the output, L
represents labor input, Krepresents capital input, and αand βare positive
constants. Suppose the marginal product of labor is given by MPL=αK
Lβ.
Find the marginal product of capital MPK.
Solution
In order to find the marginal product of capital MPK, we need to differentiate
the production function with respect to K.
MPK=∂Q
∂K
=∂
∂K (LαKβ)
=βLαKβ−1
Given that the marginal product of labor MPL=αK
Lβ, we can express
K
Las K
L=MPL
α
1
β.
Substitute this into the equation for MPK:
MPK=βLαKβ−1
=βLα LM PL
α
1
β!β−1
=βLαLβ−1M PL
α
=βα β−1
βLα+β−1·MPL
8
Therefore, the marginal product of capital MPK=βα β−1
βLα+β−1·MPL.
Question 10
Question
Consider a production function Q=L1/3K2/3where Qis the total output, Lis
the quantity of labor input, and Kis the quantity of capital input. If the price
of labor is wand the price of capital is r, find the cost minimization condition
for a firm using this production function.
Solution
Let the total cost of the firm be given by C=wL +rK, where wis the wage
rate and ris the rental rate of capital.
Step 1:
To minimize the cost of production, the firm must choose the combination of
labor and capital that gives the firm the maximum output for a given level of
cost. Thus, the firm’s objective is to maximize output Qsubject to the cost
constraint C.
Step 2:
The firm’s problem can be formulated as:
Maximize Q=L1/3K2/3
Subject to C=wL +rK.
Step 3:
To solve this optimization problem, we first convert it to a single-variable prob-
lem by substituting the cost constraint into the production function:
Q=L1/3K2/3
=rK −wL
r1/3
K2/3
=(rK −wL)1/3K2/3
r.
Step 4:
Differentiating the production function with respect to Land setting it equal
to zero to find the critical point:
∂Q
∂L =−1
3rK −wL
r−2/3
·−w
rK2/3+2
3rK −wL
r1/3
K2/3
= 0.
9
Step 5:
Solving for the cost minimization condition leads to the condition:
w
r=1
2(K/L).
Therefore, the cost minimization condition for the firm is that the marginal
rate of technical substitution of capital for labor should be equal to the ratio of
input prices (w/r).
Question 11
Question
Suppose a firm has a production function given by Q= 4L0.5K0.5, where Q
is the quantity of output, Lis the quantity of labor, and Kis the quantity of
capital. If the firm is currently using 9 units of labor (L= 9), how many units
of capital (K) should the firm use to maximize output?
Solution
Step 1: To find the quantity of capital that maximizes output, we need to
differentiate the production function with respect to Kand set the derivative
equal to zero to find the critical point.
∂Q
∂K = 0.5×4L0.5K−0.5
∂Q
∂K = 2L0.5K−0.5
Setting this derivative equal to zero:
2L0.5K−0.5= 0
Step 2: Since L= 9 in this problem, we can substitute L= 9 into the
equation obtained in Step 1.
2(9)0.5K−0.5= 0
18K−0.5= 0
Step 3: Solving for K,
K−0.5= 0
1/K0.5= 0
K0.5=∞
K=∞
Step 4: From the calculations, we see that the quantity of capital required
for maximizing the output is infinite. In practice, this result may indicate that
the firm could continue to increase capital indefinitely without affecting output,
due to diminishing returns not being applied in this specific case.
10
Question 12
Question
Suppose a firm has a production function given by Q= 2L0.5K0.5, where Q
represents output, Lrepresents labor input, and Krepresents capital input. If
the firm currently has 16 units of capital, how many units of labor must be
employed to produce 64 units of output?
Solution
Step 1: We are given the production function Q= 2L0.5K0.5, and we know
that Q= 64 and K= 16. We can substitute these values into the production
function to determine the value of L.
64 = 2L0.5(16)0.5
64 = 2L0.5·4
16 = L0.5
Step 2: To find the value of L, we need to square both sides of the equation.
162= (L0.5)2
256 = L
Therefore, 256 units of labor must be employed to produce 64 units of output
when the firm has 16 units of capital.
Question 13
Question
Consider a production function given by Q= 5LK2, where Qrepresents the
quantity of output, Lrepresents the quantity of labor, and Krepresents the
quantity of capital. If the price of labor is w= 10 and the price of capital is
r= 5, find the cost-minimizing combination of labor and capital to produce 100
units of output.
Solution
Step 1: The cost of production is given by the total cost function:
C=wL +rK
Step 2: Since we need to produce 100 units of output, we can use the pro-
duction function to find the input combination (L, K):
100 = 5LK2
11
Step 3: We want to minimize the cost C= 10L+ 5Ksubject to the pro-
duction constraint 100 = 5LK2. To do so, we can first solve the production
constraint for Lin terms of K:
L=100
5K2=20
K2
Step 4: Substitute L=20
K2into the cost function C= 10L+ 5Kto get the
cost function in terms of K:
C(K) = 10 20
K2+ 5K
Step 5: Simplify the cost function to get:
C(K) = 200
K2+ 5K
Step 6: To minimize the cost, we need to find the critical points. Take the
derivative of the cost function with respect to Kand set it equal to 0:
C′(K) = −400
K3+ 5 = 0
Step 7: Solve the equation −400
K3+ 5 = 0 to find K. We have:
400
K3= 5 ⇒K3=400
5= 80 ⇒K=3
√80 = 4
Step 8: Substitute K= 4 back into L=20
K2to find L:
L=20
42=20
16 =5
4
Therefore, the cost-minimizing combination of labor and capital to produce
100 units of output is L=5
4and K= 4.
Question 14
Question
Consider a production function given by Q= 5L0.5K0.5, where Lrepresents
labor input and Krepresents capital input.
If the marginal product of labor is given by MPL= 2K0.5, find the marginal
product of capital MPK.
12
Solution
Step 1: Calculate the total product of labor (T PL) by taking the partial deriva-
tive of the production function with respect to L.
∂Q
∂L = 2.5L−0.5K0.5
Step 2: Calculate the marginal product of capital (MPK) by taking the
partial derivative of the total product of labor with respect to K.
MPK=∂
∂K (2.5L−0.5K0.5)=1.25L−0.5K−0.5
Step 3: Substitute the given MPLinto the expression for MPKto express
MPKin terms of MPLusing the production function.
MPK= 1.25 ·2K0.5
5L0.5−0.5
= 1.25 ·2−0.5·5
L
Step 4: Simplify the expression to find the final expression for MPK.
MPK= 0.625 ·5
L=3.125
L
Therefore, the marginal product of capital is given by M PK=3.125
Lin terms
of labor input L.
Question 15
Question
Consider a production function Q= 8L0.5K0.3, where Qis the total output, L
is the amount of labor, and Kis the amount of capital. If the price of labor is
wand the price of capital is r, find the cost-minimizing combination of labor
and capital to produce 64 units of output, given that w= 4 and r= 9.
Solution
Step 1: Set up the cost-minimization problem by defining the total cost function
C(w, r, L, K), which represents the cost of producing a given level of output Q
using labor Land capital K.
C(w, r, L, K) = wL +rK
Step 2: Substitute the given values of w,r, and Qinto the total cost function
to get C(L, K).
C(L, K)=4L+ 9K
13
Step 3: We also know the total output function Qand want to produce 64
units of output. Substitute Q= 64 into the production function to get:
64 = 8L0.5K0.3
Step 4: Solve for Lin terms of Kfrom the output equation:
L=64
8K0.32
=64
64K0.3=1
K0.3
Step 5: Substitute the expression for Lback into the cost function C(L, K)
from Step 2 to get the cost function in terms of Konly.
C(K)=41
K0.3+ 9K=4
K0.3+ 9K
Step 6: To find the minimum cost combination, take the derivative of the
cost function C(K) with respect to Kand set it equal to 0.
dC
dK =−1.2K−1.3+ 9 = 0
−1.2K−1.3=−9
K−1.3= 7.5
K= (7.5)−1/1.3
K≈2.695
Step 7: Substitute the value of Kback into the equation L=1
K0.3to find
L.
L=1
(2.695)0.3
L≈3.063
Therefore, the cost-minimizing combination of labor and capital to produce
64 units of output is approximately L≈3.063 and K≈2.695.
Question 16
Question
Let Q(L, K)=4L0.5K0.5be a production function where Lrepresents labor
and Krepresents capital. Determine the level of output Qwhen labor L= 16
and capital K= 25.
14
Solution
Step 1: Substitute L= 16 and K= 25 into the production function Q(L, K) =
4L0.5K0.5.Q(16,25) = 4(16)0.5(25)0.5
= 4(4)(5)
= 80
Therefore, when labor L= 16 and capital K= 25, the output level Qis 80.
Question 17
Question
Let Q=f(K, L) = K1/3L2/3represent a production function where Qis the
total output, Kis the amount of capital input, and Lis the amount of labor
input. Find the average product of labor and the marginal product of labor.
Solution
Step 1: To find the average product of labor, we divide the total output Q=
K1/3L2/3by the amount of labor input L. Step 2: Therefore, the average
product of labor (APL) is given by:
APL=Q
L=K1/3L2/3
L=K1/3L−1/3
Step 3: To find the marginal product of labor, we take the partial derivative
of the production function with respect to labor. That is, we find ∂Q
∂L . Step 4:
Taking the partial derivative, we have:
∂Q
∂L =2
3K1/3L−1/3
Therefore, the average product of labor is K1/3L−1/3and the marginal prod-
uct of labor is 2
3K1/3L−1/3.
Question 18
Question
Let Q=K3/4L1/4be the production function for a firm, where Qrepresents
output, Kis capital input, and Lis labor input. If the firm currently has 64
units of capital and 16 units of labor, how much will production increase if they
increase labor input by 25% while keeping capital input constant?
15
Solution
Step 1: Calculate the initial level of production with 64 units of capital and 16
units of labor:
Q= 643/4·161/4= 43·2 = 64
Step 2: Calculate the new level of production with 64 units of capital and
20 units of labor (after a 25% increase in labor input):
Q′= 643/4·201/4= 43·2.5 = 100
Step 3: Calculate the increase in production:
∆Q=Q′−Q= 100 −64 = 36
Therefore, the production will increase by 36 units if the firm increases labor
input by 25% while keeping capital input constant.
Question 19
Question
Suppose a firm has a production function given by Q= 5K0.5L0.5, where Q
represents the quantity of output, Kis the amount of capital, and Lis the
amount of labor. If the price of capital is r= 4 and the price of labor is w= 2,
what is the minimum cost of producing 100 units of output?
Solution
Step 1: To determine the minimum cost of producing 100 units of output, we
need to minimize the cost function C=rK +wL subject to the production
function constraint Q= 5K0.5L0.5and the given output requirement Q= 100.
Step 2: From the production function, we know that Q= 5K0.5L0.5. Sub-
stituting Q= 100 into the production function, we get:
100 = 5K0.5L0.5
Step 3: Solving for L, we have:
L=100
5K0.5= 20K−0.5
Step 4: Substituting the expression for Linto the cost function, we have:
C= 4K+ 2(20K−0.5)
Step 5: To find the minimum cost, differentiate Cwith respect to K, set the
derivative equal to zero, and solve for K. Let’s do that:
dC
dK = 4 −40K−1.5= 0
16
40K−1.5= 4
K−1.5= 0.1
K= 102/3
Step 6: Substitute K= 102/3back into the expression for Lto find the
corresponding amount of labor L:
L= 20(102/3)−0.5= 101/3
Step 7: Finally, calculate the minimum cost by plugging in the optimum
values of Kand Linto the cost function:
C= 4(102/3) + 2(101/3)
Therefore, the minimum cost of producing 100 units of output is 4 ×102/3+
2×101/3.
Question 20
Question
Consider a production function given by f(K, L) = K1
2L1
2. Find the marginal
product of labor.
Solution
Step 1: The marginal product of labor (MPL) is defined as the partial derivative
of the production function with respect to labor. Therefore, we need to find ∂f
∂L .
Step 2: Let’s find ∂f
∂L :
∂f
∂L =1
2K1
2·1
L1
2
=1
2·K1
2
L1
2
=1
2·√K
√L
Step 3: Therefore, the marginal product of labor is 1
2·√K
√L.
Question 21
Question
Consider a production function Q= 5L0.6K0.4, 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= 10 and the rental rate for capital is r= 20,
find the cost-minimizing combination of labor and capital inputs to produce 100
units of output.
17
Solution
Step 1: The cost of production is given by the cost of labor (wL) plus the cost
of capital (rK).
Step 2: We can write the cost function as C=wL +rK.
Step 3: To find the cost-minimizing combination of inputs, we need to min-
imize the cost function subject to the production function constraint and the
output level constraint.
Step 4: The Lagrangian function is given by L=wL +rK +λ(100 −
5L0.6K0.4).
Step 5: Taking partial derivatives with respect to L,K, and λ, we have:
∂L
∂L :w−3λL0.6K0.4= 0
∂L
∂K :r−2λL0.6K0.4= 0
∂L
∂λ : 100 −5L0.6K0.4= 0
Step 6: Solving the system of equations, we find L= 10 and K= 25.
Therefore, the cost-minimizing combination of labor and capital inputs to
produce 100 units of output is 10 units of labor and 25 units of capital.
Question 22
Question
Consider the production function Q(K, L) = K0.4L0.6, where Qis the output
quantity, Kis the capital input, and Lis the labor input. If the price of capital
is r= 3 and the price of labor is w= 2, find the cost-minimizing combination
of capital and labor inputs required to produce 100 units of output.
Solution
Step 1: Set up the cost minimization problem.
Let C=rK +wL be the total cost function. The cost minimization problem
is to minimize Csubject to the constraint Q(K, L) = 100:
Minimize C= 3K+ 2L
Subject to K0.4L0.6= 100
Step 2: Find the Lagrange function.
Define the Lagrange function J(K, L, λ):
J(K, L, λ)=3K+ 2L+λ(100 −K0.4L0.6)
Step 3: Find the first-order conditions.
Calculate the partial derivatives of Jwith respect to K, L, and λ:
18
∂J
∂K = 3 −0.4λK−0.6L0.6= 0
∂J
∂L = 2 −0.6λK0.4L−0.4= 0
∂J
∂λ = 100 −K0.4L0.6= 0
Step 4: Solve the first-order conditions.
From the first two FOCs, we have:
0.4λK−0.6L0.6= 3
0.6λK0.4L−0.4= 2
Dividing these equations gives:
0.4λK−0.6L0.6
0.6λK0.4L−0.4=3
2
Solving for K
L, we get:
L
K=2
3
4
5
Step 5: Calculate the optimal input combination.
Substitute K
L=2
3
4
5into the constraint K0.4L0.6= 100 to find the optimal
values of Kand L.
K0.4
K
2
3
4
5
0.6
= 100
K0.43
2
24
25
K0.6= 100
K= 100 ·2
3
24
25 !
1
2
Similarly, find the value of Lusing K
L=2
3
4
5.
Therefore, the cost-minimizing combination of capital and labor inputs re-
quired to produce 100 units of output can be calculated using the above expres-
sions.
Question 23
Question
Consider a firm with a production function given by Q=L0.5K0.5, where Q
represents quantity produced, Lrepresents labor input, and Krepresents capital
19
input. The firm currently has 10 units of capital. If the marginal product of
labor is 4, what is the firm’s current level of labor input?
Solution
Step 1: Find the expression for the marginal product of labor. The marginal
product of labor (MP L) is given by the derivative of the production function
with respect to labor:
MP L =∂Q
∂L = 0.5L−0.5K0.5
Step 2: Substitute the values given in the question into the expression for
MP L. Given that K= 10 and M P L = 4, we have:
4=0.5L−0.5(10)0.5= 0.5L−0.5(3)
Step 3: Solve for L.
4=1.5L−0.5
L−0.5=4
1.5=8
3
L= (8
3)−2= (3
8)2=9
64
Therefore, the firm’s current level of labor input is 9
64 .
Question 24
Question
Consider a production function f(x, y) = x2+ 4xy +y2. Find the marginal
product of xand the marginal product of y.
Solution
To find the marginal product of x, we need to calculate the partial derivative of
the production function f(x, y) with respect to x. Similarly, to find the marginal
product of y, we calculate the partial derivative of f(x, y) with respect to y.
Step 1: Find the marginal product of x To find the marginal product
of x, we calculate the partial derivative of f(x, y) with respect to x:
∂f
∂x =∂
∂x(x2+ 4xy +y2)
= 2x+ 4y
Therefore, the marginal product of xis 2x+ 4y.
20
Step 2: Find the marginal product of y To find the marginal product
of y, we calculate the partial derivative of f(x, y) with respect to y:
∂f
∂y =∂
∂y (x2+ 4xy +y2)
= 4x+ 2y
Therefore, the marginal product of yis 4x+ 2y.
Question 25
Question
Given a production function Q= 4KL −3K2, where Qis the total output,
Kis the quantity of capital, and Lis the quantity of labor, find the marginal
products of labor and capital.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function Qwith respect to L, keeping Kconstant:
∂Q
∂L = 4K−3K2.
Step 2: This gives us the marginal product of labor:
MPL=∂Q
∂L = 4K−3K2.
Step 3: To find the marginal product of capital (MPK), we differentiate the
production function Qwith respect to K, keeping Lconstant:
∂Q
∂K = 4L−6K.
Step 4: This gives us the marginal product of capital:
MPK=∂Q
∂K = 4L−6K.
Therefore, the marginal product of labor is MPL= 4K−3K2, and the
marginal product of capital is MPK= 4L−6K.
Question 26
Question
Suppose a firm’s production function is given by Q= 4L0.5K0.5, where Lrep-
resents labor input and Krepresents capital input. If the firm has 16 units of
capital, what amount of labor input is required to produce 64 units of output?
21
Solution
Step 1: Start by substituting the given values (Q= 64 and K= 16) into the
production function Q= 4L0.5K0.5. Substitute Q= 64 and K= 16 into the
production function:
64 = 4L0.5(16)0.5
Step 2: Simplify the expression by evaluating (16)0.5.
64 = 4L0.5(4)
Step 3: Divide both sides by 4 to solve for L.
16 = L0.5(4)
Step 4: Divide both sides by 4 to solve for L.
4 = L0.5
Step 5: Square both sides to solve for L.
16 = L
Therefore, the firm would need 16 units of labor input to produce 64 units
of output.
Question 27
Question
Let qbe the output produced by a firm and Lbe the quantity of labor input.
The production function for the firm is given by q= 5L2−2L3. Calculate the
marginal product of labor when L= 4 and determine whether the production
function exhibits increasing, decreasing, or constant returns to scale in this case.
Solution
Step 1: Calculate the marginal product of labor. To find the marginal product
of labor (MPL), we need to take the derivative of the production function with
respect to labor, dq
dL .
dq
dL =d
dL(5L2−2L3)
dq
dL = 10L−6L2
Step 2: Substituting L= 4 to find the marginal product of labor. Now,
substitute L= 4 into the derivative we found in Step 1 to get the marginal
product of labor at L= 4.
dq
dL = 10 ×4−6×42
22
dq
dL = 40 −96
dq
dL =−56
Therefore, the marginal product of labor when L= 4 is −56.
Step 3: Determine returns to scale. To determine the returns to scale,
we need to examine the behavior of the production function with respect to
changes in the scale of input. Since the exponent of labor is greater than 1
in the production function q= 5L2−2L3, the production function exhibits
decreasing returns to scale.
Question 28
Question
Consider the production function Q= 5L0.5K0.5, where Qrepresents the quan-
tity of output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. Given that the price of labor is w= 10 and the price of capital is
r= 20, what is the cost-minimizing input combination if the firm wants to
produce 100 units of output?
Solution
Step 1: The cost-minimizing condition for the firm is that the marginal rate of
technical substitution (MRTS) equals the ratio of input prices. The MRTS is
given by the ratio of the marginal products of labor and capital:
MRT S =M P L
MP K =5×0.5L−0.5K0.5
5×L0.5K−0.5=0.5
0.5= 1
Step 2: Given that MRT S =w
r, we have 1 = 10
20 . This implies that the
cost-minimizing input combination is L:K= 1 : 2.
Step 3: To produce 100 units of output, we can use the production function
Q= 5L0.5K0.5:
100 = 5L0.5K0.5
Step 4: Substituting the cost-minimizing input combination into the pro-
duction function, we get:
100 = 5(1)0.5(2)0.5= 5√2
Thus, the cost-minimizing input combination for producing 100 units of output
is L= 1 and K= 2.
23
Question 29
Question
Consider a production function given by f(K, L) = K1
3·L2
3, where Krepresents
capital input and Lrepresents labor input. Find the marginal product of labor.
Solution
Step 1: To find the marginal product of labor (MPL), we must first find the
partial derivative of the production function with respect to labor (L). This is
represented by ∂f
∂L .
Step 2: Taking the partial derivative of the production function f(K, L) =
K1
3·L2
3with respect to L, we get:
∂f
∂L =2
3K1
3·L−1
3
Step 3: This expression represents the marginal product of labor: MPL=
2
3K1
3·L−1
3.
Question 30
Question
Suppose a firm’s production function is given by Q=K0.4L0.6, where Qis the
total output, Kis the amount of capital used, and Lis the amount of labor
employed. If the firm currently has K= 10 units of capital and L= 5 units
of labor, find the rate at which output is changing with respect to labor when
L= 5.
Solution
Step 1: Calculate the marginal product of labor (MPL)
The marginal product of labor (MPL) is given by the partial derivative of the
production function with respect to labor:
MP L =∂Q
∂L = 0.6K0.4L−0.4
Step 2: Substitute the given values K= 10 and L= 5 into the MPL formula:
MP L(10,5) = 0.6(10)0.4(5)−0.4
MP L(10,5) = 0.6(2.5118864315)(0.632455532)
MP L(10,5) ≈0.947
24
Step 3: Calculate the rate at which output is changing with respect to labor
The rate at which output is changing with respect to labor is given by the MPL:
dQ
dL =MP L(10,5) = 0.947
Therefore, the rate at which output is changing with respect to labor when
L= 5 is approximately 0.947 units of output per unit of labor.
Question 31
Question
Let f(L, K)=4L1/3K2/3represent a production function where Lis the amount
of labor input and Kis the amount of capital input. Determine the marginal
product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we need to calculate the par-
tial derivative of the production function with respect to labor, holding capital
constant. ∂f
∂L =4
3L−2/3K2/3
Step 2: Simplifying the expression, we get:
∂f
∂L =4
3K
L2/3
Step 3: Therefore, the marginal product of labor is 4
3K
L2/3.
Step 4: Next, to find the marginal product of capital, we need to calculate
the partial derivative of the production function with respect to capital, holding
labor constant. ∂f
∂K =8
3L1/3K−1/3
Step 5: Simplifying the expression, we get:
∂f
∂K =8
3L
K1/3
Step 6: Therefore, the marginal product of capital is 8
3L
K1/3.
25
Question 32
Question
Suppose a firm has the following production function: Q(K, L) = KαLβ, where
Qis the quantity of output, Kis the quantity of capital, Lis the quantity of
labor, and α, β > 0. Show that the firm’s production function exhibits constant
returns to scale.
Solution
Step 1: To determine if the production function exhibits constant returns to
scale, we need to check how the output changes when both capital and labor
are scaled up by a factor λ > 0. Let’s examine the output Q(λK, λL):
Q(λK, λL)=(λK)α(λL)β
=λαKαλβLβ
=λα+βKαLβ
=λα+βQ(K, L)
Step 2: Since α > 0 and β > 0, we have α+β > 0. Thus, when both capital
and labor are scaled up by a factor λ, the output is scaled up by a factor of
λα+β.
Step 3: A production function exhibits constant returns to scale if the output
is scaled up by the same proportion as the inputs. Since α+β > 0, we can
conclude that the firm’s production function Q(K, L) = KαLβexhibits constant
returns to scale.
Question 33
Question
Consider a production function given by Q=L0.7K0.3, where Qrepresents the
quantity of output, Lrepresents labor input, and Krepresents capital input.
Determine the marginal product of labor and the marginal product of capital.
Solution
Let’s first find the marginal product of labor (MPL) by taking the partial deriva-
tive of the production function with respect to labor input L, holding capital
input Kconstant.
MPL=∂Q
∂L = 0.7L−0.3K0.3
Step 1: Calculate the marginal product of labor.
MPL= 0.7L−0.3K0.3
26
Next, let’s find the marginal product of capital (MPK) by taking the partial
derivative of the production function with respect to capital input K, holding
labor input Lconstant.
MPK=∂Q
∂K = 0.3L0.7K−0.7
Step 2: Calculate the marginal product of capital.
MPK= 0.3L0.7K−0.7
Therefore, the marginal product of labor is MPL= 0.7L−0.3K0.3and the
marginal product of capital is MPK= 0.3L0.7K−0.7.
Question 34
Question
Suppose a production function is given by Q= 4L3/4K1/4, where Qis the
output, Lis the amount of labor input, and Kis the amount of capital input.
If the amount of capital input is fixed at K= 16, find the marginal product of
labor.
Solution
To find the marginal product of labor, we need to calculate the partial derivative
of the production function with respect to labor.
Step 1: Calculate the total product of labor:
Q= 4L3/4K1/4
Step 2: Substitute K= 16 into the production function:
Q= 4L3/4161/4
Step 3: Simplify the expression:
Q= 4L3/42
Step 4: Calculate the total product of labor by expanding the expression:
Q= 8L3/4
Step 5: Calculate the marginal product of labor (MPL) by taking the
derivative of the total product with respect to labor:
MPL=dQ
dL =d
dL(8L3/4)
27
Step 6: Differentiate 8L3/4with respect to L:
MPL=3
4·8L−1/4
Step 7: Simplify the expression for the marginal product of labor:
MPL= 6L−1/4=6
4
√L
Therefore, the marginal product of labor is 6
4
√L.
Question 35
Question
Let Qbe the quantity of output and Lbe the quantity of labor input in a
production process described by the production function Q= 10L−0.1L2. Find
the level of labor input that maximizes output and determine the corresponding
maximum level of output.
Solution
Step 1: To find the level of labor input that maximizes output, we first need
to find the critical points of the production function. These occur where the
derivative of the production function is equal to zero.
Step 1: Q= 10L−0.1L2
Step 2: dQ
dL = 10 −0.2L
Step 3: Set dQ
dL = 0 to find critical points
Step 4: 10 −0.2L= 0
Step 5: Solve for L:L=10
0.2= 50
So, the critical point is L= 50.
Step 2: To determine if this critical point corresponds to a maximum, we
will use the second derivative test.
Step 6: d2Q
dL2=−0.2
Step 7: Evaluate the second derivative at the critical point L= 50
Step 8: d2Q
dL2L=50 =−0.2<0
28
Question 2
Question
Consider a production function given by Q= 2L0.5K0.5where Qis the quantity
produced, Lis the quantity of labor, and Kis the quantity of capital. Deter-
mine the marginal product of labor (M PL) 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.
∂Q
∂L =∂(2L0.5K0.5)
∂L
Step 2: Taking the partial derivative with respect to L, we get:
∂Q
∂L = 1L−0.5K0.5
Step 3: Simplifying the expression, we find the marginal product of labor:
MPL=K0.5
L0.5
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.∂Q
∂K =∂(2L0.5K0.5)
∂K
Step 5: Taking the partial derivative with respect to K, we get:
∂Q
∂K = 1L0.5K−0.5
Step 6: Simplifying the expression, we find the marginal product of capital:
MPK=L0.5
K0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=L0.5
K0.5.
Question 3
Question
Consider a production function given by Q= 10L3/4K1/4, where Qis the
quantity of output, Lis the quantity of labor input, and Kis 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.
2
Solution
Step 1: The cost of production (C) can be expressed as C=wL +rK, where
wis the price of labor and ris the price of capital.
Step 2: To minimize cost, we need to minimize the cost function Csubject
to the constraint that the production function Q= 10L3/4K1/4produces 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:
4 + 3
4λL−7/4K−1/4= 0
2 + 1
4λL−3/4K−5/4= 0
100L−3/4K−1/4−10 = 0
Step 6: Solve the system of equations to find the values of Land K. Once
you have found Land K, substitute these values back into the cost function
C= 4L+ 2Kto calculate the minimum cost of producing 100 units of output.
Question 4
Question
Consider a production function Q= 2L0.5K0.5, where Qis the quantity of
output produced, Lis the quantity of labor input, and Kis 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 Qwith respect to L:
MPL=∂Q
∂L = 0.5×2×L−0.5×K0.5=K0.5
Step 2: Substitute L= 16 and K= 25 into the formula to find MPL:
MPL= 250.5= 5
3
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 Qwith respect to K:
MPK=∂Q
∂K = 0.5×2×L0.5×K−0.5=L0.5
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 Qdenote the total output of a firm and Kand Lrepresent 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 T V C =wL.
Step 3: To determine the equation for the firm’s TVC in terms of L, we need
to find the expression for wfirst.
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 (MP L): w=∂Q/∂L
MP L .
Step 5: Calculate the partial derivative of Qwith respect to L:
∂Q
∂L = 0.6K0.4L−0.4
= 0.6K0.4
L0.4
4
Step 6: Calculate the marginal product of labor (M P L) by taking the deriva-
tive of the production function with respect to L:
MP L =∂Q
∂L = 0.6K0.4L−0.4
= 0.6K0.4
L0.4
Step 7: Now, we can find the wage rate (w):
w=
0.6K0.4
L0.4
0.6K0.4
L0.4= 1
Step 8: Substitute w= 1 back into the TVC function T V C =wL to get the
TVC function in terms of L:
T V C = 1 ×L=L
Step 9: Therefore, the equation for the firm’s total variable cost function in
terms of Lis T V C =L.
Question 6
Question
Suppose a firm uses the production function Q=L1/3K1/2to produce output,
where Qis the quantity of output, Lis the quantity of labor, and Kis the
quantity of capital. If the firm currently has 16 units of capital and is producing
64 units of output, determine:
1. The marginal product of labor.
2. The average product of labor.
3. The rate at which the firm can substitute labor for capital while keeping
output constant.
Solution
1. To find the marginal product of labor, we need to calculate the derivative of
the production function with respect to labor L:
dQ
dL =1
3L−2/3K1/2
Given that L= 643/2and K= 16, we can substitute the values into the
derivative to find the marginal product of labor.
5
2. To find the average product of labor, we need to divide the total product
by the quantity of labor. The average product of labor (AP L) can be calculated
as:
AP L =Q
L=L1/3K1/2
L
Substitute the given values of Land Kin to find AP L.
3. The rate at which the firm can substitute labor for capital while keeping
output constant can be found using the MRTS (Marginal Rate of Technical
Substitution) formula:
MRT S =−M PL
MPK
Calculate the marginal products of labor and capital individually, then substi-
tute them into the MRTS formula to find the rate at which labor and capital
can be substituted while keeping output constant.
Question 7
Question
Suppose a production function is given by Q= 4K0.5L0.5. If the wage rate is
w= 10 and the rental rate of capital is r= 5, find the minimum cost to produce
100 units of output.
Solution
Step 1: Given the production function Q= 4K0.5L0.5, we need to minimize the
cost to produce 100 units of output. The cost function is given by C=wL+rK.
Step 2: We know that Q= 100, so substitute this into the production
function to get: 100 = 4K0.5L0.5.
Step 3: Rewrite the production function in terms of K:
K=100
4L2
=625
L2
Step 4: Substitute the value of Kback into the cost function:
C= 10L+ 5 625
L2
Step 5: To minimize cost, take the derivative of the cost function with respect
to Land set it equal to zero:
dC
dL = 10 −10 625
L3= 0
Step 6: Solve for L:
10 = 6250
L3
6
L3= 625
L= 5
Step 7: Substitute the value of Lback into the equation for K:
K=625
52= 25
Step 8: Substitute the values of Kand Lback into the cost function to find
the minimum cost:
C= 10(5) + 5(25) = 50 + 125 = 175
Therefore, the minimum cost to produce 100 units of output is 175.
Question 8
Question
Consider the production function Q= 10LK −0.1L2K2, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK). Then, find the values of Land Kthat maximize output.
Solution
Step 1: To find the marginal product of labor (MPL), we need to differentiate
the production function with respect to labor L:
MP L =∂Q
∂L = 10K−0.2LK2
Step 2: Similarly, to find the marginal product of capital (MPK), we differ-
entiate the production function with respect to capital K:
MP K =∂Q
∂K = 10L−0.2L2K
Step 3: To maximize output, we need to find the values of Land Kthat
satisfy the first-order conditions:
∂Q
∂L = 0 and ∂Q
∂K = 0
Step 4: Equating the MPL to zero:
10K−0.2LK2= 0
10 −0.2LK = 0
7
L=10
0.2K
Step 5: Substituting the expression for Linto the equation for MPK and
solving for K:
MP K = 10 10
0.2K−0.210
0.2K2
= 0
100 −2·100 = 0
100 −200 = 0
−100 = 0
Step 6: Since the equation −100 = 0 has no solutions, there seems to be an
error in the calculations. Double-check the derivatives and the steps to identify
and correct the mistake.
Question 9
Question
Consider a production function Q=LαKβ, where Qrepresents the output, L
represents labor input, Krepresents capital input, and αand βare positive
constants. Suppose the marginal product of labor is given by MPL=αK
Lβ.
Find the marginal product of capital MPK.
Solution
In order to find the marginal product of capital MPK, we need to differentiate
the production function with respect to K.
MPK=∂Q
∂K
=∂
∂K (LαKβ)
=βLαKβ−1
Given that the marginal product of labor MPL=αK
Lβ, we can express
K
Las K
L=MPL
α
1
β.
Substitute this into the equation for MPK:
MPK=βLαKβ−1
=βLα LM PL
α
1
β!β−1
=βLαLβ−1M PL
α
=βα β−1
βLα+β−1·MPL
8
Therefore, the marginal product of capital MPK=βα β−1
βLα+β−1·MPL.
Question 10
Question
Consider a production function Q=L1/3K2/3where Qis the total output, Lis
the quantity of labor input, and Kis the quantity of capital input. If the price
of labor is wand the price of capital is r, find the cost minimization condition
for a firm using this production function.
Solution
Let the total cost of the firm be given by C=wL +rK, where wis the wage
rate and ris the rental rate of capital.
Step 1:
To minimize the cost of production, the firm must choose the combination of
labor and capital that gives the firm the maximum output for a given level of
cost. Thus, the firm’s objective is to maximize output Qsubject to the cost
constraint C.
Step 2:
The firm’s problem can be formulated as:
Maximize Q=L1/3K2/3
Subject to C=wL +rK.
Step 3:
To solve this optimization problem, we first convert it to a single-variable prob-
lem by substituting the cost constraint into the production function:
Q=L1/3K2/3
=rK −wL
r1/3
K2/3
=(rK −wL)1/3K2/3
r.
Step 4:
Differentiating the production function with respect to Land setting it equal
to zero to find the critical point:
∂Q
∂L =−1
3rK −wL
r−2/3
·−w
rK2/3+2
3rK −wL
r1/3
K2/3
= 0.
9
Step 5:
Solving for the cost minimization condition leads to the condition:
w
r=1
2(K/L).
Therefore, the cost minimization condition for the firm is that the marginal
rate of technical substitution of capital for labor should be equal to the ratio of
input prices (w/r).
Question 11
Question
Suppose a firm has a production function given by Q= 4L0.5K0.5, where Q
is the quantity of output, Lis the quantity of labor, and Kis the quantity of
capital. If the firm is currently using 9 units of labor (L= 9), how many units
of capital (K) should the firm use to maximize output?
Solution
Step 1: To find the quantity of capital that maximizes output, we need to
differentiate the production function with respect to Kand set the derivative
equal to zero to find the critical point.
∂Q
∂K = 0.5×4L0.5K−0.5
∂Q
∂K = 2L0.5K−0.5
Setting this derivative equal to zero:
2L0.5K−0.5= 0
Step 2: Since L= 9 in this problem, we can substitute L= 9 into the
equation obtained in Step 1.
2(9)0.5K−0.5= 0
18K−0.5= 0
Step 3: Solving for K,
K−0.5= 0
1/K0.5= 0
K0.5=∞
K=∞
Step 4: From the calculations, we see that the quantity of capital required
for maximizing the output is infinite. In practice, this result may indicate that
the firm could continue to increase capital indefinitely without affecting output,
due to diminishing returns not being applied in this specific case.
10
Question 12
Question
Suppose a firm has a production function given by Q= 2L0.5K0.5, where Q
represents output, Lrepresents labor input, and Krepresents capital input. If
the firm currently has 16 units of capital, how many units of labor must be
employed to produce 64 units of output?
Solution
Step 1: We are given the production function Q= 2L0.5K0.5, and we know
that Q= 64 and K= 16. We can substitute these values into the production
function to determine the value of L.
64 = 2L0.5(16)0.5
64 = 2L0.5·4
16 = L0.5
Step 2: To find the value of L, we need to square both sides of the equation.
162= (L0.5)2
256 = L
Therefore, 256 units of labor must be employed to produce 64 units of output
when the firm has 16 units of capital.
Question 13
Question
Consider a production function given by Q= 5LK2, where Qrepresents the
quantity of output, Lrepresents the quantity of labor, and Krepresents the
quantity of capital. If the price of labor is w= 10 and the price of capital is
r= 5, find the cost-minimizing combination of labor and capital to produce 100
units of output.
Solution
Step 1: The cost of production is given by the total cost function:
C=wL +rK
Step 2: Since we need to produce 100 units of output, we can use the pro-
duction function to find the input combination (L, K):
100 = 5LK2
11
Step 3: We want to minimize the cost C= 10L+ 5Ksubject to the pro-
duction constraint 100 = 5LK2. To do so, we can first solve the production
constraint for Lin terms of K:
L=100
5K2=20
K2
Step 4: Substitute L=20
K2into the cost function C= 10L+ 5Kto get the
cost function in terms of K:
C(K) = 10 20
K2+ 5K
Step 5: Simplify the cost function to get:
C(K) = 200
K2+ 5K
Step 6: To minimize the cost, we need to find the critical points. Take the
derivative of the cost function with respect to Kand set it equal to 0:
C′(K) = −400
K3+ 5 = 0
Step 7: Solve the equation −400
K3+ 5 = 0 to find K. We have:
400
K3= 5 ⇒K3=400
5= 80 ⇒K=3
√80 = 4
Step 8: Substitute K= 4 back into L=20
K2to find L:
L=20
42=20
16 =5
4
Therefore, the cost-minimizing combination of labor and capital to produce
100 units of output is L=5
4and K= 4.
Question 14
Question
Consider a production function given by Q= 5L0.5K0.5, where Lrepresents
labor input and Krepresents capital input.
If the marginal product of labor is given by MPL= 2K0.5, find the marginal
product of capital MPK.
12
Solution
Step 1: Calculate the total product of labor (T PL) by taking the partial deriva-
tive of the production function with respect to L.
∂Q
∂L = 2.5L−0.5K0.5
Step 2: Calculate the marginal product of capital (MPK) by taking the
partial derivative of the total product of labor with respect to K.
MPK=∂
∂K (2.5L−0.5K0.5)=1.25L−0.5K−0.5
Step 3: Substitute the given MPLinto the expression for MPKto express
MPKin terms of MPLusing the production function.
MPK= 1.25 ·2K0.5
5L0.5−0.5
= 1.25 ·2−0.5·5
L
Step 4: Simplify the expression to find the final expression for MPK.
MPK= 0.625 ·5
L=3.125
L
Therefore, the marginal product of capital is given by M PK=3.125
Lin terms
of labor input L.
Question 15
Question
Consider a production function Q= 8L0.5K0.3, where Qis the total output, L
is the amount of labor, and Kis the amount of capital. If the price of labor is
wand the price of capital is r, find the cost-minimizing combination of labor
and capital to produce 64 units of output, given that w= 4 and r= 9.
Solution
Step 1: Set up the cost-minimization problem by defining the total cost function
C(w, r, L, K), which represents the cost of producing a given level of output Q
using labor Land capital K.
C(w, r, L, K) = wL +rK
Step 2: Substitute the given values of w,r, and Qinto the total cost function
to get C(L, K).
C(L, K)=4L+ 9K
13
Step 3: We also know the total output function Qand want to produce 64
units of output. Substitute Q= 64 into the production function to get:
64 = 8L0.5K0.3
Step 4: Solve for Lin terms of Kfrom the output equation:
L=64
8K0.32
=64
64K0.3=1
K0.3
Step 5: Substitute the expression for Lback into the cost function C(L, K)
from Step 2 to get the cost function in terms of Konly.
C(K)=41
K0.3+ 9K=4
K0.3+ 9K
Step 6: To find the minimum cost combination, take the derivative of the
cost function C(K) with respect to Kand set it equal to 0.
dC
dK =−1.2K−1.3+ 9 = 0
−1.2K−1.3=−9
K−1.3= 7.5
K= (7.5)−1/1.3
K≈2.695
Step 7: Substitute the value of Kback into the equation L=1
K0.3to find
L.
L=1
(2.695)0.3
L≈3.063
Therefore, the cost-minimizing combination of labor and capital to produce
64 units of output is approximately L≈3.063 and K≈2.695.
Question 16
Question
Let Q(L, K)=4L0.5K0.5be a production function where Lrepresents labor
and Krepresents capital. Determine the level of output Qwhen labor L= 16
and capital K= 25.
14
Solution
Step 1: Substitute L= 16 and K= 25 into the production function Q(L, K) =
4L0.5K0.5.Q(16,25) = 4(16)0.5(25)0.5
= 4(4)(5)
= 80
Therefore, when labor L= 16 and capital K= 25, the output level Qis 80.
Question 17
Question
Let Q=f(K, L) = K1/3L2/3represent a production function where Qis the
total output, Kis the amount of capital input, and Lis the amount of labor
input. Find the average product of labor and the marginal product of labor.
Solution
Step 1: To find the average product of labor, we divide the total output Q=
K1/3L2/3by the amount of labor input L. Step 2: Therefore, the average
product of labor (APL) is given by:
APL=Q
L=K1/3L2/3
L=K1/3L−1/3
Step 3: To find the marginal product of labor, we take the partial derivative
of the production function with respect to labor. That is, we find ∂Q
∂L . Step 4:
Taking the partial derivative, we have:
∂Q
∂L =2
3K1/3L−1/3
Therefore, the average product of labor is K1/3L−1/3and the marginal prod-
uct of labor is 2
3K1/3L−1/3.
Question 18
Question
Let Q=K3/4L1/4be the production function for a firm, where Qrepresents
output, Kis capital input, and Lis labor input. If the firm currently has 64
units of capital and 16 units of labor, how much will production increase if they
increase labor input by 25% while keeping capital input constant?
15
Solution
Step 1: Calculate the initial level of production with 64 units of capital and 16
units of labor:
Q= 643/4·161/4= 43·2 = 64
Step 2: Calculate the new level of production with 64 units of capital and
20 units of labor (after a 25% increase in labor input):
Q′= 643/4·201/4= 43·2.5 = 100
Step 3: Calculate the increase in production:
∆Q=Q′−Q= 100 −64 = 36
Therefore, the production will increase by 36 units if the firm increases labor
input by 25% while keeping capital input constant.
Question 19
Question
Suppose a firm has a production function given by Q= 5K0.5L0.5, where Q
represents the quantity of output, Kis the amount of capital, and Lis the
amount of labor. If the price of capital is r= 4 and the price of labor is w= 2,
what is the minimum cost of producing 100 units of output?
Solution
Step 1: To determine the minimum cost of producing 100 units of output, we
need to minimize the cost function C=rK +wL subject to the production
function constraint Q= 5K0.5L0.5and the given output requirement Q= 100.
Step 2: From the production function, we know that Q= 5K0.5L0.5. Sub-
stituting Q= 100 into the production function, we get:
100 = 5K0.5L0.5
Step 3: Solving for L, we have:
L=100
5K0.5= 20K−0.5
Step 4: Substituting the expression for Linto the cost function, we have:
C= 4K+ 2(20K−0.5)
Step 5: To find the minimum cost, differentiate Cwith respect to K, set the
derivative equal to zero, and solve for K. Let’s do that:
dC
dK = 4 −40K−1.5= 0
16
40K−1.5= 4
K−1.5= 0.1
K= 102/3
Step 6: Substitute K= 102/3back into the expression for Lto find the
corresponding amount of labor L:
L= 20(102/3)−0.5= 101/3
Step 7: Finally, calculate the minimum cost by plugging in the optimum
values of Kand Linto the cost function:
C= 4(102/3) + 2(101/3)
Therefore, the minimum cost of producing 100 units of output is 4 ×102/3+
2×101/3.
Question 20
Question
Consider a production function given by f(K, L) = K1
2L1
2. Find the marginal
product of labor.
Solution
Step 1: The marginal product of labor (MPL) is defined as the partial derivative
of the production function with respect to labor. Therefore, we need to find ∂f
∂L .
Step 2: Let’s find ∂f
∂L :
∂f
∂L =1
2K1
2·1
L1
2
=1
2·K1
2
L1
2
=1
2·√K
√L
Step 3: Therefore, the marginal product of labor is 1
2·√K
√L.
Question 21
Question
Consider a production function Q= 5L0.6K0.4, 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= 10 and the rental rate for capital is r= 20,
find the cost-minimizing combination of labor and capital inputs to produce 100
units of output.
17
Solution
Step 1: The cost of production is given by the cost of labor (wL) plus the cost
of capital (rK).
Step 2: We can write the cost function as C=wL +rK.
Step 3: To find the cost-minimizing combination of inputs, we need to min-
imize the cost function subject to the production function constraint and the
output level constraint.
Step 4: The Lagrangian function is given by L=wL +rK +λ(100 −
5L0.6K0.4).
Step 5: Taking partial derivatives with respect to L,K, and λ, we have:
∂L
∂L :w−3λL0.6K0.4= 0
∂L
∂K :r−2λL0.6K0.4= 0
∂L
∂λ : 100 −5L0.6K0.4= 0
Step 6: Solving the system of equations, we find L= 10 and K= 25.
Therefore, the cost-minimizing combination of labor and capital inputs to
produce 100 units of output is 10 units of labor and 25 units of capital.
Question 22
Question
Consider the production function Q(K, L) = K0.4L0.6, where Qis the output
quantity, Kis the capital input, and Lis the labor input. If the price of capital
is r= 3 and the price of labor is w= 2, find the cost-minimizing combination
of capital and labor inputs required to produce 100 units of output.
Solution
Step 1: Set up the cost minimization problem.
Let C=rK +wL be the total cost function. The cost minimization problem
is to minimize Csubject to the constraint Q(K, L) = 100:
Minimize C= 3K+ 2L
Subject to K0.4L0.6= 100
Step 2: Find the Lagrange function.
Define the Lagrange function J(K, L, λ):
J(K, L, λ)=3K+ 2L+λ(100 −K0.4L0.6)
Step 3: Find the first-order conditions.
Calculate the partial derivatives of Jwith respect to K, L, and λ:
18
∂J
∂K = 3 −0.4λK−0.6L0.6= 0
∂J
∂L = 2 −0.6λK0.4L−0.4= 0
∂J
∂λ = 100 −K0.4L0.6= 0
Step 4: Solve the first-order conditions.
From the first two FOCs, we have:
0.4λK−0.6L0.6= 3
0.6λK0.4L−0.4= 2
Dividing these equations gives:
0.4λK−0.6L0.6
0.6λK0.4L−0.4=3
2
Solving for K
L, we get:
L
K=2
3
4
5
Step 5: Calculate the optimal input combination.
Substitute K
L=2
3
4
5into the constraint K0.4L0.6= 100 to find the optimal
values of Kand L.
K0.4
K
2
3
4
5
0.6
= 100
K0.43
2
24
25
K0.6= 100
K= 100 ·2
3
24
25 !
1
2
Similarly, find the value of Lusing K
L=2
3
4
5.
Therefore, the cost-minimizing combination of capital and labor inputs re-
quired to produce 100 units of output can be calculated using the above expres-
sions.
Question 23
Question
Consider a firm with a production function given by Q=L0.5K0.5, where Q
represents quantity produced, Lrepresents labor input, and Krepresents capital
19
input. The firm currently has 10 units of capital. If the marginal product of
labor is 4, what is the firm’s current level of labor input?
Solution
Step 1: Find the expression for the marginal product of labor. The marginal
product of labor (MP L) is given by the derivative of the production function
with respect to labor:
MP L =∂Q
∂L = 0.5L−0.5K0.5
Step 2: Substitute the values given in the question into the expression for
MP L. Given that K= 10 and M P L = 4, we have:
4=0.5L−0.5(10)0.5= 0.5L−0.5(3)
Step 3: Solve for L.
4=1.5L−0.5
L−0.5=4
1.5=8
3
L= (8
3)−2= (3
8)2=9
64
Therefore, the firm’s current level of labor input is 9
64 .
Question 24
Question
Consider a production function f(x, y) = x2+ 4xy +y2. Find the marginal
product of xand the marginal product of y.
Solution
To find the marginal product of x, we need to calculate the partial derivative of
the production function f(x, y) with respect to x. Similarly, to find the marginal
product of y, we calculate the partial derivative of f(x, y) with respect to y.
Step 1: Find the marginal product of x To find the marginal product
of x, we calculate the partial derivative of f(x, y) with respect to x:
∂f
∂x =∂
∂x(x2+ 4xy +y2)
= 2x+ 4y
Therefore, the marginal product of xis 2x+ 4y.
20
Step 2: Find the marginal product of y To find the marginal product
of y, we calculate the partial derivative of f(x, y) with respect to y:
∂f
∂y =∂
∂y (x2+ 4xy +y2)
= 4x+ 2y
Therefore, the marginal product of yis 4x+ 2y.
Question 25
Question
Given a production function Q= 4KL −3K2, where Qis the total output,
Kis the quantity of capital, and Lis the quantity of labor, find the marginal
products of labor and capital.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function Qwith respect to L, keeping Kconstant:
∂Q
∂L = 4K−3K2.
Step 2: This gives us the marginal product of labor:
MPL=∂Q
∂L = 4K−3K2.
Step 3: To find the marginal product of capital (MPK), we differentiate the
production function Qwith respect to K, keeping Lconstant:
∂Q
∂K = 4L−6K.
Step 4: This gives us the marginal product of capital:
MPK=∂Q
∂K = 4L−6K.
Therefore, the marginal product of labor is MPL= 4K−3K2, and the
marginal product of capital is MPK= 4L−6K.
Question 26
Question
Suppose a firm’s production function is given by Q= 4L0.5K0.5, where Lrep-
resents labor input and Krepresents capital input. If the firm has 16 units of
capital, what amount of labor input is required to produce 64 units of output?
21
Solution
Step 1: Start by substituting the given values (Q= 64 and K= 16) into the
production function Q= 4L0.5K0.5. Substitute Q= 64 and K= 16 into the
production function:
64 = 4L0.5(16)0.5
Step 2: Simplify the expression by evaluating (16)0.5.
64 = 4L0.5(4)
Step 3: Divide both sides by 4 to solve for L.
16 = L0.5(4)
Step 4: Divide both sides by 4 to solve for L.
4 = L0.5
Step 5: Square both sides to solve for L.
16 = L
Therefore, the firm would need 16 units of labor input to produce 64 units
of output.
Question 27
Question
Let qbe the output produced by a firm and Lbe the quantity of labor input.
The production function for the firm is given by q= 5L2−2L3. Calculate the
marginal product of labor when L= 4 and determine whether the production
function exhibits increasing, decreasing, or constant returns to scale in this case.
Solution
Step 1: Calculate the marginal product of labor. To find the marginal product
of labor (MPL), we need to take the derivative of the production function with
respect to labor, dq
dL .
dq
dL =d
dL(5L2−2L3)
dq
dL = 10L−6L2
Step 2: Substituting L= 4 to find the marginal product of labor. Now,
substitute L= 4 into the derivative we found in Step 1 to get the marginal
product of labor at L= 4.
dq
dL = 10 ×4−6×42
22
dq
dL = 40 −96
dq
dL =−56
Therefore, the marginal product of labor when L= 4 is −56.
Step 3: Determine returns to scale. To determine the returns to scale,
we need to examine the behavior of the production function with respect to
changes in the scale of input. Since the exponent of labor is greater than 1
in the production function q= 5L2−2L3, the production function exhibits
decreasing returns to scale.
Question 28
Question
Consider the production function Q= 5L0.5K0.5, where Qrepresents the quan-
tity of output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. Given that the price of labor is w= 10 and the price of capital is
r= 20, what is the cost-minimizing input combination if the firm wants to
produce 100 units of output?
Solution
Step 1: The cost-minimizing condition for the firm is that the marginal rate of
technical substitution (MRTS) equals the ratio of input prices. The MRTS is
given by the ratio of the marginal products of labor and capital:
MRT S =M P L
MP K =5×0.5L−0.5K0.5
5×L0.5K−0.5=0.5
0.5= 1
Step 2: Given that MRT S =w
r, we have 1 = 10
20 . This implies that the
cost-minimizing input combination is L:K= 1 : 2.
Step 3: To produce 100 units of output, we can use the production function
Q= 5L0.5K0.5:
100 = 5L0.5K0.5
Step 4: Substituting the cost-minimizing input combination into the pro-
duction function, we get:
100 = 5(1)0.5(2)0.5= 5√2
Thus, the cost-minimizing input combination for producing 100 units of output
is L= 1 and K= 2.
23
Question 29
Question
Consider a production function given by f(K, L) = K1
3·L2
3, where Krepresents
capital input and Lrepresents labor input. Find the marginal product of labor.
Solution
Step 1: To find the marginal product of labor (MPL), we must first find the
partial derivative of the production function with respect to labor (L). This is
represented by ∂f
∂L .
Step 2: Taking the partial derivative of the production function f(K, L) =
K1
3·L2
3with respect to L, we get:
∂f
∂L =2
3K1
3·L−1
3
Step 3: This expression represents the marginal product of labor: MPL=
2
3K1
3·L−1
3.
Question 30
Question
Suppose a firm’s production function is given by Q=K0.4L0.6, where Qis the
total output, Kis the amount of capital used, and Lis the amount of labor
employed. If the firm currently has K= 10 units of capital and L= 5 units
of labor, find the rate at which output is changing with respect to labor when
L= 5.
Solution
Step 1: Calculate the marginal product of labor (MPL)
The marginal product of labor (MPL) is given by the partial derivative of the
production function with respect to labor:
MP L =∂Q
∂L = 0.6K0.4L−0.4
Step 2: Substitute the given values K= 10 and L= 5 into the MPL formula:
MP L(10,5) = 0.6(10)0.4(5)−0.4
MP L(10,5) = 0.6(2.5118864315)(0.632455532)
MP L(10,5) ≈0.947
24
Step 3: Calculate the rate at which output is changing with respect to labor
The rate at which output is changing with respect to labor is given by the MPL:
dQ
dL =MP L(10,5) = 0.947
Therefore, the rate at which output is changing with respect to labor when
L= 5 is approximately 0.947 units of output per unit of labor.
Question 31
Question
Let f(L, K)=4L1/3K2/3represent a production function where Lis the amount
of labor input and Kis the amount of capital input. Determine the marginal
product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we need to calculate the par-
tial derivative of the production function with respect to labor, holding capital
constant. ∂f
∂L =4
3L−2/3K2/3
Step 2: Simplifying the expression, we get:
∂f
∂L =4
3K
L2/3
Step 3: Therefore, the marginal product of labor is 4
3K
L2/3.
Step 4: Next, to find the marginal product of capital, we need to calculate
the partial derivative of the production function with respect to capital, holding
labor constant. ∂f
∂K =8
3L1/3K−1/3
Step 5: Simplifying the expression, we get:
∂f
∂K =8
3L
K1/3
Step 6: Therefore, the marginal product of capital is 8
3L
K1/3.
25
Question 32
Question
Suppose a firm has the following production function: Q(K, L) = KαLβ, where
Qis the quantity of output, Kis the quantity of capital, Lis the quantity of
labor, and α, β > 0. Show that the firm’s production function exhibits constant
returns to scale.
Solution
Step 1: To determine if the production function exhibits constant returns to
scale, we need to check how the output changes when both capital and labor
are scaled up by a factor λ > 0. Let’s examine the output Q(λK, λL):
Q(λK, λL)=(λK)α(λL)β
=λαKαλβLβ
=λα+βKαLβ
=λα+βQ(K, L)
Step 2: Since α > 0 and β > 0, we have α+β > 0. Thus, when both capital
and labor are scaled up by a factor λ, the output is scaled up by a factor of
λα+β.
Step 3: A production function exhibits constant returns to scale if the output
is scaled up by the same proportion as the inputs. Since α+β > 0, we can
conclude that the firm’s production function Q(K, L) = KαLβexhibits constant
returns to scale.
Question 33
Question
Consider a production function given by Q=L0.7K0.3, where Qrepresents the
quantity of output, Lrepresents labor input, and Krepresents capital input.
Determine the marginal product of labor and the marginal product of capital.
Solution
Let’s first find the marginal product of labor (MPL) by taking the partial deriva-
tive of the production function with respect to labor input L, holding capital
input Kconstant.
MPL=∂Q
∂L = 0.7L−0.3K0.3
Step 1: Calculate the marginal product of labor.
MPL= 0.7L−0.3K0.3
26
Next, let’s find the marginal product of capital (MPK) by taking the partial
derivative of the production function with respect to capital input K, holding
labor input Lconstant.
MPK=∂Q
∂K = 0.3L0.7K−0.7
Step 2: Calculate the marginal product of capital.
MPK= 0.3L0.7K−0.7
Therefore, the marginal product of labor is MPL= 0.7L−0.3K0.3and the
marginal product of capital is MPK= 0.3L0.7K−0.7.
Question 34
Question
Suppose a production function is given by Q= 4L3/4K1/4, where Qis the
output, Lis the amount of labor input, and Kis the amount of capital input.
If the amount of capital input is fixed at K= 16, find the marginal product of
labor.
Solution
To find the marginal product of labor, we need to calculate the partial derivative
of the production function with respect to labor.
Step 1: Calculate the total product of labor:
Q= 4L3/4K1/4
Step 2: Substitute K= 16 into the production function:
Q= 4L3/4161/4
Step 3: Simplify the expression:
Q= 4L3/42
Step 4: Calculate the total product of labor by expanding the expression:
Q= 8L3/4
Step 5: Calculate the marginal product of labor (MPL) by taking the
derivative of the total product with respect to labor:
MPL=dQ
dL =d
dL(8L3/4)
27
Step 6: Differentiate 8L3/4with respect to L:
MPL=3
4·8L−1/4
Step 7: Simplify the expression for the marginal product of labor:
MPL= 6L−1/4=6
4
√L
Therefore, the marginal product of labor is 6
4
√L.
Question 35
Question
Let Qbe the quantity of output and Lbe the quantity of labor input in a
production process described by the production function Q= 10L−0.1L2. Find
the level of labor input that maximizes output and determine the corresponding
maximum level of output.
Solution
Step 1: To find the level of labor input that maximizes output, we first need
to find the critical points of the production function. These occur where the
derivative of the production function is equal to zero.
Step 1: Q= 10L−0.1L2
Step 2: dQ
dL = 10 −0.2L
Step 3: Set dQ
dL = 0 to find critical points
Step 4: 10 −0.2L= 0
Step 5: Solve for L:L=10
0.2= 50
So, the critical point is L= 50.
Step 2: To determine if this critical point corresponds to a maximum, we
will use the second derivative test.
Step 6: d2Q
dL2=−0.2
Step 7: Evaluate the second derivative at the critical point L= 50
Step 8: d2Q
dL2L=50 =−0.2<0
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Since the second derivative is negative at the critical point, the production
function has a local maximum at L= 50.
Step 3: Finally, we will find the corresponding maximum level of output by
substituting L= 50 back into the production function.
Step 9: Q= 10(50) −0.1(50)2
Step 10: Q= 500 −0.1(2500)
Step 11: Q= 500 −250 = 250
Therefore, the level of labor input that maximizes output is L= 50 and the
corresponding maximum level of output is Q= 250.
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