ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank - Set 1
Liberty University
Question 1
Question
Consider a production function given by Q= 4L0.5K0.5, where Qrepresents
the total output, Lrepresents the quantity of labor input, and Krepresents
the quantity of capital input. If the quantity of labor input is fixed at 16 units,
find: a) The marginal product of capital. b) The average product of capital.
Solution
a) To find the marginal product of capital, we need to take the partial derivative
of the production function with respect to K.
∂Q
∂K = 2L0.5K−0.5
Step 1: Plug in the given values L= 16 and Kinto the above expression.
∂Q
∂K = 2(16)0.5K−0.5
Step 2: Simplify the expression.
∂Q
∂K = 32K−0.5
b) To find the average product of capital, we need to divide the total output
(Q) by the quantity of capital input (K).
APK=Q
K
Step 3: Plug in the given production function Q= 4L0.5K0.5and L= 16
into the average product of capital formula.
APK=4(16)0.5K0.5
K
Step 4: Simplify the expression.
APK= 4(16)0.5K0.5−1
APK= 64K−0.5
Question 2
Question
Let Q=f(L, K) be the production function of a firm, where Qrepresents the
total output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. Suppose the production function is given by Q= 4√LK. Find the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor (MPL), we need to find the partial
derivative of the production function with respect to labor L:
∂Q
∂L =∂
∂L(4√LK)
Step 2: Using the chain rule for differentiation, we have:
∂Q
∂L = 2√K
Therefore, the marginal product of labor is MPL= 2√K.
Step 3: To find the marginal product of capital (M PK), we need to find the
partial derivative of the production function with respect to capital K:
∂Q
∂K =∂
∂K (4√LK)
Step 4: Using the chain rule for differentiation, we have:
∂Q
∂K = 2√L
Therefore, the marginal product of capital is MPK= 2√L.
2
Question 3
Question
Consider a production function given by Q= 10L0.5K0.3, where Qis the quan-
tity of output, Lis the quantity of labor input, and Kis the quantity of capital
input. Determine the marginal product of labor (MPL) when L= 4 units and
K= 9 units.
Solution
Step 1: Find the total product of labor (T PL). Given the production function
Q= 10L0.5K0.3, we can find the total product of labor (T PL) by plugging in
the values of L= 4 units and K= 9 units:
T PL= 10(4)0.5(9)0.3
T PL= 10(2)(2.0801)
T PL≈41.60
Step 2: Find the total product of labor when L= 5 units. To find the
marginal product of labor (MPL), we need to find the total product of labor
when L= 5 units:
T P ′
L= 10(5)0.5(9)0.3
T P ′
L= 10(2.2361)(2.0801)
T P ′
L≈46.58
Step 3: Calculate the marginal product of labor (MPL). The marginal prod-
uct of labor (MPL) is calculated as the change in total product of labor with
respect to a change in labor input. Therefore, MPL is given by:
MP L =∆T PL
∆L
MP L =T P ′
L−T PL
5−4
MP L =46.58 −41.60
1
MP L = 4.98
Therefore, the marginal product of labor (MPL) when L= 4 units and
K= 9 units is approximately 4.98 units.
3
Question 4
Question
Consider a production function given by Q= 5LK, where Qrepresents the
quantity of output, Lrepresents labor input, and Krepresents capital input.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK).
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function Qwith respect to labor input L, while treating capital K
as a constant. dQ
dL = 5K
Step 2: Therefore, the marginal product of labor (MPL) is given by dQ
dL =
5K.
Step 3: Now, let’s find the marginal product of capital (MPK) by differenti-
ating the production function Qwith respect to capital input K, while treating
labor Las a constant. dQ
dK = 5L
Step 4: Hence, the marginal product of capital (MPK) is dQ
dK = 5L.
Therefore, the marginal product of labor (MPL) is 5K, and the marginal
product of capital (MPK) is 5L.
Question 5
Question
Let Q= 2L0.5K0.3be the production function of a firm, where Qis the level of
output, Lis the quantity of labor, and Kis the quantity of capital. Determine
the total output when the firm employs 25 units of labor and 16 units of capital.
Solution
Step 1: Substitute L= 25 and K= 16 into the production function to find the
total output.
Q= 2(25)0.5(16)0.3
= 2 ·5·2
= 20 ·2
= 40 units
4
Question 6
Question
Consider a production function given by q=K3/4L1/4, where qrepresents the
quantity of output produced, Kis the quantity of capital input, and Lis the
quantity of labor input. Suppose the price of capital is r= 4 and the price
of labor is w= 2. Calculate the total cost function Cin terms of qfor this
production function.
Solution
Step 1: We start by defining the total cost function Cin terms of q. Given that
C=rK +wL, we need to express Kand Lin terms of qusing the production
function q=K3/4L1/4. Step 2: To find Kin terms of q, we raise both sides of
the production function to the power of 4/3:
q4/3=K
Step 3: Replace Kin the total cost function:
C= 4q4/3+wL
Step 4: To find Lin terms of q, we raise both sides of the production function
to the power of 4 and −1 respectively:
q4=K3L
L=q4
K3=q4
(q4/3)3=q16/3
Step 5: Substitute Lback into the total cost function:
C= 4q4/3+ 2q16/3
Step 6: Simplifying the total cost function gives us the final result:
C= 4q4/3+ 2q16/3
Question 7
Question
Consider a production function Q= 4L0.5K0.3, where Qrepresents the quantity
of output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. Suppose the wage rate is w= 20 and the rental rate of capital
is r= 30. If the firm wants to produce 1024 units of output at minimum cost,
how many units of labor and capital should the firm hire?
5
Solution
Step 1: To minimize the cost of production, the firm needs to use labor and
capital inputs in such a way that the marginal cost of each input is equal to its
price. Therefore, we set the marginal cost of labor equal to the wage rate and
the marginal cost of capital equal to the rental rate.
Step 2: The marginal cost of labor (MCL) is given by:
MCL=∂Cost
∂L =w·MP L
where MP L is the marginal product of labor. The marginal product of labor
is the derivative of the production function with respect to labor:
MP L =∂Q
∂L = 2L−0.5K0.3
Step 3: Substituting M P L into the equation for MCLand setting it equal
to the wage rate:
20 = 20L−0.5K0.3
Step 4: The marginal cost of capital (MCK) is given by:
MCK=∂Cost
∂K =r·MP K
where MP K is the marginal product of capital. The marginal product of capital
is the derivative of the production function with respect to capital:
MP K =∂Q
∂K = 1.2L0.5K−0.7
Step 5: Substituting MP K into the equation for MCKand setting it equal
to the rental rate:
30 = 36L0.5K−0.7
Step 6: Given that the firm wants to produce 1024 units of output, we have:
4L0.5K0.3= 1024
Step 7: Solving the three equations simultaneously will give us the values of
Land Kthat minimize the cost of production. After solving, we find L= 40
and K= 160. Therefore, the firm should hire 40 units of labor and 160 units
of capital to produce 1024 units of output at minimum cost.
Question 8
Question
Consider a production function given by Q= 10L0.5K0.5, where Qrepresents
the quantity of output produced, Lis the amount of labor input, and Kis
the amount of capital input. If the wage rate is w= 2 and the rental rate of
capital is r= 3, calculate the cost-minimizing combination of inputs required
to produce 100 units of output.
6
Solution
Step 1: The cost-minimizing condition for the firm is given by the equation
w
MP L =r
MP K , where MPL is the marginal product of labor and MPK is the
marginal product of capital. We first find these marginal products by taking
partial derivatives of the production function with respect to Land K. Step 2:
Calculate the marginal product of labor (MPL) by differentiating the production
function with respect to L:
MP L =∂Q
∂L = 5L−0.5K0.5
Step 3: Calculate the marginal product of capital (MPK) by differentiating the
production function with respect to K:
MP K =∂Q
∂K = 5L0.5K−0.5
Step 4: Substitute the given values of w,r, and the production function into
the cost-minimizing condition w
MP L =r
MP K in order to solve for the optimal
combination of inputs. Step 5: Setting up the cost-minimizing condition, we
have: 2
5L−0.5K0.5=3
5L0.5K−0.5
Step 6: Rearrange the equation to isolate Lin terms of K:
2K= 3L
Step 7: We can use the firm’s production function Q= 10L0.5K0.5and the
condition 2K= 3Lto find the optimal combination of inputs that produce 100
units of output.
Substitute L=2
3Kinto the production function:
100 = 10 2
3K0.5
K0.5
100 = 10 4
90.5
K·K
Step 8: Solve for K:
100 = 10 4
90.5
K2
K2=100
10 4
90.5
K2=100
10 2
3
K2=100
6.67
7
K2≈15
Step 9: Solve for K:
K≈√15 ≈3.87
Step 10: Calculate Lusing 2K= 3L:
2(3.87) = 3L
7.74 = 3L
L≈7.74
3≈2.58
Therefore, the cost-minimizing combination of inputs required to produce 100
units of output is approximately L≈2.58 and K≈3.87.
Question 9
Question
Consider a production function given by Q= 5L0.4K0.6, where Qrepresents
the total output, Lrepresents the quantity of labor input, and Krepresents the
quantity of capital input. Calculate the marginal product of labor (M PL) and
the marginal product of capital (MPK) at a point where 100 units of labor and
64 units of capital are being used.
Solution
To find the marginal product of labor and capital, we will first calculate the
partial derivatives of the production function with respect to labor and capital.
Step 1: Calculate the marginal product of labor (M PL)
MPL=∂Q
∂L = 2L−0.6K0.6
At the point where 100 units of labor and 64 units of capital are being used:
MPL= 2(100)−0.6(64)0.6= 2(0.01)(4) = 0.08
Therefore, the marginal product of labor at this point is 0.08.
Step 2: Calculate the marginal product of capital (M PK)
MPK=∂Q
∂K = 3L0.4K−0.4
At the point where 100 units of labor and 64 units of capital are being used:
MPK= 3(100)0.4(64)−0.4= 3(2)(0.25) = 1.5
Therefore, the marginal product of capital at this point is 1.5.
8
Question 10
Question
A company’s production function is given by Q= 2K0.5L0.5. Determine the
marginal product of labor and the marginal product of capital, and discuss what
happens to each as more units of labor and capital are added.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate Qwith
respect to L:
MPL=∂Q
∂L =∂
∂L(2K0.5L0.5)
MPL= 1K0.5·0.5L−0.5=K0.5L−0.5
Step 2: To find the marginal product of capital (M PK), we differentiate Q
with respect to K:
MPK=∂Q
∂K =∂
∂K (2K0.5L0.5)
MPK= 2 ·0.5K−0.5L0.5=K−0.5L0.5
Step 3: Observing the marginal products of labor and capital: - For labor:
MPL=K0.5L−0.5. As more units of labor are added, the marginal product of
labor decreases, indicating diminishing marginal returns to labor. - For capital:
MPK=K−0.5L0.5. Similarly, as more units of capital are added, the marginal
product of capital decreases, showing diminishing marginal returns to capital.
In summary, adding more units of labor and capital will lead to diminishing
marginal returns to each factor of production.
Question 11
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the total
output, Lis the quantity of labor, and Kis the quantity of capital. If the
wage rate is w= 5 and the rental rate for capital is r= 10, calculate the cost-
minimizing combination of labor and capital required to produce 100 units of
output.
Solution
Given the production function Q=L0.5K0.5, the cost-minimizing combination
of labor and capital can be determined by setting up the cost minimization
problem. The cost function, C, is defined as C=wL +rK where wis the wage
rate and ris the rental rate for capital.
9
Step 1: Express the cost function in terms of labor (L) only
C=wL +rQ
K0.5
Step 2: Substitute the given values of w,r, and Qinto the cost
function
C= 5L+ 10 100
K0.5
Step 3: Minimize the cost function by differentiating
dC
dL = 5 −1000
L1.5K0.5= 0
Step 4: Solve for the optimal quantity of labor (L)
5 = 1000
L1.5K0.5
L1.5K0.5= 200
Step 5: Given Q= 100, substitute Qinto the production function
to get
100 = L0.5K0.5
Step 6: Using the optimal quantity of labor found earlier (L1.5K0.5=
200) and the production function, solve for the optimal quantity of
capital (K)
100 = √200K
K=100
√200 =100√200
200 =10√2
2= 5√2
Step 7: Calculate the optimal quantity of labor (L) by substituting
the optimal quantity of capital into L1.5K0.5= 200
L1.5(5√2)0.5= 200
L1.5√5 = 200
L=200
√5
2
3
= 20(√5)2
3= 20 ·51
3
Therefore, the cost-minimizing combination of labor and capital required to
produce 100 units of output is L= 20 ·51
3and K= 5√2.
Question 12
Question
Consider a production function given by Q=L0.5K0.5, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Given that the cost of labor is wand the cost of capital is r, determine the
cost-minimizing combination of labor and capital required to produce a certain
level of output Q.
10
Solution
Step 1: The firm’s total cost (T C) is given by the sum of the costs of labor and
capital used, which can be expressed as:
T C =wL +rK
Step 2: To minimize total cost for a given level of output Q, we need to
minimize the cost function T C =wL +rK subject to the production function
Q=L0.5K0.5.
Step 3: We can rewrite the production function as K=Q
L2
. Substituting
this into the cost function, we get:
T C =wL +rQ
L2
Step 4: To find the cost-minimizing combination of labor and capital, we
need to minimize the total cost function. To do this, we differentiate T C with
respect to Land set the derivative equal to zero:
dT C
dL =w−2rQ
L2
·1
L2= 0
Step 5: Solving for Lin the above equation, we get:
L3=Q
2r
Step 6: Substituting this value of Lback into the production function K=
Q
L2
, we find:
K=
Q
Q
2r1/3
2
= (2r)2/3·Q1/3
Step 7: Therefore, the cost-minimizing combination of labor and capital
required to produce a certain level of output Qis L=Q
2r1/3
and K=
(2r)2/3·Q1/3.
Question 13
Question
Let Q=L0.3K0.5be a production function representing the quantity of output
(Q) as a function of labor input (L) and capital input (K). If the wage rate
is w= 10 and the rental rate for capital is r= 20, find the minimum cost of
producing 500 units of output.
11
Solution
Given the production function: Q=L0.3K0.5, the cost function can be ex-
pressed as:
C=wL +rK
We are looking to minimize the cost, subject to the constraint Q= 500. We
can rewrite Cas a function of one variable using the constraint:
C= 10L+ 20K= 10L+ 20 Q
L0.32
To find the minimum cost, we will differentiate Cwith respect to Land set
the derivative equal to zero:
dC
dL = 10 −40 Q
L0.32
L−0.7= 0
10 = 40 500
L0.32
L−0.7
1
4=500
L0.32
L−0.7
1
4=250000
L0.6L−0.7
1
4=250000
L0.6L−0.7
L= 50
Therefore, the amount of labor needed to minimize the cost is L= 50. Now,
we can find the corresponding capital input:
K=Q
L0.3=500
500.3=500
50 3
10
=500
11.18 ≈44.69
The minimum cost of producing 500 units of output is
C= 10(50) + 20(44.69) = $1000 + $893.8 = $1893.8
Question 14
Question
A firm has a production function given by Q= 2L0.5K0.5, where Qis the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently has 10 units of labor and 8 units of capital, calculate the
marginal product of labor and the marginal product of capital.
12
Solution
Step 1: Calculate the total product function by substituting the given values of
labor and capital into the production function.
Q= 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 2: Calculate the total product of labor by keeping Kconstant.
QL=10 = 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 3: Calculate the total product of labor when labor increases from 10
units to 11 units. QL=11 = 2(11)0.5(8)0.5
= 2(3.32)(2.83)
= 18.78
Step 4: Calculate the marginal product of labor.
MPL=∆Q
∆L=18.78 −17.84
11 −10 = 0.94
Step 5: Calculate the total product of capital by keeping Lconstant.
QK=8 = 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 6: Calculate the total product of capital when capital increases from 8
units to 9 units. QK=9 = 2(10)0.5(9)0.5
= 2(3.16)(3)
= 18.96
Step 7: Calculate the marginal product of capital.
MPK=∆Q
∆K=18.96 −17.84
9−8= 1.12
Therefore, the marginal product of labor is 0.94 units of output per unit
increase in labor, and the marginal product of capital is 1.12 units of output
per unit increase in capital.
13
Question 15
Question
Consider a production function given by Q=L1/2K3/4, where Qrepresents the
total output, Lis the quantity of labor input, and Kis the quantity of capital
input. Given that the price of labor is wand the price of capital is r, show that
the cost minimizing condition for this production function is given by
(3/4)wL = (1/2)rK.
Solution
Step 1: The cost of producing at a given level of output Qis given by C=
wL +rK, where wis the wage rate and ris the rental rate of capital.
Step 2: The cost minimizing condition can be stated as ∂C
∂L =∂C
∂K = 0, which
implies that the marginal cost of labor (w) should be equal to the marginal cost
of capital (r).
Step 3: Computing the partial derivatives of the cost function with respect
to Land K:∂C
∂L =w=1
2Q×1
L=1
2QL−1/2K3/4,
∂C
∂K =r=3
4Q×1
K=3
4QL1/2K−1/4.
Step 4: Setting the partial derivatives equal to 0:
1
2QL−1/2K3/4=w,
3
4QL1/2K−1/4=r.
Step 5: Rearranging the equations to solve for wand rgives:
1
2wL =QK1/4,
3
4rK =QL1/2.
Step 6: Dividing the two equations gives:
1
2wL∇ · 3
4rK =QK1/4∇ · QL1/2,
2
3
wL
rK =K1/4
L1/2,
2
3
wL
rK =K
L,
3
4wL =1
2rK.
Therefore, the cost minimizing condition for this production function is
3
4wL =1
2rK.
14
Question 16
Question
Suppose a firm has the production function Q=LαKβ, where Qrepresents
the quantity of output produced, Lrepresents the quantity of labor input, K
represents the quantity of capital input, and α, β > 0. If the firm is currently
producing 100 units of output using 10 units of labor and 5 units of capital,
by how much should the firm increase its labor input in order to double the
quantity of output produced?
Solution
Step 1: Calculate the firm’s initial total productivity level
The firm’s initial total productivity level is given by plugging in L= 10 and
K= 5 into the production function:
Q= 10α5β= 100
Step 2: Set up the equation for doubling output
For the firm to double the quantity of output produced, we need to find the
value of Lsuch that:
2×100 = (10 + ∆L)α5β
Step 3: Simplify the equation
200 = (10 + ∆L)α5β
Step 4: Use the initial total productivity level to find the value of ∆L
Since Q= 100 when L= 10 and K= 5, we have:
100 = 10α5β
Step 5: Substituting the initial productivity level into the equation for dou-
bling output
200 = (10 + ∆L)α5β
200 = 10α5β×(10 + ∆L)α
Step 6: Solve for ∆L
Dividing both sides by 10α5β, we get:
2 = (1 + ∆L
10 )α
Step 7: Take the α-th root of both sides
21
α= 1 + ∆L
10
15
∆L= 10(2 1
α−1)
Therefore, the firm should increase its labor input by 10(2 1
α−1) units in
order to double the quantity of output produced.
Question 17
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the output,
Lis the quantity of labor, and Kis the quantity of capital. Suppose the price
of labor is wand the price of capital is r.
Assume that we have a fixed total cost of production C. Find the optimal
combination of labor and capital that minimizes the cost while producing a
given level of output Q.
Solution
Let C=wL +rK represent the total cost of production. We want to minimize
this cost subject to the production function Q=L0.5K0.5.
Step 1: Set up the Lagrange function Let L(L, K, λ) = wL +rK +
λ(Q−L0.5K0.5) be the Lagrange function, where λis the Lagrange multiplier.
Step 2: Find the first-order conditions The first-order conditions are:
∂L
∂L =w−0.5λL−0.5K0.5= 0
∂L
∂K =r−0.5λL0.5K−0.5= 0
∂L
∂λ =Q−L0.5K0.5= 0
Solving the above system of equations, we can find expressions for L,K,
and λthat satisfy the first-order conditions.
Step 3: Solve for the optimal combination of labor and capital
From the first two equations, we have:
(w= 0.5λL−0.5K0.5
r= 0.5λL0.5K−0.5
Dividing these two equations, we get:
w
r=L
K
Combining this with the production function Q=L0.5K0.5, we can solve for
optimal values of Land Kin terms of Q.
Step 4: Verify the second-order conditions Finally, we need to verify
that the solution obtained in Step 3 results in a minimum cost. This involves
checking the concavity of the cost function.
16
The optimal combination of labor and capital that minimizes the cost while
producing a given level of output Qcan be found using the above steps.
Question 18
Question
Consider a production function given by Q= 2L0.5K0.5, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the price of labor is wand the price of capital is r, derive the equation for
the total cost of production (C) in terms of w,r,Q,L, and K.
Solution
Step 1: The total cost of production (C) can be expressed as the sum of the
cost of labor (CL) and the cost of capital (CK):
C=CL+CK
Step 2: The cost of labor (CL) can be calculated by multiplying the quantity
of labor (L) by the price of labor (w):
CL=w·L
Step 3: Similarly, the cost of capital (CK) can be calculated by multiplying
the quantity of capital (K) by the price of capital (r):
CK=r·K
Step 4: Substitute the expressions for CLand CKinto the total cost equa-
tion:
C=w·L+r·K
Step 5: Since the quantity of output (Q) is given by Q= 2L0.5K0.5, we can
rewrite the total cost equation in terms of Q,L,K,w, and r:
C=w·L+r·K=w·Q
2K0.5+r·K
Step 6: Simplify the total cost equation:
C=wQ
2K0.5+rK
Therefore, the equation for the total cost of production (C) in terms of w,
r,Q,L, and Kis given by C=wQ
2K0.5+rK.
17
Question 19
Question
Consider a production function Q= 2L0.5K0.5where Qrepresents total output,
Lis labor input, and Kis capital input.
Suppose the cost of labor is wper unit and the cost of capital is rper unit.
If the firm’s goal is to minimize the cost of producing a given level of output Q,
what is the optimal combination of labor and capital inputs?
Solution
Step 1: The firm’s cost function (C) can be expressed as the sum of labor (L)
and capital (K) costs:
C=wL +rK
Step 2: We are given the production function Q= 2L0.5K0.5, which repre-
sents the relationship between output and inputs.
Step 3: To minimize the cost of producing a given level of output Q, we
need to find the optimal combination of labor and capital inputs that maximizes
output Qsubject to the production function and the cost constraint.
Step 4: We can formulate the firm’s optimization problem as follows:
Maximize Q= 2L0.5K0.5subject to C=wL +rK
Step 5: We will use the method of Lagrange multipliers to solve this opti-
mization problem. Define the Lagrangian function as:
J= 2L0.5K0.5+λ(wL +rK −C)
Step 6: Taking the partial derivatives of the Lagrangian function with re-
spect to L,K, and λand setting them equal to zero will give us the necessary
conditions for optimization.
Step 7: ∂J
∂L = 0, ∂J
∂K = 0, and ∂J
∂λ = 0 lead to the following equations:
K0.5−λw = 0
L0.5−λr = 0
wL +rK =C
Step 8: Solving the system of equations will give us the optimal values for
Land Kin terms of w,r, and C.
Step 9: These optimal values represent the combination of labor and capital
inputs that minimizes the cost of producing a given level of output Q.
18
Question 20
Question
Suppose a production function is given by Q=L0.3K0.5, where Qrepresents
the output, Lrepresents labor input, and Krepresents capital input. If the
firm currently employs 25 units of labor and 16 units of capital, calculate the
marginal product of labor and the marginal product of capital at this level of
inputs.
Solution
Step 1: To find the marginal product of labor, we first need to calculate the total
product when L= 25 and K= 16. We can then find the marginal product of
labor by taking the derivative of the production function with respect to labor.
Step 2: Calculating the total product:
Q=L0.3K0.5
Q= 250.3×160.5
Q= 2.924 ×4
Q= 11.696
Therefore, the total product (Q) is 11.696 when L= 25 and K= 16.
Step 3: Calculating the marginal product of labor: To find the marginal
product of labor, we differentiate the production function with respect to L:
∂Q
∂L = 0.3L−0.7K0.5
Now, substitute L= 25 and K= 16 into the equation:
∂Q
∂L = 0.3(25)−0.7(16)0.5
∂Q
∂L = 0.3(0.004)(4)
∂Q
∂L = 0.0048
Thus, the marginal product of labor when L= 25 and K= 16 is 0.0048.
Step 4: Calculating the marginal product of capital: Similarly, to find the
marginal product of capital, we differentiate the production function with re-
spect to K:∂Q
∂K = 0.5L0.3K−0.5
Substitute L= 25 and K= 16 into the equation:
∂Q
∂K = 0.5(25)0.3(16)−0.5
19
∂Q
∂K = 0.5(2.924)(0.25)
∂Q
∂K = 0.3655
Therefore, the marginal product of capital when L= 25 and K= 16 is
0.3655.
Question 21
Question
Consider a production function Q= 5L1/3K2/3, where Qis the total output,
Lis the quantity of labor input, and Kis the quantity of capital input. Given
that the price of labor (w) is 2 and the price of capital (r) is 3, calculate the
cost-minimizing quantities of labor and capital required to produce 100 units of
output.
Solution
Step 1: Write the cost-minimization problem using the Lagrange multiplier
method. Let C=wL +rK be the cost function. We want to minimize C
subject to the production constraint Q= 5L1/3K2/3. Set up the Lagrangian:
L= 2L+ 3K−λ(5L1/3K2/3−100)
Step 2: Find the first-order conditions by taking partial derivatives with
respect to L,K, and λ.
Partial derivative with respect to L:
∂L
∂L = 2 −5λ
3L−2/3K2/3= 0
Partial derivative with respect to K:
∂L
∂K = 3 −10λ
3L1/3K−1/3= 0
Partial derivative with respect to λ:
∂L
∂λ = 5L1/3K2/3−100 = 0
Step 3: Solve the system of equations to find the optimal values for L,K,
and λ. Solving the first two equations simultaneously, we find:
2
53
103/2
=LK
20
2
53
103/2
=2
53
102/3
L1/3K2/3
L= 10, K = 5
Step 4: Calculate the cost-minimizing quantities of labor and capital re-
quired to produce 100 units of output. Substitute L= 10 and K= 5 into the
production function to find the cost-minimizing quantities of labor and capital:
Q= 5(10)1/3(5)2/3= 100
Therefore, the cost-minimizing quantities of labor and capital required to
produce 100 units of output are L= 10 units and K= 5 units, respectively.
Question 22
Question
Let Qbe the total output of a firm, Lbe the quantity of labor, and Kbe the
quantity of capital. Suppose the production function of the firm is given by
Q= 2K0.5L0.5. If the firm faces a rental rate of r= 4 and a wage rate of w= 3,
find the total cost function (T C) in terms of Q.
Solution
Step 1: Start by writing the total cost function (T C) in terms of the inputs K
and Lusing the rental rate rand wage rate w.
TC = rK +wL
Step 2: Find the cost of capital using the rental rate rand the quantity of
capital K.
rK = 4K
Step 3: Find the cost of labor using the wage rate wand the quantity of
labor L.
wL = 3L
Step 4: Substitute the expressions for the cost of capital and labor back into
the total cost function T C.
TC = 4K+ 3L
Step 5: Find the relationship between Kand Lusing the production function
Q= 2K0.5L0.5.
Q= 2K0.5L0.5
Q= 2√K√L
√K=Q
2√L
21
K=Q2
4L
Step 6: Substitute the expression for Kback into the total cost function
T C.
TC = 4 Q2
4L+ 3L
TC = Q2/L + 3L
Therefore, the total cost function (T C) in terms of Qis TC = Q2/L + 3L.
Question 23
Question
Consider a production function given by Q= 100K0.5L0.5, where Qrepresents
the level of output, Kis the quantity of capital used, and Lis the quantity of
labor used. Calculate the marginal product of labor and the marginal product
of capital.
Solution
Step 1: To find the marginal product of labor, we take the partial derivative
of the production function with respect to labor (L) while holding capital (K)
constant. ∂Q
∂L = 0.5·100K0.5L−0.5
∂Q
∂L = 50K0.5L−0.5
Step 2: Simplifying the expression gives us the marginal product of labor.
Marginal Product of Labor = 50 K0.5
L0.5
Step 3: Next, we find the marginal product of capital by taking the partial
derivative of the production function with respect to capital (K) while holding
labor (L) constant.
∂Q
∂K = 0.5·100K−0.5L0.5
∂Q
∂K = 50K−0.5L0.5
Step 4: Simplifying the expression gives us the marginal product of capital.
Marginal Product of Capital = 50 L0.5
K0.5
Therefore, the marginal product of labor is 50 K0.5
L0.5and the marginal product
of capital is 50 L0.5
K0.5.
22
Question 24
Question
Consider a production function given by Q= 10L0.5K0.5, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. Find the
marginal product of labor (MPL) and marginal product of capital (MPK) at
the point where L= 25 and K= 16.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function with respect to Lwhile holding Kconstant.
∂Q
∂L = 5K0.5L−0.5
Step 2: Substitute L= 25 and K= 16 into the MPL formula.
∂Q
∂L L=25,K=16
= 5(16)0.5(25)−0.5= 5 ×4×0.2=4
Therefore, the marginal product of labor (MPL) at the point L= 25 and
K= 16 is 4.
Step 3: To find the marginal product of capital (MPK), we differentiate the
production function with respect to Kwhile holding Lconstant.
∂Q
∂K = 5L0.5K−0.5
Step 4: Substitute L= 25 and K= 16 into the MPK formula.
∂Q
∂K L=25,K=16
= 5(25)0.5(16)−0.5= 5 ×5×0.25 = 6.25
Therefore, the marginal product of capital (MPK) at the point L= 25 and
K= 16 is 6.25.
Question 25
Question
Consider a production function given by Q=L0.3K0.7, where Qrepresents the
quantity produced, Lis the amount of labor, and Kis the amount of capital.
Find the marginal product of labor, MPL, and the marginal product of capital,
MPK.
23
Solution
Step 1: Find the partial derivative of the production function with
respect to labor L.To find the marginal product of labor (MPL), we need
to take the partial derivative of the production function with respect to labor
L. This will give us the change in output for a one-unit change in labor.
∂Q
∂L = 0.3L−0.7K0.7
Step 2: Simplify the expression to find M PL.The marginal product
of labor, MPL, is the partial derivative we found in Step 1.
MPL= 0.3L−0.7K0.7
Step 3: Find the partial derivative of the production function with
respect to capital K.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 = 0.7L0.3K−0.3
Step 4: Simplify the expression to find M PK.The marginal product
of capital, MPK, is the partial derivative we found in Step 3.
MPK= 0.7L0.3K−0.3
Therefore, the marginal product of labor is MPL= 0.3L−0.7K0.7and the
marginal product of capital is MPK= 0.7L0.3K−0.3.
Question 26
Question
Let Qbe the total output of a firm, Lbe the amount of labor used, and Kbe
the amount of capital used. Consider the production function Q= 3K0.5L0.5.
If the firm can hire labor at a rate of w= 10 and rent capital at a rate of r= 20,
find the rate at which the firm is willing to substitute labor for capital such that
the total cost is minimized.
Solution
Step 1: The total cost of production (C) is given by the expression C=rK+wL.
Step 2: We can rewrite the total cost in terms of only Kor Lusing the
production function Q= 3K0.5L0.5. Solving for Kin terms of Qand L, we
have:
K=Q2
9L
24
Step 3: Substitute K=Q2
9Linto the cost function to express Cin terms of
Qand L:
C= 20 Q2
9L+ 10L
Step 4: To minimize total cost, we need to find the critical point of the
function Cwith respect to L. To do this, differentiate Cwith respect to Land
set it equal to 0:
dC
dL =−20Q2
9L2+ 10 = 0
Step 5: Solve the equation 20Q2
9L2= 10 for L:
20Q2
9L2= 10
L=r20Q2
9×10 =r2Q2
9=Q√2
3
Step 6: To find the rate at which the firm is willing to substitute labor for
capital, we need to calculate the ratio MRT S =MPL
MPK.
Step 7: Calculate the marginal product of labor (M PL) and marginal prod-
uct of capital (MPK) using the production function Q= 3K0.5L0.5:
MPL=∂Q
∂L =3
2√K=3
2rQ2
9L=Q
2√2L
MPK=∂Q
∂K =3
2√L=3
2rQ2
9K=Q
2√2K
Step 8: Calculate the MRTS:
MRT S =M PL
MPK
=
Q
2√2L
Q
2√2K
=K
L=
Q2
9L
L=Q2
9L2
Step 9: Substitute the value of Lwe found earlier into M RT S:
MRT S =Q2
9Q√2
32=Q2
9×2Q2
9
=1
2
Therefore, the firm is willing to substitute labor for capital at a rate of 1
2in
order to minimize total cost.
Question 27
Question
Let Q=f(K, L) = K1/3L2/3be a production function, where Qis the quantity
of output, Kis the quantity of capital, and Lis the quantity of labor. Determine
the marginal products of capital and labor, and the returns to scale of this
production function.
25
Solution
Step 1: To find the marginal product of capital (MPK), we take the partial
derivative of the production function with respect to K:
MPK=∂Q
∂K =1
3K−2/3L2/3
Step 2: Similarly, to find the marginal product of labor (MPL), we take the
partial derivative of the production function with respect to L:
MPL=∂Q
∂L =2
3K1/3L−1/3
Step 3: The returns to scale of a production function can be determined by
evaluating f(tK,tL)
f(K,L), where tis a positive constant. Let’s calculate f(2K, 2L):
f(2K, 2L) = (2K)1/3(2L)2/3= 21/3K1/3·22/3L2/3= 2K1/3L2/3= 2Q
Step 4: Now we calculate f(2K,2L)
f(K,L):
f(2K, 2L)
f(K, L)=2Q
Q= 2
Therefore, since f(tK,tL)
f(K,L)= 2 for all positive t, this production function
exhibits increasing returns to scale.
Question 28
Question
Consider a firm with the production function Q=L1/3K2/3, where Qrepresents
the output, Lis the labor input, and Kis the capital input. Suppose the firm
wants to increase output Qby 16 units while keeping the ratio of labor to capital
constant. What percentage increase in the labor input Lis needed to achieve
this increase in output?
Solution
Step 1: Calculate the initial levels of labor and capital inputs necessary to
produce the original output Q. Given the production function Q=L1/3K2/3,
we have Q=L1/3K2/3. Let the initial levels of labor and capital inputs be
denoted as L0and K0, respectively. Thus, Q=L1/3
0K2/3
0.
Step 2: Calculate the new levels of labor and capital inputs necessary to
produce the increased output Q+ 16. To increase the output by 16 units, the
new output will be Q+ 16. Therefore, we have (Q+ 16) = L1/3K2/3.
26
Step 3: Determine the new levels of labor and capital inputs required to
achieve Q+ 16 with the same labor-to-capital ratio as before. Since the labor-
to-capital ratio is to remain constant, we can write L=kK for some constant
k. Substitute this into the equation (Q+ 16) = L1/3K2/3:
(Q+ 16) = (kK)1/3K2/3.
Step 4: Find the value of kand calculate the new labor input Lto achieve
Q+ 16 units of output. From the equation above, we have:
(Q+ 16) = k1/3K1/3K2/3
(Q+ 16) = kK.
Since Q=L1/3
0K2/3
0and Q+ 16 = kK, let’s solve for kin terms of the initial
levels L0and K0:
L1/3
0K2/3
0+ 16 = kK0.
k=L1/3
0K2/3
0+ 16
K0
.
Step 5: Calculate the new labor input Lneeded to achieve Q+ 16 units of
output. To determine the new labor input, substitute kback into the labor-to-
capital ratio equation:
L=L1/3
0K2/3
0+ 16
K0×K.
L=L1/3
0 L1/3
0K2/3
0+ 16
K0!.
Step 6: Calculate the percentage increase in labor input Lneeded to achieve
Q+16 units of output. The percentage increase in labor input can be calculated
as:
% increase in L=L−L0
L0×100.
Question 29
Question
Consider a production function given by Q= 3L1/3K2/3, where Qrepresents
the quantity of a product produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the cost of labor is wand the cost of capital is
r, express the total cost of production (C) as a function of Q,w, and r.
27
Solution
Step 1: The total cost of production (C) can be expressed as the sum of the
cost of labor and the cost of capital:
C=wL +rK
Step 2: To express total cost (C) in terms of Q, we first need to substitute
the production function into the cost function. This involves solving for Land
Kin terms of Q.
Step 3: From the production function Q= 3L1/3K2/3, we can find Kin
terms of Qand L:
K=Q
3L1/33/2
Step 4: Next, we substitute Kback into the cost function to express it solely
in terms of Qand L:
C=wL +rQ
3L1/33/2
Step 5: To eliminate Lfrom the cost function, we can use the constraint
provided by the production function Q= 3L1/3K2/3to solve for Lin terms of
Q:
L=Q
3K2/33
Step 6: Substituting Lback into the cost function, we express the total cost
of production (C) solely in terms of Q:
C=wQ
3K2/33
+r
Q
3Q
3K2/32/3
3/2
Step 7: Simplifying the expression yields the total cost of production (C) as
a function of Q,w, and r:C=wQ3
27K2+rQ3/2
9
Question 30
Question
Consider a production function Q=L3/7K3/7, 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= 16 and the rental rate of capital is r= 25,
determine the equations for the isoquant and isocost lines that represent the
optimal combination of labor and capital inputs when the firm is minimizing its
cost subject to producing a fixed quantity of output.
28
Solution
Step 1: The isoquant equation represents all possible combinations of labor and
capital input that yield the same level of output. In this case, the isoquant
equation is given by:
Q=L3/7K3/7
Step 2: To determine the optimal combination of labor and capital inputs
that minimize cost while producing a fixed quantity of output, we need to set
up the Lagrangian function:
L=wL +rK −λ(Q−L3/7K3/7)
Step 3: Calculate the partial derivatives of the Lagrangian function with
respect to L,K, and λ, and set them equal to 0 to find the optimal values of L
and K:∂L
∂L = 0 =⇒3
7L−4/7K3/7−λ3
7L−4/7K3/7= 0
∂L
∂K = 0 =⇒3
7L3/7K−4/7−λ3
7L3/7K−4/7= 0
∂L
∂λ = 0 =⇒Q−L3/7K3/7= 0
Step 4: Solve the system of equations to find the optimal values of Land K.
Then, substitute these values into the isoquant equation to obtain the equation
for the optimal isoquant line.
Step 5: The isocost line equation represents all combinations of labor and
capital inputs that exhaust the total budget. It can be expressed as:
wL +rK =C
where Cis the total cost.
Step 6: To find the equation for the isocost line that corresponds to the
optimal combination of labor and capital inputs, substitute the optimal values
of Land Kinto the isocost equation.
Therefore, the equations for the optimal isoquant and isocost lines can be
determined using the Lagrangian method to find the optimal combination of
labor and capital inputs that minimize cost while producing a fixed quantity of
output.
Question 31
Question
Given a production function Q=L3/4K1/4, where Qrepresents the quantity
of output produced, Lrepresents the quantity of labor input, and Krepresents
the quantity of capital input, determine the marginal product of labor (MPL)
and the marginal product of capital (MPK).
29
Solution
Step 1: Find the marginal product of labor (MPL) by taking the partial deriva-
tive of the production function with respect to L.
MPL=∂Q
∂L
Step 2: Calculate MPL.
∂Q
∂L =3
4L−1/4K1/4
Step 3: Simplify the expression for MPL.
MPL=3
4·K1/4
L1/4=3
4·K
L1/4
Step 4: Find the marginal product of capital (MPK) by taking the partial
derivative of the production function with respect to K.
MPK=∂Q
∂K
Step 5: Calculate MPK.
∂Q
∂K =1
4L3/4K−3/4
Step 6: Simplify the expression for MPK.
MPK=1
4·L3/4
K3/4=1
4·L
K3/4
Therefore, the marginal product of labor is 3
4·K
L1/4and the marginal
product of capital is 1
4·L
K3/4.
Question 32
Question
Consider a firm with the production function Q= 10L0.4K0.6. If the price
of labor is W= 10 and the price of capital is R= 20, determine the cost-
minimizing combination of labor and capital that produces 100 units of output.
30
Solution
Step 1: The cost function is given by C=W·L+R·K, where Land Kare
the quantities of labor and capital, respectively.
Step 2: We need to find the total cost function in terms of either Lor K.
Step 3: Given the production function Q= 10L0.4K0.6and the desired
output level Q= 100, we can solve for Kin terms of Las follows:
100 = 10L0.4K0.6
10 = L0.4K0.6
10
L0.4=K0.6
10
L0.4
1
0.6
=K
Step 4: Substitute the expression for Kback into the cost function C=
10L+ 20K:
C= 10L+ 20 10
L0.4
1
0.6
Step 5: To minimize the cost, we need to find the minimum value of Cby
taking the derivative of the cost function with respect to Land setting it equal
to zero:
dC
dL = 10 −40
L1.4= 0
10 = 40
L1.4
L1.4= 4
L= 41/1.4
Step 6: With the optimal value of L, we can find the corresponding value of
Kusing the earlier expression:
K=10
(41/1.4)0.4
1
0.6
Step 7: Calculate the specific values for Land Kto determine the cost-
minimizing combination of labor and capital.
31
Question 33
Question
Consider a production function given by Q=L0.4K0.6, where Qrepresents the
level of output, Lis the quantity of labor input, and Kis the quantity of capital
input. Find the marginal product of labor (MPL) and the marginal product of
capital (MPK) at the point (5,10) where L= 5 units and K= 10 units.
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 L, holding Kconstant.
MPL=∂Q
∂L = 0.4L−0.6K0.6
Step 2: Substitute the values L= 5 and K= 10 into the expression for
MPLto find the marginal product of labor at the point (5,10).
MPL(5,10) = 0.4(5)−0.6(10)0.6= 0.4(0.1132)(10) = 0.4528
Step 3: Now, let’s find the marginal product of capital (M PK) by taking
the partial derivative of the production function with respect to K, holding L
constant.
MPK=∂Q
∂K = 0.6L0.4K−0.4
Step 4: Substitute the values L= 5 and K= 10 into the expression for
MPKto find the marginal product of capital at the point (5,10).
MPK(5,10) = 0.6(5)0.4(10)−0.4= 0.6(1.583)0.6309 = 0.5961
Therefore, at the point (5,10), the marginal product of labor is 0.4528 units
of output per unit increase in labor, and the marginal product of capital is
0.5961 units of output per unit increase in capital.
Question 34
Question
Given a production function Q= 50L0.5K0.5, where Qrepresents the quantity of
output, Lrepresents the quantity of labor input, and Krepresents the quantity
of capital input. Determine the marginal product of labor and the marginal
product of capital.
32
Solution
Step 1: To find the marginal product of labor (MPL), we take the partial
derivative of the production function with respect to labor (L):
MPL=∂Q
∂L = 0.5×50 ×L−0.5×K0.5
Step 2: Simplify the expression to get the marginal product of labor:
MPL= 25 ×K0.5×L−0.5
Step 3: To find the marginal product of capital (MPK), we take the partial
derivative of the production function with respect to capital (K):
MPK=∂Q
∂K = 0.5×50 ×L0.5×K−0.5
Step 4: Simplify the expression to get the marginal product of capital:
MPK= 25 ×L0.5×K−0.5
Therefore, the marginal product of labor is 25×K0.5×L−0.5and the marginal
product of capital is 25 ×L0.5×K−0.5.
Question 35
Question
Consider a production function given by Q=L0.4K0.6, where Qrepresents the
level of output, Lis the quantity of labor, and Kis the quantity of capital.
If the price of labor is w= 10 and the price of capital is r= 20, what is the
cost-minimizing combination of labor and capital required to produce 100 units
of output?
Solution
Step 1: The cost function for producing Qunits of output is given by C=
wL +rK.
Step 2: We want to minimize the cost function C= 10L+ 20Ksubject to
the constraint L0.4K0.6= 100.
Step 3: We can rewrite the constraint as K=100
L0.41/0.6=100
L2/55/3=
1005/3L1/3.
Step 4: Substituting K= 1005/3L1/3into the cost function gives C=
10L+ 20(1005/3L1/3) = 10L+ 20(1005/3)L1/3.
Step 5: We can simplify the cost function to C= 10L+ 20(1005/3)L1/3=
10L+ 20(1005/3)L1/3.
Step 6: To minimize the cost function, we take the derivative with respect
to Land set it equal to zero: dC
dL = 10 + 20
3(1005/3)L−2/3= 0.
33
Step 3: Plug in the given production function Q= 4L0.5K0.5and L= 16
into the average product of capital formula.
APK=4(16)0.5K0.5
K
Step 4: Simplify the expression.
APK= 4(16)0.5K0.5−1
APK= 64K−0.5
Question 2
Question
Let Q=f(L, K) be the production function of a firm, where Qrepresents the
total output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. Suppose the production function is given by Q= 4√LK. Find the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor (MPL), we need to find the partial
derivative of the production function with respect to labor L:
∂Q
∂L =∂
∂L(4√LK)
Step 2: Using the chain rule for differentiation, we have:
∂Q
∂L = 2√K
Therefore, the marginal product of labor is MPL= 2√K.
Step 3: To find the marginal product of capital (M PK), we need to find the
partial derivative of the production function with respect to capital K:
∂Q
∂K =∂
∂K (4√LK)
Step 4: Using the chain rule for differentiation, we have:
∂Q
∂K = 2√L
Therefore, the marginal product of capital is MPK= 2√L.
2
Question 3
Question
Consider a production function given by Q= 10L0.5K0.3, where Qis the quan-
tity of output, Lis the quantity of labor input, and Kis the quantity of capital
input. Determine the marginal product of labor (MPL) when L= 4 units and
K= 9 units.
Solution
Step 1: Find the total product of labor (T PL). Given the production function
Q= 10L0.5K0.3, we can find the total product of labor (T PL) by plugging in
the values of L= 4 units and K= 9 units:
T PL= 10(4)0.5(9)0.3
T PL= 10(2)(2.0801)
T PL≈41.60
Step 2: Find the total product of labor when L= 5 units. To find the
marginal product of labor (MPL), we need to find the total product of labor
when L= 5 units:
T P ′
L= 10(5)0.5(9)0.3
T P ′
L= 10(2.2361)(2.0801)
T P ′
L≈46.58
Step 3: Calculate the marginal product of labor (MPL). The marginal prod-
uct of labor (MPL) is calculated as the change in total product of labor with
respect to a change in labor input. Therefore, MPL is given by:
MP L =∆T PL
∆L
MP L =T P ′
L−T PL
5−4
MP L =46.58 −41.60
1
MP L = 4.98
Therefore, the marginal product of labor (MPL) when L= 4 units and
K= 9 units is approximately 4.98 units.
3
Question 4
Question
Consider a production function given by Q= 5LK, where Qrepresents the
quantity of output, Lrepresents labor input, and Krepresents capital input.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK).
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function Qwith respect to labor input L, while treating capital K
as a constant. dQ
dL = 5K
Step 2: Therefore, the marginal product of labor (MPL) is given by dQ
dL =
5K.
Step 3: Now, let’s find the marginal product of capital (MPK) by differenti-
ating the production function Qwith respect to capital input K, while treating
labor Las a constant. dQ
dK = 5L
Step 4: Hence, the marginal product of capital (MPK) is dQ
dK = 5L.
Therefore, the marginal product of labor (MPL) is 5K, and the marginal
product of capital (MPK) is 5L.
Question 5
Question
Let Q= 2L0.5K0.3be the production function of a firm, where Qis the level of
output, Lis the quantity of labor, and Kis the quantity of capital. Determine
the total output when the firm employs 25 units of labor and 16 units of capital.
Solution
Step 1: Substitute L= 25 and K= 16 into the production function to find the
total output.
Q= 2(25)0.5(16)0.3
= 2 ·5·2
= 20 ·2
= 40 units
4
Question 6
Question
Consider a production function given by q=K3/4L1/4, where qrepresents the
quantity of output produced, Kis the quantity of capital input, and Lis the
quantity of labor input. Suppose the price of capital is r= 4 and the price
of labor is w= 2. Calculate the total cost function Cin terms of qfor this
production function.
Solution
Step 1: We start by defining the total cost function Cin terms of q. Given that
C=rK +wL, we need to express Kand Lin terms of qusing the production
function q=K3/4L1/4. Step 2: To find Kin terms of q, we raise both sides of
the production function to the power of 4/3:
q4/3=K
Step 3: Replace Kin the total cost function:
C= 4q4/3+wL
Step 4: To find Lin terms of q, we raise both sides of the production function
to the power of 4 and −1 respectively:
q4=K3L
L=q4
K3=q4
(q4/3)3=q16/3
Step 5: Substitute Lback into the total cost function:
C= 4q4/3+ 2q16/3
Step 6: Simplifying the total cost function gives us the final result:
C= 4q4/3+ 2q16/3
Question 7
Question
Consider a production function Q= 4L0.5K0.3, where Qrepresents the quantity
of output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. Suppose the wage rate is w= 20 and the rental rate of capital
is r= 30. If the firm wants to produce 1024 units of output at minimum cost,
how many units of labor and capital should the firm hire?
5
Solution
Step 1: To minimize the cost of production, the firm needs to use labor and
capital inputs in such a way that the marginal cost of each input is equal to its
price. Therefore, we set the marginal cost of labor equal to the wage rate and
the marginal cost of capital equal to the rental rate.
Step 2: The marginal cost of labor (MCL) is given by:
MCL=∂Cost
∂L =w·MP L
where MP L is the marginal product of labor. The marginal product of labor
is the derivative of the production function with respect to labor:
MP L =∂Q
∂L = 2L−0.5K0.3
Step 3: Substituting M P L into the equation for MCLand setting it equal
to the wage rate:
20 = 20L−0.5K0.3
Step 4: The marginal cost of capital (MCK) is given by:
MCK=∂Cost
∂K =r·MP K
where MP K is the marginal product of capital. The marginal product of capital
is the derivative of the production function with respect to capital:
MP K =∂Q
∂K = 1.2L0.5K−0.7
Step 5: Substituting MP K into the equation for MCKand setting it equal
to the rental rate:
30 = 36L0.5K−0.7
Step 6: Given that the firm wants to produce 1024 units of output, we have:
4L0.5K0.3= 1024
Step 7: Solving the three equations simultaneously will give us the values of
Land Kthat minimize the cost of production. After solving, we find L= 40
and K= 160. Therefore, the firm should hire 40 units of labor and 160 units
of capital to produce 1024 units of output at minimum cost.
Question 8
Question
Consider a production function given by Q= 10L0.5K0.5, where Qrepresents
the quantity of output produced, Lis the amount of labor input, and Kis
the amount of capital input. If the wage rate is w= 2 and the rental rate of
capital is r= 3, calculate the cost-minimizing combination of inputs required
to produce 100 units of output.
6
Solution
Step 1: The cost-minimizing condition for the firm is given by the equation
w
MP L =r
MP K , where MPL is the marginal product of labor and MPK is the
marginal product of capital. We first find these marginal products by taking
partial derivatives of the production function with respect to Land K. Step 2:
Calculate the marginal product of labor (MPL) by differentiating the production
function with respect to L:
MP L =∂Q
∂L = 5L−0.5K0.5
Step 3: Calculate the marginal product of capital (MPK) by differentiating the
production function with respect to K:
MP K =∂Q
∂K = 5L0.5K−0.5
Step 4: Substitute the given values of w,r, and the production function into
the cost-minimizing condition w
MP L =r
MP K in order to solve for the optimal
combination of inputs. Step 5: Setting up the cost-minimizing condition, we
have: 2
5L−0.5K0.5=3
5L0.5K−0.5
Step 6: Rearrange the equation to isolate Lin terms of K:
2K= 3L
Step 7: We can use the firm’s production function Q= 10L0.5K0.5and the
condition 2K= 3Lto find the optimal combination of inputs that produce 100
units of output.
Substitute L=2
3Kinto the production function:
100 = 10 2
3K0.5
K0.5
100 = 10 4
90.5
K·K
Step 8: Solve for K:
100 = 10 4
90.5
K2
K2=100
10 4
90.5
K2=100
10 2
3
K2=100
6.67
7
K2≈15
Step 9: Solve for K:
K≈√15 ≈3.87
Step 10: Calculate Lusing 2K= 3L:
2(3.87) = 3L
7.74 = 3L
L≈7.74
3≈2.58
Therefore, the cost-minimizing combination of inputs required to produce 100
units of output is approximately L≈2.58 and K≈3.87.
Question 9
Question
Consider a production function given by Q= 5L0.4K0.6, where Qrepresents
the total output, Lrepresents the quantity of labor input, and Krepresents the
quantity of capital input. Calculate the marginal product of labor (M PL) and
the marginal product of capital (MPK) at a point where 100 units of labor and
64 units of capital are being used.
Solution
To find the marginal product of labor and capital, we will first calculate the
partial derivatives of the production function with respect to labor and capital.
Step 1: Calculate the marginal product of labor (M PL)
MPL=∂Q
∂L = 2L−0.6K0.6
At the point where 100 units of labor and 64 units of capital are being used:
MPL= 2(100)−0.6(64)0.6= 2(0.01)(4) = 0.08
Therefore, the marginal product of labor at this point is 0.08.
Step 2: Calculate the marginal product of capital (M PK)
MPK=∂Q
∂K = 3L0.4K−0.4
At the point where 100 units of labor and 64 units of capital are being used:
MPK= 3(100)0.4(64)−0.4= 3(2)(0.25) = 1.5
Therefore, the marginal product of capital at this point is 1.5.
8
Question 10
Question
A company’s production function is given by Q= 2K0.5L0.5. Determine the
marginal product of labor and the marginal product of capital, and discuss what
happens to each as more units of labor and capital are added.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate Qwith
respect to L:
MPL=∂Q
∂L =∂
∂L(2K0.5L0.5)
MPL= 1K0.5·0.5L−0.5=K0.5L−0.5
Step 2: To find the marginal product of capital (M PK), we differentiate Q
with respect to K:
MPK=∂Q
∂K =∂
∂K (2K0.5L0.5)
MPK= 2 ·0.5K−0.5L0.5=K−0.5L0.5
Step 3: Observing the marginal products of labor and capital: - For labor:
MPL=K0.5L−0.5. As more units of labor are added, the marginal product of
labor decreases, indicating diminishing marginal returns to labor. - For capital:
MPK=K−0.5L0.5. Similarly, as more units of capital are added, the marginal
product of capital decreases, showing diminishing marginal returns to capital.
In summary, adding more units of labor and capital will lead to diminishing
marginal returns to each factor of production.
Question 11
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the total
output, Lis the quantity of labor, and Kis the quantity of capital. If the
wage rate is w= 5 and the rental rate for capital is r= 10, calculate the cost-
minimizing combination of labor and capital required to produce 100 units of
output.
Solution
Given the production function Q=L0.5K0.5, the cost-minimizing combination
of labor and capital can be determined by setting up the cost minimization
problem. The cost function, C, is defined as C=wL +rK where wis the wage
rate and ris the rental rate for capital.
9
Step 1: Express the cost function in terms of labor (L) only
C=wL +rQ
K0.5
Step 2: Substitute the given values of w,r, and Qinto the cost
function
C= 5L+ 10 100
K0.5
Step 3: Minimize the cost function by differentiating
dC
dL = 5 −1000
L1.5K0.5= 0
Step 4: Solve for the optimal quantity of labor (L)
5 = 1000
L1.5K0.5
L1.5K0.5= 200
Step 5: Given Q= 100, substitute Qinto the production function
to get
100 = L0.5K0.5
Step 6: Using the optimal quantity of labor found earlier (L1.5K0.5=
200) and the production function, solve for the optimal quantity of
capital (K)
100 = √200K
K=100
√200 =100√200
200 =10√2
2= 5√2
Step 7: Calculate the optimal quantity of labor (L) by substituting
the optimal quantity of capital into L1.5K0.5= 200
L1.5(5√2)0.5= 200
L1.5√5 = 200
L=200
√5
2
3
= 20(√5)2
3= 20 ·51
3
Therefore, the cost-minimizing combination of labor and capital required to
produce 100 units of output is L= 20 ·51
3and K= 5√2.
Question 12
Question
Consider a production function given by Q=L0.5K0.5, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
Given that the cost of labor is wand the cost of capital is r, determine the
cost-minimizing combination of labor and capital required to produce a certain
level of output Q.
10
Solution
Step 1: The firm’s total cost (T C) is given by the sum of the costs of labor and
capital used, which can be expressed as:
T C =wL +rK
Step 2: To minimize total cost for a given level of output Q, we need to
minimize the cost function T C =wL +rK subject to the production function
Q=L0.5K0.5.
Step 3: We can rewrite the production function as K=Q
L2
. Substituting
this into the cost function, we get:
T C =wL +rQ
L2
Step 4: To find the cost-minimizing combination of labor and capital, we
need to minimize the total cost function. To do this, we differentiate T C with
respect to Land set the derivative equal to zero:
dT C
dL =w−2rQ
L2
·1
L2= 0
Step 5: Solving for Lin the above equation, we get:
L3=Q
2r
Step 6: Substituting this value of Lback into the production function K=
Q
L2
, we find:
K=
Q
Q
2r1/3
2
= (2r)2/3·Q1/3
Step 7: Therefore, the cost-minimizing combination of labor and capital
required to produce a certain level of output Qis L=Q
2r1/3
and K=
(2r)2/3·Q1/3.
Question 13
Question
Let Q=L0.3K0.5be a production function representing the quantity of output
(Q) as a function of labor input (L) and capital input (K). If the wage rate
is w= 10 and the rental rate for capital is r= 20, find the minimum cost of
producing 500 units of output.
11
Solution
Given the production function: Q=L0.3K0.5, the cost function can be ex-
pressed as:
C=wL +rK
We are looking to minimize the cost, subject to the constraint Q= 500. We
can rewrite Cas a function of one variable using the constraint:
C= 10L+ 20K= 10L+ 20 Q
L0.32
To find the minimum cost, we will differentiate Cwith respect to Land set
the derivative equal to zero:
dC
dL = 10 −40 Q
L0.32
L−0.7= 0
10 = 40 500
L0.32
L−0.7
1
4=500
L0.32
L−0.7
1
4=250000
L0.6L−0.7
1
4=250000
L0.6L−0.7
L= 50
Therefore, the amount of labor needed to minimize the cost is L= 50. Now,
we can find the corresponding capital input:
K=Q
L0.3=500
500.3=500
50 3
10
=500
11.18 ≈44.69
The minimum cost of producing 500 units of output is
C= 10(50) + 20(44.69) = $1000 + $893.8 = $1893.8
Question 14
Question
A firm has a production function given by Q= 2L0.5K0.5, where Qis the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently has 10 units of labor and 8 units of capital, calculate the
marginal product of labor and the marginal product of capital.
12
Solution
Step 1: Calculate the total product function by substituting the given values of
labor and capital into the production function.
Q= 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 2: Calculate the total product of labor by keeping Kconstant.
QL=10 = 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 3: Calculate the total product of labor when labor increases from 10
units to 11 units. QL=11 = 2(11)0.5(8)0.5
= 2(3.32)(2.83)
= 18.78
Step 4: Calculate the marginal product of labor.
MPL=∆Q
∆L=18.78 −17.84
11 −10 = 0.94
Step 5: Calculate the total product of capital by keeping Lconstant.
QK=8 = 2(10)0.5(8)0.5
= 2(3.16)(2.83)
= 17.84
Step 6: Calculate the total product of capital when capital increases from 8
units to 9 units. QK=9 = 2(10)0.5(9)0.5
= 2(3.16)(3)
= 18.96
Step 7: Calculate the marginal product of capital.
MPK=∆Q
∆K=18.96 −17.84
9−8= 1.12
Therefore, the marginal product of labor is 0.94 units of output per unit
increase in labor, and the marginal product of capital is 1.12 units of output
per unit increase in capital.
13
Question 15
Question
Consider a production function given by Q=L1/2K3/4, where Qrepresents the
total output, Lis the quantity of labor input, and Kis the quantity of capital
input. Given that the price of labor is wand the price of capital is r, show that
the cost minimizing condition for this production function is given by
(3/4)wL = (1/2)rK.
Solution
Step 1: The cost of producing at a given level of output Qis given by C=
wL +rK, where wis the wage rate and ris the rental rate of capital.
Step 2: The cost minimizing condition can be stated as ∂C
∂L =∂C
∂K = 0, which
implies that the marginal cost of labor (w) should be equal to the marginal cost
of capital (r).
Step 3: Computing the partial derivatives of the cost function with respect
to Land K:∂C
∂L =w=1
2Q×1
L=1
2QL−1/2K3/4,
∂C
∂K =r=3
4Q×1
K=3
4QL1/2K−1/4.
Step 4: Setting the partial derivatives equal to 0:
1
2QL−1/2K3/4=w,
3
4QL1/2K−1/4=r.
Step 5: Rearranging the equations to solve for wand rgives:
1
2wL =QK1/4,
3
4rK =QL1/2.
Step 6: Dividing the two equations gives:
1
2wL∇ · 3
4rK =QK1/4∇ · QL1/2,
2
3
wL
rK =K1/4
L1/2,
2
3
wL
rK =K
L,
3
4wL =1
2rK.
Therefore, the cost minimizing condition for this production function is
3
4wL =1
2rK.
14
Question 16
Question
Suppose a firm has the production function Q=LαKβ, where Qrepresents
the quantity of output produced, Lrepresents the quantity of labor input, K
represents the quantity of capital input, and α, β > 0. If the firm is currently
producing 100 units of output using 10 units of labor and 5 units of capital,
by how much should the firm increase its labor input in order to double the
quantity of output produced?
Solution
Step 1: Calculate the firm’s initial total productivity level
The firm’s initial total productivity level is given by plugging in L= 10 and
K= 5 into the production function:
Q= 10α5β= 100
Step 2: Set up the equation for doubling output
For the firm to double the quantity of output produced, we need to find the
value of Lsuch that:
2×100 = (10 + ∆L)α5β
Step 3: Simplify the equation
200 = (10 + ∆L)α5β
Step 4: Use the initial total productivity level to find the value of ∆L
Since Q= 100 when L= 10 and K= 5, we have:
100 = 10α5β
Step 5: Substituting the initial productivity level into the equation for dou-
bling output
200 = (10 + ∆L)α5β
200 = 10α5β×(10 + ∆L)α
Step 6: Solve for ∆L
Dividing both sides by 10α5β, we get:
2 = (1 + ∆L
10 )α
Step 7: Take the α-th root of both sides
21
α= 1 + ∆L
10
15
∆L= 10(2 1
α−1)
Therefore, the firm should increase its labor input by 10(2 1
α−1) units in
order to double the quantity of output produced.
Question 17
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the output,
Lis the quantity of labor, and Kis the quantity of capital. Suppose the price
of labor is wand the price of capital is r.
Assume that we have a fixed total cost of production C. Find the optimal
combination of labor and capital that minimizes the cost while producing a
given level of output Q.
Solution
Let C=wL +rK represent the total cost of production. We want to minimize
this cost subject to the production function Q=L0.5K0.5.
Step 1: Set up the Lagrange function Let L(L, K, λ) = wL +rK +
λ(Q−L0.5K0.5) be the Lagrange function, where λis the Lagrange multiplier.
Step 2: Find the first-order conditions The first-order conditions are:
∂L
∂L =w−0.5λL−0.5K0.5= 0
∂L
∂K =r−0.5λL0.5K−0.5= 0
∂L
∂λ =Q−L0.5K0.5= 0
Solving the above system of equations, we can find expressions for L,K,
and λthat satisfy the first-order conditions.
Step 3: Solve for the optimal combination of labor and capital
From the first two equations, we have:
(w= 0.5λL−0.5K0.5
r= 0.5λL0.5K−0.5
Dividing these two equations, we get:
w
r=L
K
Combining this with the production function Q=L0.5K0.5, we can solve for
optimal values of Land Kin terms of Q.
Step 4: Verify the second-order conditions Finally, we need to verify
that the solution obtained in Step 3 results in a minimum cost. This involves
checking the concavity of the cost function.
16
The optimal combination of labor and capital that minimizes the cost while
producing a given level of output Qcan be found using the above steps.
Question 18
Question
Consider a production function given by Q= 2L0.5K0.5, where Qrepresents the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the price of labor is wand the price of capital is r, derive the equation for
the total cost of production (C) in terms of w,r,Q,L, and K.
Solution
Step 1: The total cost of production (C) can be expressed as the sum of the
cost of labor (CL) and the cost of capital (CK):
C=CL+CK
Step 2: The cost of labor (CL) can be calculated by multiplying the quantity
of labor (L) by the price of labor (w):
CL=w·L
Step 3: Similarly, the cost of capital (CK) can be calculated by multiplying
the quantity of capital (K) by the price of capital (r):
CK=r·K
Step 4: Substitute the expressions for CLand CKinto the total cost equa-
tion:
C=w·L+r·K
Step 5: Since the quantity of output (Q) is given by Q= 2L0.5K0.5, we can
rewrite the total cost equation in terms of Q,L,K,w, and r:
C=w·L+r·K=w·Q
2K0.5+r·K
Step 6: Simplify the total cost equation:
C=wQ
2K0.5+rK
Therefore, the equation for the total cost of production (C) in terms of w,
r,Q,L, and Kis given by C=wQ
2K0.5+rK.
17
Question 19
Question
Consider a production function Q= 2L0.5K0.5where Qrepresents total output,
Lis labor input, and Kis capital input.
Suppose the cost of labor is wper unit and the cost of capital is rper unit.
If the firm’s goal is to minimize the cost of producing a given level of output Q,
what is the optimal combination of labor and capital inputs?
Solution
Step 1: The firm’s cost function (C) can be expressed as the sum of labor (L)
and capital (K) costs:
C=wL +rK
Step 2: We are given the production function Q= 2L0.5K0.5, which repre-
sents the relationship between output and inputs.
Step 3: To minimize the cost of producing a given level of output Q, we
need to find the optimal combination of labor and capital inputs that maximizes
output Qsubject to the production function and the cost constraint.
Step 4: We can formulate the firm’s optimization problem as follows:
Maximize Q= 2L0.5K0.5subject to C=wL +rK
Step 5: We will use the method of Lagrange multipliers to solve this opti-
mization problem. Define the Lagrangian function as:
J= 2L0.5K0.5+λ(wL +rK −C)
Step 6: Taking the partial derivatives of the Lagrangian function with re-
spect to L,K, and λand setting them equal to zero will give us the necessary
conditions for optimization.
Step 7: ∂J
∂L = 0, ∂J
∂K = 0, and ∂J
∂λ = 0 lead to the following equations:
K0.5−λw = 0
L0.5−λr = 0
wL +rK =C
Step 8: Solving the system of equations will give us the optimal values for
Land Kin terms of w,r, and C.
Step 9: These optimal values represent the combination of labor and capital
inputs that minimizes the cost of producing a given level of output Q.
18
Question 20
Question
Suppose a production function is given by Q=L0.3K0.5, where Qrepresents
the output, Lrepresents labor input, and Krepresents capital input. If the
firm currently employs 25 units of labor and 16 units of capital, calculate the
marginal product of labor and the marginal product of capital at this level of
inputs.
Solution
Step 1: To find the marginal product of labor, we first need to calculate the total
product when L= 25 and K= 16. We can then find the marginal product of
labor by taking the derivative of the production function with respect to labor.
Step 2: Calculating the total product:
Q=L0.3K0.5
Q= 250.3×160.5
Q= 2.924 ×4
Q= 11.696
Therefore, the total product (Q) is 11.696 when L= 25 and K= 16.
Step 3: Calculating the marginal product of labor: To find the marginal
product of labor, we differentiate the production function with respect to L:
∂Q
∂L = 0.3L−0.7K0.5
Now, substitute L= 25 and K= 16 into the equation:
∂Q
∂L = 0.3(25)−0.7(16)0.5
∂Q
∂L = 0.3(0.004)(4)
∂Q
∂L = 0.0048
Thus, the marginal product of labor when L= 25 and K= 16 is 0.0048.
Step 4: Calculating the marginal product of capital: Similarly, to find the
marginal product of capital, we differentiate the production function with re-
spect to K:∂Q
∂K = 0.5L0.3K−0.5
Substitute L= 25 and K= 16 into the equation:
∂Q
∂K = 0.5(25)0.3(16)−0.5
19
∂Q
∂K = 0.5(2.924)(0.25)
∂Q
∂K = 0.3655
Therefore, the marginal product of capital when L= 25 and K= 16 is
0.3655.
Question 21
Question
Consider a production function Q= 5L1/3K2/3, where Qis the total output,
Lis the quantity of labor input, and Kis the quantity of capital input. Given
that the price of labor (w) is 2 and the price of capital (r) is 3, calculate the
cost-minimizing quantities of labor and capital required to produce 100 units of
output.
Solution
Step 1: Write the cost-minimization problem using the Lagrange multiplier
method. Let C=wL +rK be the cost function. We want to minimize C
subject to the production constraint Q= 5L1/3K2/3. Set up the Lagrangian:
L= 2L+ 3K−λ(5L1/3K2/3−100)
Step 2: Find the first-order conditions by taking partial derivatives with
respect to L,K, and λ.
Partial derivative with respect to L:
∂L
∂L = 2 −5λ
3L−2/3K2/3= 0
Partial derivative with respect to K:
∂L
∂K = 3 −10λ
3L1/3K−1/3= 0
Partial derivative with respect to λ:
∂L
∂λ = 5L1/3K2/3−100 = 0
Step 3: Solve the system of equations to find the optimal values for L,K,
and λ. Solving the first two equations simultaneously, we find:
2
53
103/2
=LK
20
2
53
103/2
=2
53
102/3
L1/3K2/3
L= 10, K = 5
Step 4: Calculate the cost-minimizing quantities of labor and capital re-
quired to produce 100 units of output. Substitute L= 10 and K= 5 into the
production function to find the cost-minimizing quantities of labor and capital:
Q= 5(10)1/3(5)2/3= 100
Therefore, the cost-minimizing quantities of labor and capital required to
produce 100 units of output are L= 10 units and K= 5 units, respectively.
Question 22
Question
Let Qbe the total output of a firm, Lbe the quantity of labor, and Kbe the
quantity of capital. Suppose the production function of the firm is given by
Q= 2K0.5L0.5. If the firm faces a rental rate of r= 4 and a wage rate of w= 3,
find the total cost function (T C) in terms of Q.
Solution
Step 1: Start by writing the total cost function (T C) in terms of the inputs K
and Lusing the rental rate rand wage rate w.
TC = rK +wL
Step 2: Find the cost of capital using the rental rate rand the quantity of
capital K.
rK = 4K
Step 3: Find the cost of labor using the wage rate wand the quantity of
labor L.
wL = 3L
Step 4: Substitute the expressions for the cost of capital and labor back into
the total cost function T C.
TC = 4K+ 3L
Step 5: Find the relationship between Kand Lusing the production function
Q= 2K0.5L0.5.
Q= 2K0.5L0.5
Q= 2√K√L
√K=Q
2√L
21
K=Q2
4L
Step 6: Substitute the expression for Kback into the total cost function
T C.
TC = 4 Q2
4L+ 3L
TC = Q2/L + 3L
Therefore, the total cost function (T C) in terms of Qis TC = Q2/L + 3L.
Question 23
Question
Consider a production function given by Q= 100K0.5L0.5, where Qrepresents
the level of output, Kis the quantity of capital used, and Lis the quantity of
labor used. Calculate the marginal product of labor and the marginal product
of capital.
Solution
Step 1: To find the marginal product of labor, we take the partial derivative
of the production function with respect to labor (L) while holding capital (K)
constant. ∂Q
∂L = 0.5·100K0.5L−0.5
∂Q
∂L = 50K0.5L−0.5
Step 2: Simplifying the expression gives us the marginal product of labor.
Marginal Product of Labor = 50 K0.5
L0.5
Step 3: Next, we find the marginal product of capital by taking the partial
derivative of the production function with respect to capital (K) while holding
labor (L) constant.
∂Q
∂K = 0.5·100K−0.5L0.5
∂Q
∂K = 50K−0.5L0.5
Step 4: Simplifying the expression gives us the marginal product of capital.
Marginal Product of Capital = 50 L0.5
K0.5
Therefore, the marginal product of labor is 50 K0.5
L0.5and the marginal product
of capital is 50 L0.5
K0.5.
22
Question 24
Question
Consider a production function given by Q= 10L0.5K0.5, where Qis the total
output, Lis the quantity of labor, and Kis the quantity of capital. Find the
marginal product of labor (MPL) and marginal product of capital (MPK) at
the point where L= 25 and K= 16.
Solution
Step 1: To find the marginal product of labor (MPL), we differentiate the
production function with respect to Lwhile holding Kconstant.
∂Q
∂L = 5K0.5L−0.5
Step 2: Substitute L= 25 and K= 16 into the MPL formula.
∂Q
∂L L=25,K=16
= 5(16)0.5(25)−0.5= 5 ×4×0.2=4
Therefore, the marginal product of labor (MPL) at the point L= 25 and
K= 16 is 4.
Step 3: To find the marginal product of capital (MPK), we differentiate the
production function with respect to Kwhile holding Lconstant.
∂Q
∂K = 5L0.5K−0.5
Step 4: Substitute L= 25 and K= 16 into the MPK formula.
∂Q
∂K L=25,K=16
= 5(25)0.5(16)−0.5= 5 ×5×0.25 = 6.25
Therefore, the marginal product of capital (MPK) at the point L= 25 and
K= 16 is 6.25.
Question 25
Question
Consider a production function given by Q=L0.3K0.7, where Qrepresents the
quantity produced, Lis the amount of labor, and Kis the amount of capital.
Find the marginal product of labor, MPL, and the marginal product of capital,
MPK.
23
Solution
Step 1: Find the partial derivative of the production function with
respect to labor L.To find the marginal product of labor (MPL), we need
to take the partial derivative of the production function with respect to labor
L. This will give us the change in output for a one-unit change in labor.
∂Q
∂L = 0.3L−0.7K0.7
Step 2: Simplify the expression to find M PL.The marginal product
of labor, MPL, is the partial derivative we found in Step 1.
MPL= 0.3L−0.7K0.7
Step 3: Find the partial derivative of the production function with
respect to capital K.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 = 0.7L0.3K−0.3
Step 4: Simplify the expression to find M PK.The marginal product
of capital, MPK, is the partial derivative we found in Step 3.
MPK= 0.7L0.3K−0.3
Therefore, the marginal product of labor is MPL= 0.3L−0.7K0.7and the
marginal product of capital is MPK= 0.7L0.3K−0.3.
Question 26
Question
Let Qbe the total output of a firm, Lbe the amount of labor used, and Kbe
the amount of capital used. Consider the production function Q= 3K0.5L0.5.
If the firm can hire labor at a rate of w= 10 and rent capital at a rate of r= 20,
find the rate at which the firm is willing to substitute labor for capital such that
the total cost is minimized.
Solution
Step 1: The total cost of production (C) is given by the expression C=rK+wL.
Step 2: We can rewrite the total cost in terms of only Kor Lusing the
production function Q= 3K0.5L0.5. Solving for Kin terms of Qand L, we
have:
K=Q2
9L
24
Step 3: Substitute K=Q2
9Linto the cost function to express Cin terms of
Qand L:
C= 20 Q2
9L+ 10L
Step 4: To minimize total cost, we need to find the critical point of the
function Cwith respect to L. To do this, differentiate Cwith respect to Land
set it equal to 0:
dC
dL =−20Q2
9L2+ 10 = 0
Step 5: Solve the equation 20Q2
9L2= 10 for L:
20Q2
9L2= 10
L=r20Q2
9×10 =r2Q2
9=Q√2
3
Step 6: To find the rate at which the firm is willing to substitute labor for
capital, we need to calculate the ratio MRT S =MPL
MPK.
Step 7: Calculate the marginal product of labor (M PL) and marginal prod-
uct of capital (MPK) using the production function Q= 3K0.5L0.5:
MPL=∂Q
∂L =3
2√K=3
2rQ2
9L=Q
2√2L
MPK=∂Q
∂K =3
2√L=3
2rQ2
9K=Q
2√2K
Step 8: Calculate the MRTS:
MRT S =M PL
MPK
=
Q
2√2L
Q
2√2K
=K
L=
Q2
9L
L=Q2
9L2
Step 9: Substitute the value of Lwe found earlier into M RT S:
MRT S =Q2
9Q√2
32=Q2
9×2Q2
9
=1
2
Therefore, the firm is willing to substitute labor for capital at a rate of 1
2in
order to minimize total cost.
Question 27
Question
Let Q=f(K, L) = K1/3L2/3be a production function, where Qis the quantity
of output, Kis the quantity of capital, and Lis the quantity of labor. Determine
the marginal products of capital and labor, and the returns to scale of this
production function.
25
Solution
Step 1: To find the marginal product of capital (MPK), we take the partial
derivative of the production function with respect to K:
MPK=∂Q
∂K =1
3K−2/3L2/3
Step 2: Similarly, to find the marginal product of labor (MPL), we take the
partial derivative of the production function with respect to L:
MPL=∂Q
∂L =2
3K1/3L−1/3
Step 3: The returns to scale of a production function can be determined by
evaluating f(tK,tL)
f(K,L), where tis a positive constant. Let’s calculate f(2K, 2L):
f(2K, 2L) = (2K)1/3(2L)2/3= 21/3K1/3·22/3L2/3= 2K1/3L2/3= 2Q
Step 4: Now we calculate f(2K,2L)
f(K,L):
f(2K, 2L)
f(K, L)=2Q
Q= 2
Therefore, since f(tK,tL)
f(K,L)= 2 for all positive t, this production function
exhibits increasing returns to scale.
Question 28
Question
Consider a firm with the production function Q=L1/3K2/3, where Qrepresents
the output, Lis the labor input, and Kis the capital input. Suppose the firm
wants to increase output Qby 16 units while keeping the ratio of labor to capital
constant. What percentage increase in the labor input Lis needed to achieve
this increase in output?
Solution
Step 1: Calculate the initial levels of labor and capital inputs necessary to
produce the original output Q. Given the production function Q=L1/3K2/3,
we have Q=L1/3K2/3. Let the initial levels of labor and capital inputs be
denoted as L0and K0, respectively. Thus, Q=L1/3
0K2/3
0.
Step 2: Calculate the new levels of labor and capital inputs necessary to
produce the increased output Q+ 16. To increase the output by 16 units, the
new output will be Q+ 16. Therefore, we have (Q+ 16) = L1/3K2/3.
26
Step 3: Determine the new levels of labor and capital inputs required to
achieve Q+ 16 with the same labor-to-capital ratio as before. Since the labor-
to-capital ratio is to remain constant, we can write L=kK for some constant
k. Substitute this into the equation (Q+ 16) = L1/3K2/3:
(Q+ 16) = (kK)1/3K2/3.
Step 4: Find the value of kand calculate the new labor input Lto achieve
Q+ 16 units of output. From the equation above, we have:
(Q+ 16) = k1/3K1/3K2/3
(Q+ 16) = kK.
Since Q=L1/3
0K2/3
0and Q+ 16 = kK, let’s solve for kin terms of the initial
levels L0and K0:
L1/3
0K2/3
0+ 16 = kK0.
k=L1/3
0K2/3
0+ 16
K0
.
Step 5: Calculate the new labor input Lneeded to achieve Q+ 16 units of
output. To determine the new labor input, substitute kback into the labor-to-
capital ratio equation:
L=L1/3
0K2/3
0+ 16
K0×K.
L=L1/3
0 L1/3
0K2/3
0+ 16
K0!.
Step 6: Calculate the percentage increase in labor input Lneeded to achieve
Q+16 units of output. The percentage increase in labor input can be calculated
as:
% increase in L=L−L0
L0×100.
Question 29
Question
Consider a production function given by Q= 3L1/3K2/3, where Qrepresents
the quantity of a product produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the cost of labor is wand the cost of capital is
r, express the total cost of production (C) as a function of Q,w, and r.
27
Solution
Step 1: The total cost of production (C) can be expressed as the sum of the
cost of labor and the cost of capital:
C=wL +rK
Step 2: To express total cost (C) in terms of Q, we first need to substitute
the production function into the cost function. This involves solving for Land
Kin terms of Q.
Step 3: From the production function Q= 3L1/3K2/3, we can find Kin
terms of Qand L:
K=Q
3L1/33/2
Step 4: Next, we substitute Kback into the cost function to express it solely
in terms of Qand L:
C=wL +rQ
3L1/33/2
Step 5: To eliminate Lfrom the cost function, we can use the constraint
provided by the production function Q= 3L1/3K2/3to solve for Lin terms of
Q:
L=Q
3K2/33
Step 6: Substituting Lback into the cost function, we express the total cost
of production (C) solely in terms of Q:
C=wQ
3K2/33
+r
Q
3Q
3K2/32/3
3/2
Step 7: Simplifying the expression yields the total cost of production (C) as
a function of Q,w, and r:C=wQ3
27K2+rQ3/2
9
Question 30
Question
Consider a production function Q=L3/7K3/7, 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= 16 and the rental rate of capital is r= 25,
determine the equations for the isoquant and isocost lines that represent the
optimal combination of labor and capital inputs when the firm is minimizing its
cost subject to producing a fixed quantity of output.
28
Solution
Step 1: The isoquant equation represents all possible combinations of labor and
capital input that yield the same level of output. In this case, the isoquant
equation is given by:
Q=L3/7K3/7
Step 2: To determine the optimal combination of labor and capital inputs
that minimize cost while producing a fixed quantity of output, we need to set
up the Lagrangian function:
L=wL +rK −λ(Q−L3/7K3/7)
Step 3: Calculate the partial derivatives of the Lagrangian function with
respect to L,K, and λ, and set them equal to 0 to find the optimal values of L
and K:∂L
∂L = 0 =⇒3
7L−4/7K3/7−λ3
7L−4/7K3/7= 0
∂L
∂K = 0 =⇒3
7L3/7K−4/7−λ3
7L3/7K−4/7= 0
∂L
∂λ = 0 =⇒Q−L3/7K3/7= 0
Step 4: Solve the system of equations to find the optimal values of Land K.
Then, substitute these values into the isoquant equation to obtain the equation
for the optimal isoquant line.
Step 5: The isocost line equation represents all combinations of labor and
capital inputs that exhaust the total budget. It can be expressed as:
wL +rK =C
where Cis the total cost.
Step 6: To find the equation for the isocost line that corresponds to the
optimal combination of labor and capital inputs, substitute the optimal values
of Land Kinto the isocost equation.
Therefore, the equations for the optimal isoquant and isocost lines can be
determined using the Lagrangian method to find the optimal combination of
labor and capital inputs that minimize cost while producing a fixed quantity of
output.
Question 31
Question
Given a production function Q=L3/4K1/4, where Qrepresents the quantity
of output produced, Lrepresents the quantity of labor input, and Krepresents
the quantity of capital input, determine the marginal product of labor (MPL)
and the marginal product of capital (MPK).
29
Solution
Step 1: Find the marginal product of labor (MPL) by taking the partial deriva-
tive of the production function with respect to L.
MPL=∂Q
∂L
Step 2: Calculate MPL.
∂Q
∂L =3
4L−1/4K1/4
Step 3: Simplify the expression for MPL.
MPL=3
4·K1/4
L1/4=3
4·K
L1/4
Step 4: Find the marginal product of capital (MPK) by taking the partial
derivative of the production function with respect to K.
MPK=∂Q
∂K
Step 5: Calculate MPK.
∂Q
∂K =1
4L3/4K−3/4
Step 6: Simplify the expression for MPK.
MPK=1
4·L3/4
K3/4=1
4·L
K3/4
Therefore, the marginal product of labor is 3
4·K
L1/4and the marginal
product of capital is 1
4·L
K3/4.
Question 32
Question
Consider a firm with the production function Q= 10L0.4K0.6. If the price
of labor is W= 10 and the price of capital is R= 20, determine the cost-
minimizing combination of labor and capital that produces 100 units of output.
30
Solution
Step 1: The cost function is given by C=W·L+R·K, where Land Kare
the quantities of labor and capital, respectively.
Step 2: We need to find the total cost function in terms of either Lor K.
Step 3: Given the production function Q= 10L0.4K0.6and the desired
output level Q= 100, we can solve for Kin terms of Las follows:
100 = 10L0.4K0.6
10 = L0.4K0.6
10
L0.4=K0.6
10
L0.4
1
0.6
=K
Step 4: Substitute the expression for Kback into the cost function C=
10L+ 20K:
C= 10L+ 20 10
L0.4
1
0.6
Step 5: To minimize the cost, we need to find the minimum value of Cby
taking the derivative of the cost function with respect to Land setting it equal
to zero:
dC
dL = 10 −40
L1.4= 0
10 = 40
L1.4
L1.4= 4
L= 41/1.4
Step 6: With the optimal value of L, we can find the corresponding value of
Kusing the earlier expression:
K=10
(41/1.4)0.4
1
0.6
Step 7: Calculate the specific values for Land Kto determine the cost-
minimizing combination of labor and capital.
31
Question 33
Question
Consider a production function given by Q=L0.4K0.6, where Qrepresents the
level of output, Lis the quantity of labor input, and Kis the quantity of capital
input. Find the marginal product of labor (MPL) and the marginal product of
capital (MPK) at the point (5,10) where L= 5 units and K= 10 units.
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 L, holding Kconstant.
MPL=∂Q
∂L = 0.4L−0.6K0.6
Step 2: Substitute the values L= 5 and K= 10 into the expression for
MPLto find the marginal product of labor at the point (5,10).
MPL(5,10) = 0.4(5)−0.6(10)0.6= 0.4(0.1132)(10) = 0.4528
Step 3: Now, let’s find the marginal product of capital (M PK) by taking
the partial derivative of the production function with respect to K, holding L
constant.
MPK=∂Q
∂K = 0.6L0.4K−0.4
Step 4: Substitute the values L= 5 and K= 10 into the expression for
MPKto find the marginal product of capital at the point (5,10).
MPK(5,10) = 0.6(5)0.4(10)−0.4= 0.6(1.583)0.6309 = 0.5961
Therefore, at the point (5,10), the marginal product of labor is 0.4528 units
of output per unit increase in labor, and the marginal product of capital is
0.5961 units of output per unit increase in capital.
Question 34
Question
Given a production function Q= 50L0.5K0.5, where Qrepresents the quantity of
output, Lrepresents the quantity of labor input, and Krepresents the quantity
of capital input. Determine the marginal product of labor and the marginal
product of capital.
32
Solution
Step 1: To find the marginal product of labor (MPL), we take the partial
derivative of the production function with respect to labor (L):
MPL=∂Q
∂L = 0.5×50 ×L−0.5×K0.5
Step 2: Simplify the expression to get the marginal product of labor:
MPL= 25 ×K0.5×L−0.5
Step 3: To find the marginal product of capital (MPK), we take the partial
derivative of the production function with respect to capital (K):
MPK=∂Q
∂K = 0.5×50 ×L0.5×K−0.5
Step 4: Simplify the expression to get the marginal product of capital:
MPK= 25 ×L0.5×K−0.5
Therefore, the marginal product of labor is 25×K0.5×L−0.5and the marginal
product of capital is 25 ×L0.5×K−0.5.
Question 35
Question
Consider a production function given by Q=L0.4K0.6, where Qrepresents the
level of output, Lis the quantity of labor, and Kis the quantity of capital.
If the price of labor is w= 10 and the price of capital is r= 20, what is the
cost-minimizing combination of labor and capital required to produce 100 units
of output?
Solution
Step 1: The cost function for producing Qunits of output is given by C=
wL +rK.
Step 2: We want to minimize the cost function C= 10L+ 20Ksubject to
the constraint L0.4K0.6= 100.
Step 3: We can rewrite the constraint as K=100
L0.41/0.6=100
L2/55/3=
1005/3L1/3.
Step 4: Substituting K= 1005/3L1/3into the cost function gives C=
10L+ 20(1005/3L1/3) = 10L+ 20(1005/3)L1/3.
Step 5: We can simplify the cost function to C= 10L+ 20(1005/3)L1/3=
10L+ 20(1005/3)L1/3.
Step 6: To minimize the cost function, we take the derivative with respect
to Land set it equal to zero: dC
dL = 10 + 20
3(1005/3)L−2/3= 0.
33
Step 7: Solving for Lin 20
3(1005/3)L−2/3=−10 gives L=20
3(1005/3)3/2.
Step 8: Plug Lback into the constraint L0.4K0.6= 100 to find the corre-
sponding Kvalue.
34
