ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank - Set 3
Liberty University
Question 1
Question
Suppose a firm has the production function Q= 3L0.5K0.5, where Qis the
quantity of output, Lis the quantity of labor input, and Kis the quantity of
capital input. If the firm currently employs 16 units of labor and 16 units of
capital, what is the average product of labor at this level of input?
Solution
Step 1: To find the average product of labor, we need to first calculate the total
output produced with 16 units of labor and 16 units of capital.
Step 2: Plug in L= 16 and K= 16 into the production function Q=
3L0.5K0.5.
Q= 3(16)0.5(16)0.5
Step 3: Simplify the expression.
Q= 3(4)(4) = 48
Step 4: The total output produced with 16 units of labor and 16 units of
capital is 48 units.
Step 5: The average product of labor is calculated as the total output divided
by the quantity of labor input.
AP L =Q
L=48
16
Step 6: Calculate the average product of labor.
AP L = 3
Therefore, the average product of labor when the firm employs 16 units of
labor is 3 units of output per unit of labor.
Question 2
Question
Let Q=L1/2K1/3represent a production function where Qis the output, Lis
the amount of labor, and Kis the amount of capital used. Find the marginal
product of labor (MPL) and the marginal product of capital (MPK).
Solution
To find the marginal product of labor (MPL) and the marginal product of
capital (MPK), we will differentiate the production function Q=L1/2K1/3
with respect to labor Land capital K, respectively.
Step 1: Find MPL
MPL=∂Q
∂L
=∂
∂L(L1/2K1/3)
=1
2·L−1/2·K1/3
=1
2·1
√L·K1/3
Step 2: Find MPK
MPK=∂Q
∂K
=∂
∂K (L1/2K1/3)
=1
3·L1/2·K−2/3
=1
3·L1/2·1
K2/3
=1
3·√L·1
K2/3
Therefore, the marginal product of labor is 1
2·1
√L·K1/3and the marginal
product of capital is 1
3·√L·1
K2/3.
Question 3
Question
Consider a firm with the production function Q=K3/4L1/4, where Qis the
quantity of output, Kis the quantity of capital, and Lis the quantity of labor.
If the firm currently has 16 units of capital and 256 units of labor, calculate the
marginal product of labor at this level of input.
2
Solution
1. The marginal product of labor (MPL) is given by the derivative of the
production function with respect to labor:
MP L =∂Q
∂L =1
4K3/4L−3/4
2. Substitute the values K= 16 and L= 256 into the MPL formula:
MP L =1
4×163/4×256−3/4
3. Simplify the expression:
MP L =1
4×2×1
16
4. Calculate MPL:
MP L =1
32
Therefore, the marginal product of labor at this level of input is 1
32 .
Question 4
Question
Consider the Cobb-Douglas production function given by Q= 2L0.3K0.6, where
Qis the total output, Lis the amount of labor input, and Kis the amount of
capital input. Given that the amount of labor input is fixed at 10 units, calculate
the marginal product of capital when the amount of capital input is 25 units.
Solution
Step 1: Calculate the total output Qwhen L= 10 and K= 25.
Q= 2(10)0.3(25)0.6
Q= 2(100.3)(250.6)
Q= 2(2.154)(10)
Q= 43.08
Step 2: Calculate the total output Qwhen L= 10 and K= 24.
Q′= 2(10)0.3(24)0.6
Q′= 2(100.3)(240.6)
Q′= 2(2.154)(8.366)
3
Q′= 36.09
Step 3: Calculate the marginal product of capital (MPL) using the formula:
MP L =∆Q
∆K
Step 4: Substitute the values of ∆Qand ∆Kinto the formula.
MP L =43.08 −36.09
25 −24
MP L =6.99
1
MP L = 6.99
Therefore, the marginal product of capital when the amount of capital input
is 25 units is 6.99 units of output per unit of capital input.
Question 5
Question
Consider a firm that produces output using two inputs: labor (L) and capital
(K). The production function is given by Q= 2L0.5K0.5.
a) Determine whether this production function exhibits constant, increasing,
or decreasing returns to scale.
b) Find the marginal product of labor and the marginal product of capital.
Solution
a) To determine the returns to scale of the production function, we will examine
how output changes when both inputs are increased by a factor λ.
Q(λL, λK) = 2(λL)0.5(λK)0.5= 2λ0.5λ0.5L0.5K0.5= 2λL0.5K0.5= 2Q
Since Q(λL, λK)=2Q, the production function exhibits constant returns
to scale.
b) The marginal product of labor (MPL) is found by taking the partial
derivative of the production function with respect to L, holding Kconstant.
MPL=∂Q
∂L =∂(2L0.5K0.5)
∂L =K0.5
Similarly, the marginal product of capital (MPK) is found by taking the
partial derivative of the production function with respect to K, holding Lcon-
stant.
MPK=∂Q
∂K =∂(2L0.5K0.5)
∂K =L0.5
4
Question 6
Question
Consider a production function f(x, y) = x2
3·y1
3where xis the quantity of
input Xand yis the quantity of input Y. Find the marginal product of Xand
the marginal product of Y.
Solution
Step 1: To find the marginal product of input X, we need to calculate the
partial derivative ∂f
∂x .
∂f
∂x =2
3x−1
3·y1
3
Step 2: Simplify the expression for the marginal product of input X.
∂f
∂x =2y1
3
3x1
3
Step 3: Next, to find the marginal product of input Y, we need to calculate
the partial derivative ∂f
∂y .
∂f
∂y =x2
3·1
3y−2
3
Step 4: Simplify the expression for the marginal product of input Y.
∂f
∂y =x2
3
3y2
3
Therefore, the marginal product of input Xis 2y1
3
3x1
3
and the marginal product
of input Yis x2
3
3y2
3
for the production function f(x, y) = x2
3·y1
3.
Question 7
Question
Let Q=f(K, L) = K2/3L1/3be the production function for a certain company,
where Qis the total quantity of output produced, Kis the amount of capital
used, and Lis the amount of labor employed. If the company’s current capital
input is 64 units and the current labor input is 27 units, find the marginal
product of labor (MPL) and the marginal product of capital (MPK).
5
Solution
Step 1: Find the marginal product of labor (MPL) The marginal product of
labor (MPL) is defined as the change in output resulting from one additional
unit of labor, holding capital constant. Mathematically, it is given by
MPL=∂Q
∂L =1
3K2/3L−2/3
Step 2: Substitute the given values of Kand Linto the expression for MPL:
MPL=1
3×642/3×27−2/3
MPL=1
3×4×1
3=4
9
Therefore, the marginal product of labor is 4
9units of output for each addi-
tional unit of labor employed.
Step 3: Find the marginal product of capital (M PK) The marginal product
of capital (MPK) is defined as the change in output resulting from one additional
unit of capital, holding labor constant. Mathematically, it is given by
MPK=∂Q
∂K =2
3K−1/3L1/3
Step 4: Substitute the given values of Kand Linto the expression for MPK:
MPK=2
3×64−1/3×271/3
MPK=2
3×1
4×3 = 1
2
Therefore, the marginal product of capital is 1
2units of output for each
additional unit of capital used.
Question 8
Question
Suppose a firm has a production function given by Q= 10L0.5K0.5, where Qis
the level of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently employs 100 units of labor and 25 units of capital, how
much output is the firm currently producing?
6
Solution
Step 1: Substitute the given values of Land Kinto the production function.
Q= 10(100)0.5(25)0.5
= 10(10)(5)
= 500
Step 2: Calculate the output level. Therefore, the firm is currently producing
500 units of output.
Question 9
Question
Let Q=L1/3K2/3represent a production function where Qis the output quan-
tity, Lis the quantity of labor, and Kis the quantity of capital. Suppose the
price of labor is wand the price of capital is r. Given that the total cost function
is T C =wL +rK, find the cost-minimizing combination of labor and capital
required to produce a given level of output.
Solution
To find the cost-minimizing combination of labor and capital, we need to mini-
mize the total cost function subject to the production function constraints.
Step 1: Set up the Lagrangian Function The Lagrangian function is
defined as:
L=wL +rK +λ(Q−L1/3K2/3)
Step 2: Find the Partial Derivatives Find the partial derivatives of the
Lagrangian function with respect to L,K, and λ:
∂L
∂L =w−1
3λL−2/3K2/3
∂L
∂K =r−2
3λL1/3K−1/3
∂L
∂λ =Q−L1/3K2/3
Step 3: Set the Partial Derivatives to Zero Set the partial derivatives
to zero and solve for L,K, and λ:
w−1
3λL−2/3K2/3= 0
r−2
3λL1/3K−1/3= 0
Q−L1/3K2/3= 0
Step 4: Solve the System of Equations Solve the system of equations
to find the optimal values of L,K, and λ.
Step 5: Interpret the Results Once you have the optimal values of L
and K, you have found the cost-minimizing combination of labor and capital
required to produce the given level of output.
7
Question 10
Question
Consider a production function defined by Q=L0.3K0.7, where Qrepresents
the quantity of output, Lrepresents the quantity of labor, and Krepresents
the quantity of capital. Suppose the firm’s output level is fixed at Q= 125
units. If the wage rate is w= 10 and the rental rate of capital is r= 20, what
combination of labor and capital should the firm use to minimize its total cost
of production?
Solution
Step 1: Write the total cost function. The total cost (C) of production is given
by the sum of labor and capital costs:
C=wL +rK
Step 2: Express labor (L) in terms of capital (K) using the production
function. Given: Q=L0.3K0.7Since Q= 125, we have:
125 = L0.3K0.7
L=125
K0.71/0.3
L= 12510
3K−7
3
L= 125 ·K−7
3
Step 3: Substitute the expression for Linto the cost function.
C= 10 ·(125 ·K−7
3) + 20K
C= 1250K−7
3+ 20K
C= 1250K2
3+ 20K
Step 4: Find the minimum total cost by taking the derivative with respect
to Kand setting it to zero.
dC
dK = 1250 ·2
3K−1
3+ 20
Setting the derivative to zero:
1250 ·2
3K−1
3+ 20 = 0
833.33K−1
3=−20
K−1
3=−0.024
8
K= (−0.024)−3=1
0.0243
Step 5: Determine the optimal quantity of labor using the production func-
tion.
L= 125 ·1
0.0243−7
3
Therefore, the firm should use 1
0.0243units of capital and 125 ·1
0.0243−7
3
units of labor to minimize its total cost of production.
Question 11
Question
Suppose a firm’s production function is given by Q= 10L0.5K0.3, where Q
represents the quantity of output, Lis the quantity of labor input, and Kis the
quantity of capital input. If the firm is currently using 100 units of labor and 64
units of capital, determine the marginal product of labor at this level of inputs.
Solution
Step 1: Calculate the partial derivative of the production function with respect
to labor, L.∂Q
∂L = 0.5×10 ×L−0.5×K0.3= 5L−0.5K0.3
Step 2: Substitute the given values of L= 100 and K= 64 into the expres-
sion obtained in Step 1.
∂Q
∂L
L=100,K=64
= 5 ×100−0.5×640.3
∂Q
∂L
L=100,K=64
= 5 ×1
√100 ×10
√64
∂Q
∂L
L=100,K=64
= 5 ×1
10 ×4=2
Therefore, the marginal product of labor at 100 units of labor and 64 units
of capital is 2.
Question 12
Question
Consider a production function Q= 10L0.4K0.6, where Qrepresents the quan-
tity of output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. If the quantity of labor is fixed at 10 units, what is the average prod-
uct of capital when the quantity of capital is increased from 25 to 30 units?
9
Solution
Step 1: Calculate the total product of capital.
Q= 10(10)0.4×250.6= 10 ×5×5 = 250
Step 2: Calculate the new total product of capital after increasing the quan-
tity of capital to 30 units.
Q′= 10(10)0.4×300.6= 10 ×5×6 = 300
Step 3: Calculate the average product of capital. Average product of capital
(APK) is given by the formula:
APK=Q
K
Step 4: Calculate the average product of capital when the quantity of capital
is 25 units.
APK=250
25 = 10
Step 5: Calculate the average product of capital when the quantity of capital
is 30 units.
AP ′
K=300
30 = 10
Therefore, the average product of capital when the quantity of capital is
increased from 25 to 30 units remains constant at 10 units.
Question 13
Question
Consider a production function defined by q= 5K0.5L0.5, where qis the quantity
produced, Kis capital input, and Lis labor input. If the price of capital is
$
4
per unit and the price of labor is
$
2 per unit, determine the cost-minimizing
combination of capital and labor to produce 100 units of output.
Solution
Step 1: The cost-minimization problem can be expressed as: Minimize C=
4K+ 2Lsubject to the production function q= 5K0.5L0.5and the output level
q= 100.
Step 2: Substitute the production function into the cost function to get a
single variable function: C(K) = 4K+ 2 q
5K0.52
Step 3: Given that q= 100, the cost function becomes: C(K) = 4K+
2100
5K0.52= 4K+ 2 20
K0.52= 4K+ 2 400
K= 4K+800
K
Step 4: To find the minimum value of C(K), differentiate with respect to K
and set it equal to 0: dC
dK = 4 −800
K2= 0
10
Step 5: Solve for K: 4 = 800
K2
K2=800
4= 200
K=√200 = 10√2
Step 6: Substitute K= 10√2 back into the production function to find L:
q= 5(10√2)0.5L0.5= 100
50√2L0.5= 100
L0.5=100
50√2=2
√2=√2
L= (√2)2= 2
Therefore, the cost-minimizing combination of capital and labor to produce
100 units of output is K= 10√2 and L= 2.
Question 14
Question
Suppose a company produces two goods, Xand Y, using the production func-
tion Q= 24X0.4Y0.6. Determine the rate at which output Qchanges with
respect to Xwhen X= 5 and Y= 10.
Solution
Step 1: Calculate the partial derivative ∂Q
∂X using the power rule for differenti-
ation. ∂Q
∂X = 0.4×24X−0.6Y0.6= 9.6X−0.6Y0.6
Step 2: Substitute X= 5 and Y= 10 into the expression for ∂Q
∂X .
∂Q
∂X
(5,10) = 9.6(5)−0.6(10)0.6
∂Q
∂X
(5,10) = 9.6(0.1296)(6.4308)
∂Q
∂X
(5,10) = 7.42231
Therefore, the rate at which output Qchanges with respect to Xwhen
X= 5 and Y= 10 is approximately 7.42231.
Question 15
Question
Consider the production function f(x, y)=2x2+ 3y2−xy, where xrepresents
units of labor and yrepresents units of capital. Find the level of output when
x= 3 and y= 4.
11
Solution
To find the level of output when x= 3 and y= 4, we substitute these values
into the production function f(x, y)=2x2+ 3y2−xy.
Step 1: Substitute x= 3 and y= 4 into the production function:
f(3,4) = 2(3)2+ 3(4)2−3(4)
Step 2: Simplify the expression:
f(3,4) = 2(9) + 3(16) −12
f(3,4) = 18 + 48 −12
f(3,4) = 54
Therefore, the level of output when x= 3 and y= 4 is 54 units.
Question 16
Question
Consider a firm with a production function given by Q= 10L0.5K0.5, where Q
represents output, Lrepresents labor input, and Krepresents capital input.
Suppose the firm has 100 units of capital. If the wage rate is 3 and the price
of capital is 2, determine the level of labor input that will maximize output for
the firm.
Solution
Step 1: Calculate the total cost function. Given that the firm’s total cost
function is the sum of labor and capital costs, we can express it as:
T C =wL +rK
Where: - wis the wage rate, - ris the price of capital.
Substitute the given values of w= 3 and r= 2:
T C = 3L+ 2(100)
Step 2: Maximize output by minimizing cost. To maximize output, the
firm needs to minimize cost. In this case, we minimize the total cost function:
T C = 3L+ 200.
Step 3: Differentiate the total cost function to find the level of labor input
that minimizes cost. dT C
dL = 3
Step 4: Set the derivative equal to zero and solve for L.
3=0
12
L= 0
Step 5: Analyze the result. The result L= 0 implies that the firm should
not hire any labor in order to maximize output and minimize cost. This is a
unique case because of the specific form of the production function given.
Question 17
Question
Consider a production function given by Q= 5K0.5L0.5where Qrepresents the
output, Krepresents the units of capital, and Lrepresents the units of labor.
If the wage rate is w=
$
10 per unit of labor and the rental rate of capital is
r=
$
20 per unit of capital, determine the least-cost combination of capital and
labor required to produce 400 units of output.
Solution
Step 1: The cost minimization problem can be formulated as finding the values
of Kand Lthat minimize the cost function C=rK +wL subject to the
production function Q= 5K0.5L0.5and the output requirement Q= 400.
Step 2: Substitute the production function Q= 5K0.5L0.5into the output
requirement Q= 400 to get 400 = 5K0.5L0.5.
Step 3: Simplify the equation to get 80 = K0.5L0.5.
Step 4: Rewrite the production function in terms of one variable using the
budget constraint to minimize cost. Substitute K= 80/L into the cost function
to get C= 20(80/L) + 10L= 1600/L + 10L.
Step 5: To minimize the cost function, find the derivative of Cwith respect
to Land set it to zero: dC
dL =−1600
L2+ 10 = 0.
Step 6: Solve for Lto find the value that minimizes the cost.
1600
L2= 10
1600 = 10L2
L2= 160
L= 12.65
Step 7: Substitute L= 12.65 back into K= 80/L to find the corresponding
value of K.
K= 80/12.65
K≈6.32
Therefore, the least-cost combination of capital and labor required to pro-
duce 400 units of output is approximately 6.32 units of capital and 12.65 units
of labor.
13
Question 18
Question
Consider a production function given by Q= 2L0.5K0.5, where Qrepresents
the output quantity, Lis the quantity of labor input, and Kis the quantity of
capital input. If the price of labor (P L) is 10 and the price of capital (P K) is
20, find the cost-minimizing combination of labor and capital inputs needed to
produce 100 units of output while minimizing the total cost of production.
Solution
Step 1: The total cost of production can be calculated as the sum of the cost of
labor and the cost of capital. The cost of labor (CL) is given by CL =P L ·L,
and the cost of capital (CK) is given by CK =P K ·K.
Step 2: Since we want to minimize the total cost of production while still
producing 100 units of output, we can formulate the total cost function as
follows:
T C =P L ·L+P K ·K
Step 3: We are given the production function Q= 2L0.5K0.5and the desired
output quantity of Q= 100. Substituting the production function into the
output quantity and simplifying, we get:
100 = 2L0.5K0.5
50 = L0.5K0.5
50 = √LK
2500 = LK
Step 4: Now, we substitute the cost of labor and cost of capital back into
the total cost function:
T C = 10L+ 20K
Step 5: We can rewrite the total cost function in terms of a single variable
by substituting 2500 = LK into the cost function:
T C = 10 2500
K+ 20K
Step 6: To minimize the total cost function, we need to find the critical
points. To do this, we take the derivative of the total cost function with respect
to Kand set it equal to zero:
dT C
dK =−2500
K2+ 20 = 0
Step 7: Solving the above equation for Kgives us the optimal value of
capital. Substituting this value back into L=2500
Kwill give us the optimal
value of labor needed for minimizing the total cost of production.
14
Question 19
Question
Consider a firm with the production function Q= 3K0.5L0.5, where Qis the
quantity of output produced, Kis the quantity of capital, and Lis the quantity
of labor. If the firm is currently using 16 units of capital and 25 units of labor,
how much output is the firm currently producing?
Solution
Step 1: Plug in the given values of capital and labor into the production function.
Output (Q) = 3(16)0.5(25)0.5
Step 2: Simplify the expression by evaluating the square roots. 160.5= 4
and 250.5= 5
Output (Q) = 3(4)(5)
Step 3: Calculate the output.
Output (Q) = 60
Therefore, the firm is currently producing 60 units of output.
Question 20
Question
Consider a firm with the production function Q=L0.7K0.3, where Qis the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently employs 100 units of labor and 64 units of capital, how
much will output increase if the firm increases labor by 10% and capital by
20%?
Solution
1. First, calculate the initial output using the given quantities of labor and
capital.
Q=L0.7K0.3
Q= 1000.7×640.3
Q≈10.593 ×4.938
Q≈52.462
2. Next, calculate how much output will increase if labor increases by 10%.
New quantity of labor: 100 + 0.10 ×100 = 110
15
New output when labor increases:
Q′= 1100.7×640.3
Q′≈13.020 ×4.938
Q′≈64.213
3. Finally, calculate how much output will increase if capital increases by
20%. New quantity of capital: 64 + 0.20 ×64 = 76.8
New output when capital increases:
Q′′ = 1000.7×76.80.3
Q′′ ≈10.593 ×5.794
Q′′ ≈61.406
Therefore, if the firm increases labor by 10% and capital by 20%, the output
will increase by approximately 1.751 units.
Question 21
Question
Consider a production function given by Q= 100L0.5K0.3, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis the
quantity of capital input. Find the marginal product of labor and the marginal
product of capital.
Solution
To find the marginal product of labor (M P L) and the marginal product of
capital (MP K), we need to take the partial derivative of the production function
with respect to each input variable.
Step 1: Find the Marginal Product of Labor (M P L)
MP L =∂Q
∂L = 50L−0.5K0.3
Step 2: Find the Marginal Product of Capital (M P K)
MP K =∂Q
∂K = 30L0.5K−0.7
Therefore, the marginal product of labor is 50L−0.5K0.3and the marginal
product of capital is 30L0.5K−0.7.
16
Question 22
Question
Suppose a production function is given by Q=L3/5K2/5, where Qrepresents
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-minimizing combination of inputs needed to produce Qunits of output.
Solution
Step 1: The cost of producing the given output Qusing the inputs Land K
can be represented by the total cost function C=wL +rK.
Step 2: To minimize cost, we need to minimize C=wL +rK subject to the
production function Q=L3/5K2/5.
Step 3: Since Q=L3/5K2/5, we can rewrite the cost function as C=
wL +r(Q
L3/5)5/2.
Step 4: We can now minimize Cby taking the partial derivatives with respect
to Land Kand setting them equal to 0.
Step 5: Computing the partial derivative of Cwith respect to L, we get:
∂C
∂L =w−5
2r·Q·L−8/5= 0
Step 6: Similarly, computing the partial derivative of Cwith respect to K,
we get: ∂C
∂K =−5
2r·Q·K−3/5= 0
Step 7: Solving the above two equations simultaneously, we can find the
values of Land Kthat minimize cost.
Step 8: After solving, we find that the cost-minimizing combination of inputs
is:
L=2w
5r5/8
·Q3/8
K=2w
5r5/8
·Q2/8
Therefore, the cost-minimizing combination of inputs to produce Qunits of
output is given by L=2w
5r5/8·Q3/8and K=2w
5r5/8·Q2/8.
Question 23
Question
Consider a production function given by Q=L1
3K2
3, where Qrepresents the
output quantity, Lrepresents the quantity of labor, and Krepresents the quan-
tity of capital. Assume that the price of labor (w) is
$
10 per unit and the price
17
of capital (r) is
$
20 per unit. If the firm wants to minimize the cost of producing
100 units of output, what amount of labor and capital should the firm hire?
Solution
Step 1: The cost of producing output Qusing labor Land capital Kcan be
expressed as the total cost function: C=w·L+r·K.
Step 2: In this case, we want to minimize the cost function C= 10L+ 20K
subject to the production function Q=L1
3K2
3and the constraint Q= 100.
Step 3: We can rewrite the constraint as 100 = L1
3K2
3.
Step 4: To minimize cost, we can use the Lagrange multiplier method. Define
the Lagrangian function as:
L= 10L+ 20K+λ(100 −L1
3K2
3)
Step 5: Calculate the partial derivatives of the Lagrangian function with
respect to L,K, and λ, and set them equal to 0 to find the critical points.
Step 6: Taking the derivative of Lwith respect to L, we get:
∂L
∂L = 10 −λ
3L−2
3K2
3= 0
Step 7: Taking the derivative of Lwith respect to K, we get:
∂L
∂K = 20 −2λ
3L1
3K−1
3= 0
Step 8: Taking the derivative of Lwith respect to λ, we get:
∂L
∂λ = 100 −L1
3K2
3= 0
Step 9: Solve the system of equations formed by setting the partial deriva-
tives equal to 0 to find the values of L,K, and λ.
Step 10: Calculate the values of Land Kwhich minimize the cost function
C= 10L+ 20K, subject to the production constraint Q= 100.
Question 24
Question
Consider a production function given by Q= 2L0.5K0.5, where Qis the level of
output, Lis the amount of labor input, and Kis the amount of capital input. If
the price of labor is wand the price of capital is r, what is the cost-minimizing
combination of labor and capital inputs that will produce a given level of output
Q?
18
Solution
To find the cost-minimizing combination of labor and capital inputs, we need
to minimize the total cost function, which is given by C=wL +rK subject to
the production function Q= 2L0.5K0.5.
Step 1: Set up the Lagrangian function:
Let Λ = wL +rK +λ(Q−2L0.5K0.5), where λis the Lagrange multiplier.
Step 2: Find the first-order conditions by taking the partial derivatives of
the Lagrangian function with respect to L,K, and λand setting them equal to
zero:
∂Λ
∂L :w−λL−0.5K0.5= 0
∂Λ
∂K :r−λK−0.5L0.5= 0
∂Λ
∂λ :Q−2L0.5K0.5= 0
Step 3: Solve the system of equations to find the optimal values of L,K,
and λ:
From the first equation, we have w=λL−0.5K0.5, which implies λ=w
√LK0.5.
Substitute λinto the second equation and simplify:
r=w
√LK0.5K−0.5L0.5
r=w
√L
Solving for L, we get L=w2
r2.
Similarly, solving for K, we get K=r2
w2.
Step 4: The cost-minimizing combination of labor and capital inputs is
given by L=w2
r2and K=r2
w2. This combination minimizes the total cost while
producing the given level of output Q.
Question 25
Question
Consider a production function Q= 5L0.4K0.6, where Qrepresents the output,
Lrepresents the quantity of labor, and Krepresents the quantity of capital.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK).
Solution
Step 1: To find the marginal product of labor (MPL), we need to take the
partial derivative of the production function Qwith respect to Lwhile holding
Kconstant.
MPL=∂Q
∂L
19
Step 2: Calculate the partial derivative of Qwith respect to L.
∂Q
∂L = 0.4×5×L−0.6×K0.6= 2K0.6L−0.6
Step 3: Simplify the expression.
MPL= 2K0.6L−0.6
Step 4: To find the marginal product of capital (MPK), we need to take the
partial derivative of the production function Qwith respect to Kwhile holding
Lconstant.
MPK=∂Q
∂K
Step 5: Calculate the partial derivative of Qwith respect to K.
∂Q
∂K = 0.6×5×L0.4×K−0.4= 3L0.4K−0.4
Step 6: Simplify the expression.
MPK= 3L0.4K−0.4
Therefore, the marginal product of labor is MPL= 2K0.6L−0.6and the
marginal product of capital is MPK= 3L0.4K−0.4.
Question 26
Question
Consider a production function given by Q= 4L0.5K0.5, where Qis the total
output, Lis the amount of labor input, and Kis the amount of capital input.
If the price of labor is w= 16 and the price of capital is r= 25, find the
cost-minimizing combination of labor and capital inputs required to produce
100 units of output.
Solution
Step 1: The cost function is defined as C=wL +rK, where wis the price of
labor and ris the price of capital.
Step 2: Given the production function Q= 4L0.5K0.5and the output level
Q= 100, we can substitute Qwith 100 in the production function:
100 = 4L0.5K0.5
Step 3: To find the cost-minimizing combination of labor and capital inputs,
we need to minimize the cost function C= 16L+ 25Ksubject to the constraint
100 = 4L0.5K0.5.
20
Step 4: First, we rewrite the constraint equation in terms of one variable
using the Lagrange multiplier method:
L0.5K0.5=100
4= 25
Step 5: Set up the Lagrange function J= 16L+ 25K+λ(25 −L0.5K0.5).
Step 6: Calculate the first partial derivatives of Jwith respect to L,K, and
λ, and set them equal to 0:
∂J
∂L = 16 −λ
2√K= 0
∂J
∂K = 25 −λ
2√L= 0
∂J
∂λ = 25 −L0.5K0.5= 0
Step 7: Solve the system of equations to find the values of Land K.
Step 8: Substitute the values of Land Kinto the cost function C= 16L+
25Kto find the cost-minimizing combination of labor and capital inputs.
Question 27
Question
Let Q=L1/3K2/3represent a production function where Qis the quantity of
output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. If the wage rate is w= 6 and the rental rate of capital is r= 4,
find the minimum cost of producing 64 units of output.
Solution
Step 1: To minimize cost, we need to minimize the total cost T C =wL +rK
subject to the constraint Q= 64. We can rewrite this constraint using the
production function as 64 = L1/3K2/3.
Step 2: We can now form the Lagrangian function L=wL +rK +λ(64 −
L1/3K2/3) where λis the Lagrange multiplier.
Step 3: Taking the partial derivatives of Lwith respect to L, K, and λand
setting them equal to 0, we get:
∂L
∂L =w−1
3λL−2/3K2/3= 0
∂L
∂K =r−2
3λL1/3K−1/3= 0
∂L
∂λ = 64 −L1/3K2/3= 0
21
Step 4: Solving these equations simultaneously, we get:
w=1
3λL−2/3K2/3
r=2
3λL1/3K−1/3
64 = L1/3K2/3
Step 5: From the first equation, we have λ= 3wL2/3K−2/3= 18L2/3K−2/3.
Substitute into the second equation, we get r= 12L1/3K−1/3.
Step 6: Substituting r= 4 and w= 6 into the rand wequations, we get
L= 27 and K= 8.
Therefore, the minimum cost of producing 64 units of output is T C =wL +
rK = 6(27) + 4(8) = 162 + 32 = 194.
Question 28
Question
A production function is given by Q= 5L0.5K0.5, where Qis the level of output,
Lis the amount of labor, and Kis the amount of capital. Given the total amount
of capital is fixed at K= 25, determine the marginal product of labor when
L= 9.
Solution
Step 1: Calculate the total product of labor.
Total Product of Labor (TP) = 5L0.5K0.5= 5(9)0.5(25)0.5= 5(3)(5) = 75
Step 2: Calculate the total product of labor when L= 9 + ∆L.
Total Product of Labor (TP’) = 5(9+∆L)0.5(25)0.5= 5(3+∆L)(5) = 25+25∆L
Step 3: Calculate the marginal product of labor (MPL).
MPL = ∆TP
∆L=75 −25
9−0=50
9≈5.56
Therefore, the marginal product of labor when L= 9 is approximately 5.56
units of output.
Question 29
Question
Let f(K, L)=3K0.4L0.6be a production function where Kis capital and Lis
labor. Find the average product of labor.
22
Solution
Step 1: Calculate the total product of labor by plugging L= 1 into the produc-
tion function.
f(K, 1) = 3K0.4·10.6= 3K0.4
Step 2: Define the average product of labor as the total product of labor
divided by the quantity of labor used.
APL=f(K, L)
L
Step 3: Substitute f(K, 1) = 3K0.4into the average product of labor for-
mula.
APL=3K0.4
1= 3K0.4
Therefore, the average product of labor is 3K0.4.
Question 30
Question
Consider a production function given by Q= 4L0.5K0.3, where Qrepresents
the total output, Lrepresents the quantity of labor input, and Krepresents the
quantity of capital input. If the wage rate is w= 10 and the rental rate for
capital is r= 20, determine the minimum cost of producing 100 units of output.
Solution
Step 1: The cost function for producing Qunits of output is given by C=
wL +rK.
Step 2: We need to find the values of Land Kthat minimize the cost
function Csubject to the production function constraint 4L0.5K0.3=Q= 100.
Step 3: We can rewrite the production function as L=Q
4K0.32.
Step 4: Substitute the expression for Linto the cost function: C= 10 100
4K0.32+
20K.
Step 5: By taking the derivative of the cost function with respect to Kand
setting it equal to 0, we can find the value of Kthat minimizes the cost.
Step 6: Differentiate the cost function Cwith respect to K:
dC
dK = 10 ×2×100
4K0.3×(−0.3) ×K−0.3−1+ 20
Step 7: Simplify the derivative expression:
−60 ×100
4K0.3×K−1.3+ 20
23
Step 8: Set the derivative equal to 0 and solve for K:
−60 ×100
4K0.3×K−1.3+ 20 = 0
Step 9: After solving for K, substitute the value of Kback into the produc-
tion function L=Q
4K0.32to find the value of L.
Step 10: Finally, calculate the cost C= 10L+ 20Kusing the values of L
and Kobtained. This minimum cost will be the cost of producing 100 units of
output.
Question 31
Question
Given a production function Q= 10L3/5K2/5, where Qrepresents output, L
represents labor input, and Krepresents capital input, find the level of output
when L= 8 and K= 32.
Solution
Step 1: Substitute the given values of Land Kinto the production function.
Q= 10(8)3/5(32)2/5
= 10(83/5)(322/5)
= 10(2.2974)(4)
= 91.89
Therefore, the level of output when L= 8 and K= 32 is 91.89 units.
Question 32
Question
Let f(K, L) = K0.3L0.7be a production function where Krepresents capital
and Lrepresents labor. Determine the marginal product of labor, M PL, at the
point (10,20).
Solution
Step 1: To find the marginal product of labor M PL, we need to take the partial
derivative of the production function f(K, L) with respect to labor L.
Step 1: ∂f
∂L = 0.7K0.3L−0.3
24
Step 2: Next, evaluate the expression at the point (10,20).
Step 2: ∂f
∂L
(10,20)
= 0.7×100.3×20−0.3
Step 3: Calculate the marginal product of labor MPL.
Step 3: MPL= 0.7×100.3×20−0.3= 1.2942
Therefore, the marginal product of labor M PLat the point (10,20) is 1.2942.
Question 33
Question
Suppose a production function is given by Q(K, L) = 4K0.5L0.5. Find the
marginal product of labor.
Solution
Step 1: To find the marginal product of labor (MPL), we need to take the
partial derivative of the production function Q(K, L) with respect to labor (L).
Step 2: Differentiate Q(K, L) = 4K0.5L0.5with respect to L:
∂Q
∂L = 4 ·0.5·K0.5·L0.5−1
Step 3: Simplify the derivative:
∂Q
∂L = 2K0.5L−0.5
Step 4: Therefore, the marginal product of labor is MPL = 2K0.5L−0.5.
Question 34
Question
Let Q=f(L, K)=4L0.5K0.5represent a production function where Land K
are inputs of labor and capital, respectively. Suppose the price of labor is w= 5
and the price of capital is r= 10. Find the cost minimizing input combination
to produce 100 units of output.
25
Solution
Step 1: The cost function is given by C=wL +rK.
Step 2: We want to minimize cost, subject to the constraint Q= 100.
Step 3: Substitute the given production function into the cost function to
get C= 5L+ 10K.
Step 4: We can re-write the production function as Q= 4√L·K.
Step 5: Use the constraint Q= 100 to find the relationship between Land
K: 4√LK = 100 ⇒√LK = 25 ⇒LK = 625.
Step 6: We want to minimize C= 5L+ 10Ksubject to LK = 625.
Step 7: Rewrite the cost function as C(L)=5L+ 10(625/L).
Step 8: Take the derivative of Cwith respect to Land set it equal to zero
to find the minimum: C′(L)=5−6250/L2= 0.
Step 9: Solve for L: 5L2= 6250 ⇒L2= 1250 ⇒L=√1250 = 25√2.
Step 10: Find the corresponding value of Kusing LK = 625: K=625
25√2=
25√2.
Step 11: Therefore, the cost-minimizing input combination to produce 100
units of output is L=K= 25√2.
Question 35
Question
Let Q=f(L, K) be a production function representing the output Qproduced
by using labor Land capital K. Suppose the production function is given by
Q= 4L0.5K0.5. Determine 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 take the
partial derivative of the production function with respect to labor L:
∂Q
∂L = 2L−0.5K0.5
Step 2: Simplify the expression:
∂Q
∂L = 2K0.5
L0.5= 2K0.5
√L
Therefore, the marginal product of labor is MPL= 2K0.5
√L.
Step 3: 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 = 2L0.5K−0.5
26
Question 2
Question
Let Q=L1/2K1/3represent a production function where Qis the output, Lis
the amount of labor, and Kis the amount of capital used. Find the marginal
product of labor (MPL) and the marginal product of capital (MPK).
Solution
To find the marginal product of labor (MPL) and the marginal product of
capital (MPK), we will differentiate the production function Q=L1/2K1/3
with respect to labor Land capital K, respectively.
Step 1: Find MPL
MPL=∂Q
∂L
=∂
∂L(L1/2K1/3)
=1
2·L−1/2·K1/3
=1
2·1
√L·K1/3
Step 2: Find MPK
MPK=∂Q
∂K
=∂
∂K (L1/2K1/3)
=1
3·L1/2·K−2/3
=1
3·L1/2·1
K2/3
=1
3·√L·1
K2/3
Therefore, the marginal product of labor is 1
2·1
√L·K1/3and the marginal
product of capital is 1
3·√L·1
K2/3.
Question 3
Question
Consider a firm with the production function Q=K3/4L1/4, where Qis the
quantity of output, Kis the quantity of capital, and Lis the quantity of labor.
If the firm currently has 16 units of capital and 256 units of labor, calculate the
marginal product of labor at this level of input.
2
Solution
1. The marginal product of labor (MPL) is given by the derivative of the
production function with respect to labor:
MP L =∂Q
∂L =1
4K3/4L−3/4
2. Substitute the values K= 16 and L= 256 into the MPL formula:
MP L =1
4×163/4×256−3/4
3. Simplify the expression:
MP L =1
4×2×1
16
4. Calculate MPL:
MP L =1
32
Therefore, the marginal product of labor at this level of input is 1
32 .
Question 4
Question
Consider the Cobb-Douglas production function given by Q= 2L0.3K0.6, where
Qis the total output, Lis the amount of labor input, and Kis the amount of
capital input. Given that the amount of labor input is fixed at 10 units, calculate
the marginal product of capital when the amount of capital input is 25 units.
Solution
Step 1: Calculate the total output Qwhen L= 10 and K= 25.
Q= 2(10)0.3(25)0.6
Q= 2(100.3)(250.6)
Q= 2(2.154)(10)
Q= 43.08
Step 2: Calculate the total output Qwhen L= 10 and K= 24.
Q′= 2(10)0.3(24)0.6
Q′= 2(100.3)(240.6)
Q′= 2(2.154)(8.366)
3
Q′= 36.09
Step 3: Calculate the marginal product of capital (MPL) using the formula:
MP L =∆Q
∆K
Step 4: Substitute the values of ∆Qand ∆Kinto the formula.
MP L =43.08 −36.09
25 −24
MP L =6.99
1
MP L = 6.99
Therefore, the marginal product of capital when the amount of capital input
is 25 units is 6.99 units of output per unit of capital input.
Question 5
Question
Consider a firm that produces output using two inputs: labor (L) and capital
(K). The production function is given by Q= 2L0.5K0.5.
a) Determine whether this production function exhibits constant, increasing,
or decreasing returns to scale.
b) Find the marginal product of labor and the marginal product of capital.
Solution
a) To determine the returns to scale of the production function, we will examine
how output changes when both inputs are increased by a factor λ.
Q(λL, λK) = 2(λL)0.5(λK)0.5= 2λ0.5λ0.5L0.5K0.5= 2λL0.5K0.5= 2Q
Since Q(λL, λK)=2Q, the production function exhibits constant returns
to scale.
b) The marginal product of labor (MPL) is found by taking the partial
derivative of the production function with respect to L, holding Kconstant.
MPL=∂Q
∂L =∂(2L0.5K0.5)
∂L =K0.5
Similarly, the marginal product of capital (M PK) is found by taking the
partial derivative of the production function with respect to K, holding Lcon-
stant.
MPK=∂Q
∂K =∂(2L0.5K0.5)
∂K =L0.5
4
Question 6
Question
Consider a production function f(x, y) = x2
3·y1
3where xis the quantity of
input Xand yis the quantity of input Y. Find the marginal product of Xand
the marginal product of Y.
Solution
Step 1: To find the marginal product of input X, we need to calculate the
partial derivative ∂f
∂x .
∂f
∂x =2
3x−1
3·y1
3
Step 2: Simplify the expression for the marginal product of input X.
∂f
∂x =2y1
3
3x1
3
Step 3: Next, to find the marginal product of input Y, we need to calculate
the partial derivative ∂f
∂y .
∂f
∂y =x2
3·1
3y−2
3
Step 4: Simplify the expression for the marginal product of input Y.
∂f
∂y =x2
3
3y2
3
Therefore, the marginal product of input Xis 2y1
3
3x1
3
and the marginal product
of input Yis x2
3
3y2
3
for the production function f(x, y) = x2
3·y1
3.
Question 7
Question
Let Q=f(K, L) = K2/3L1/3be the production function for a certain company,
where Qis the total quantity of output produced, Kis the amount of capital
used, and Lis the amount of labor employed. If the company’s current capital
input is 64 units and the current labor input is 27 units, find the marginal
product of labor (MPL) and the marginal product of capital (MPK).
5
Solution
Step 1: Find the marginal product of labor (MPL) The marginal product of
labor (MPL) is defined as the change in output resulting from one additional
unit of labor, holding capital constant. Mathematically, it is given by
MPL=∂Q
∂L =1
3K2/3L−2/3
Step 2: Substitute the given values of Kand Linto the expression for MPL:
MPL=1
3×642/3×27−2/3
MPL=1
3×4×1
3=4
9
Therefore, the marginal product of labor is 4
9units of output for each addi-
tional unit of labor employed.
Step 3: Find the marginal product of capital (M PK) The marginal product
of capital (MPK) is defined as the change in output resulting from one additional
unit of capital, holding labor constant. Mathematically, it is given by
MPK=∂Q
∂K =2
3K−1/3L1/3
Step 4: Substitute the given values of Kand Linto the expression for MPK:
MPK=2
3×64−1/3×271/3
MPK=2
3×1
4×3 = 1
2
Therefore, the marginal product of capital is 1
2units of output for each
additional unit of capital used.
Question 8
Question
Suppose a firm has a production function given by Q= 10L0.5K0.5, where Qis
the level of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently employs 100 units of labor and 25 units of capital, how
much output is the firm currently producing?
6
Solution
Step 1: Substitute the given values of Land Kinto the production function.
Q= 10(100)0.5(25)0.5
= 10(10)(5)
= 500
Step 2: Calculate the output level. Therefore, the firm is currently producing
500 units of output.
Question 9
Question
Let Q=L1/3K2/3represent a production function where Qis the output quan-
tity, Lis the quantity of labor, and Kis the quantity of capital. Suppose the
price of labor is wand the price of capital is r. Given that the total cost function
is T C =wL +rK, find the cost-minimizing combination of labor and capital
required to produce a given level of output.
Solution
To find the cost-minimizing combination of labor and capital, we need to mini-
mize the total cost function subject to the production function constraints.
Step 1: Set up the Lagrangian Function The Lagrangian function is
defined as:
L=wL +rK +λ(Q−L1/3K2/3)
Step 2: Find the Partial Derivatives Find the partial derivatives of the
Lagrangian function with respect to L,K, and λ:
∂L
∂L =w−1
3λL−2/3K2/3
∂L
∂K =r−2
3λL1/3K−1/3
∂L
∂λ =Q−L1/3K2/3
Step 3: Set the Partial Derivatives to Zero Set the partial derivatives
to zero and solve for L,K, and λ:
w−1
3λL−2/3K2/3= 0
r−2
3λL1/3K−1/3= 0
Q−L1/3K2/3= 0
Step 4: Solve the System of Equations Solve the system of equations
to find the optimal values of L,K, and λ.
Step 5: Interpret the Results Once you have the optimal values of L
and K, you have found the cost-minimizing combination of labor and capital
required to produce the given level of output.
7
Question 10
Question
Consider a production function defined by Q=L0.3K0.7, where Qrepresents
the quantity of output, Lrepresents the quantity of labor, and Krepresents
the quantity of capital. Suppose the firm’s output level is fixed at Q= 125
units. If the wage rate is w= 10 and the rental rate of capital is r= 20, what
combination of labor and capital should the firm use to minimize its total cost
of production?
Solution
Step 1: Write the total cost function. The total cost (C) of production is given
by the sum of labor and capital costs:
C=wL +rK
Step 2: Express labor (L) in terms of capital (K) using the production
function. Given: Q=L0.3K0.7Since Q= 125, we have:
125 = L0.3K0.7
L=125
K0.71/0.3
L= 12510
3K−7
3
L= 125 ·K−7
3
Step 3: Substitute the expression for Linto the cost function.
C= 10 ·(125 ·K−7
3) + 20K
C= 1250K−7
3+ 20K
C= 1250K2
3+ 20K
Step 4: Find the minimum total cost by taking the derivative with respect
to Kand setting it to zero.
dC
dK = 1250 ·2
3K−1
3+ 20
Setting the derivative to zero:
1250 ·2
3K−1
3+ 20 = 0
833.33K−1
3=−20
K−1
3=−0.024
8
K= (−0.024)−3=1
0.0243
Step 5: Determine the optimal quantity of labor using the production func-
tion.
L= 125 ·1
0.0243−7
3
Therefore, the firm should use 1
0.0243units of capital and 125 ·1
0.0243−7
3
units of labor to minimize its total cost of production.
Question 11
Question
Suppose a firm’s production function is given by Q= 10L0.5K0.3, where Q
represents the quantity of output, Lis the quantity of labor input, and Kis the
quantity of capital input. If the firm is currently using 100 units of labor and 64
units of capital, determine the marginal product of labor at this level of inputs.
Solution
Step 1: Calculate the partial derivative of the production function with respect
to labor, L.∂Q
∂L = 0.5×10 ×L−0.5×K0.3= 5L−0.5K0.3
Step 2: Substitute the given values of L= 100 and K= 64 into the expres-
sion obtained in Step 1.
∂Q
∂L
L=100,K=64
= 5 ×100−0.5×640.3
∂Q
∂L
L=100,K=64
= 5 ×1
√100 ×10
√64
∂Q
∂L
L=100,K=64
= 5 ×1
10 ×4=2
Therefore, the marginal product of labor at 100 units of labor and 64 units
of capital is 2.
Question 12
Question
Consider a production function Q= 10L0.4K0.6, where Qrepresents the quan-
tity of output, Lrepresents the quantity of labor, and Krepresents the quantity
of capital. If the quantity of labor is fixed at 10 units, what is the average prod-
uct of capital when the quantity of capital is increased from 25 to 30 units?
9
Solution
Step 1: Calculate the total product of capital.
Q= 10(10)0.4×250.6= 10 ×5×5 = 250
Step 2: Calculate the new total product of capital after increasing the quan-
tity of capital to 30 units.
Q′= 10(10)0.4×300.6= 10 ×5×6 = 300
Step 3: Calculate the average product of capital. Average product of capital
(APK) is given by the formula:
APK=Q
K
Step 4: Calculate the average product of capital when the quantity of capital
is 25 units.
APK=250
25 = 10
Step 5: Calculate the average product of capital when the quantity of capital
is 30 units.
AP ′
K=300
30 = 10
Therefore, the average product of capital when the quantity of capital is
increased from 25 to 30 units remains constant at 10 units.
Question 13
Question
Consider a production function defined by q= 5K0.5L0.5, where qis the quantity
produced, Kis capital input, and Lis labor input. If the price of capital is
$
4
per unit and the price of labor is
$
2 per unit, determine the cost-minimizing
combination of capital and labor to produce 100 units of output.
Solution
Step 1: The cost-minimization problem can be expressed as: Minimize C=
4K+ 2Lsubject to the production function q= 5K0.5L0.5and the output level
q= 100.
Step 2: Substitute the production function into the cost function to get a
single variable function: C(K) = 4K+ 2 q
5K0.52
Step 3: Given that q= 100, the cost function becomes: C(K) = 4K+
2100
5K0.52= 4K+ 2 20
K0.52= 4K+ 2 400
K= 4K+800
K
Step 4: To find the minimum value of C(K), differentiate with respect to K
and set it equal to 0: dC
dK = 4 −800
K2= 0
10
Step 5: Solve for K: 4 = 800
K2
K2=800
4= 200
K=√200 = 10√2
Step 6: Substitute K= 10√2 back into the production function to find L:
q= 5(10√2)0.5L0.5= 100
50√2L0.5= 100
L0.5=100
50√2=2
√2=√2
L= (√2)2= 2
Therefore, the cost-minimizing combination of capital and labor to produce
100 units of output is K= 10√2 and L= 2.
Question 14
Question
Suppose a company produces two goods, Xand Y, using the production func-
tion Q= 24X0.4Y0.6. Determine the rate at which output Qchanges with
respect to Xwhen X= 5 and Y= 10.
Solution
Step 1: Calculate the partial derivative ∂Q
∂X using the power rule for differenti-
ation. ∂Q
∂X = 0.4×24X−0.6Y0.6= 9.6X−0.6Y0.6
Step 2: Substitute X= 5 and Y= 10 into the expression for ∂Q
∂X .
∂Q
∂X
(5,10) = 9.6(5)−0.6(10)0.6
∂Q
∂X
(5,10) = 9.6(0.1296)(6.4308)
∂Q
∂X
(5,10) = 7.42231
Therefore, the rate at which output Qchanges with respect to Xwhen
X= 5 and Y= 10 is approximately 7.42231.
Question 15
Question
Consider the production function f(x, y)=2x2+ 3y2−xy, where xrepresents
units of labor and yrepresents units of capital. Find the level of output when
x= 3 and y= 4.
11
Solution
To find the level of output when x= 3 and y= 4, we substitute these values
into the production function f(x, y)=2x2+ 3y2−xy.
Step 1: Substitute x= 3 and y= 4 into the production function:
f(3,4) = 2(3)2+ 3(4)2−3(4)
Step 2: Simplify the expression:
f(3,4) = 2(9) + 3(16) −12
f(3,4) = 18 + 48 −12
f(3,4) = 54
Therefore, the level of output when x= 3 and y= 4 is 54 units.
Question 16
Question
Consider a firm with a production function given by Q= 10L0.5K0.5, where Q
represents output, Lrepresents labor input, and Krepresents capital input.
Suppose the firm has 100 units of capital. If the wage rate is 3 and the price
of capital is 2, determine the level of labor input that will maximize output for
the firm.
Solution
Step 1: Calculate the total cost function. Given that the firm’s total cost
function is the sum of labor and capital costs, we can express it as:
T C =wL +rK
Where: - wis the wage rate, - ris the price of capital.
Substitute the given values of w= 3 and r= 2:
T C = 3L+ 2(100)
Step 2: Maximize output by minimizing cost. To maximize output, the
firm needs to minimize cost. In this case, we minimize the total cost function:
T C = 3L+ 200.
Step 3: Differentiate the total cost function to find the level of labor input
that minimizes cost. dT C
dL = 3
Step 4: Set the derivative equal to zero and solve for L.
3=0
12
L= 0
Step 5: Analyze the result. The result L= 0 implies that the firm should
not hire any labor in order to maximize output and minimize cost. This is a
unique case because of the specific form of the production function given.
Question 17
Question
Consider a production function given by Q= 5K0.5L0.5where Qrepresents the
output, Krepresents the units of capital, and Lrepresents the units of labor.
If the wage rate is w=
$
10 per unit of labor and the rental rate of capital is
r=
$
20 per unit of capital, determine the least-cost combination of capital and
labor required to produce 400 units of output.
Solution
Step 1: The cost minimization problem can be formulated as finding the values
of Kand Lthat minimize the cost function C=rK +wL subject to the
production function Q= 5K0.5L0.5and the output requirement Q= 400.
Step 2: Substitute the production function Q= 5K0.5L0.5into the output
requirement Q= 400 to get 400 = 5K0.5L0.5.
Step 3: Simplify the equation to get 80 = K0.5L0.5.
Step 4: Rewrite the production function in terms of one variable using the
budget constraint to minimize cost. Substitute K= 80/L into the cost function
to get C= 20(80/L) + 10L= 1600/L + 10L.
Step 5: To minimize the cost function, find the derivative of Cwith respect
to Land set it to zero: dC
dL =−1600
L2+ 10 = 0.
Step 6: Solve for Lto find the value that minimizes the cost.
1600
L2= 10
1600 = 10L2
L2= 160
L= 12.65
Step 7: Substitute L= 12.65 back into K= 80/L to find the corresponding
value of K.
K= 80/12.65
K≈6.32
Therefore, the least-cost combination of capital and labor required to pro-
duce 400 units of output is approximately 6.32 units of capital and 12.65 units
of labor.
13
Question 18
Question
Consider a production function given by Q= 2L0.5K0.5, where Qrepresents
the output quantity, Lis the quantity of labor input, and Kis the quantity of
capital input. If the price of labor (P L) is 10 and the price of capital (P K) is
20, find the cost-minimizing combination of labor and capital inputs needed to
produce 100 units of output while minimizing the total cost of production.
Solution
Step 1: The total cost of production can be calculated as the sum of the cost of
labor and the cost of capital. The cost of labor (CL) is given by CL =P L ·L,
and the cost of capital (CK) is given by CK =P K ·K.
Step 2: Since we want to minimize the total cost of production while still
producing 100 units of output, we can formulate the total cost function as
follows:
T C =P L ·L+P K ·K
Step 3: We are given the production function Q= 2L0.5K0.5and the desired
output quantity of Q= 100. Substituting the production function into the
output quantity and simplifying, we get:
100 = 2L0.5K0.5
50 = L0.5K0.5
50 = √LK
2500 = LK
Step 4: Now, we substitute the cost of labor and cost of capital back into
the total cost function:
T C = 10L+ 20K
Step 5: We can rewrite the total cost function in terms of a single variable
by substituting 2500 = LK into the cost function:
T C = 10 2500
K+ 20K
Step 6: To minimize the total cost function, we need to find the critical
points. To do this, we take the derivative of the total cost function with respect
to Kand set it equal to zero:
dT C
dK =−2500
K2+ 20 = 0
Step 7: Solving the above equation for Kgives us the optimal value of
capital. Substituting this value back into L=2500
Kwill give us the optimal
value of labor needed for minimizing the total cost of production.
14
Question 19
Question
Consider a firm with the production function Q= 3K0.5L0.5, where Qis the
quantity of output produced, Kis the quantity of capital, and Lis the quantity
of labor. If the firm is currently using 16 units of capital and 25 units of labor,
how much output is the firm currently producing?
Solution
Step 1: Plug in the given values of capital and labor into the production function.
Output (Q) = 3(16)0.5(25)0.5
Step 2: Simplify the expression by evaluating the square roots. 160.5= 4
and 250.5= 5
Output (Q) = 3(4)(5)
Step 3: Calculate the output.
Output (Q) = 60
Therefore, the firm is currently producing 60 units of output.
Question 20
Question
Consider a firm with the production function Q=L0.7K0.3, where Qis the
quantity of output, Lis the quantity of labor, and Kis the quantity of capital.
If the firm currently employs 100 units of labor and 64 units of capital, how
much will output increase if the firm increases labor by 10% and capital by
20%?
Solution
1. First, calculate the initial output using the given quantities of labor and
capital.
Q=L0.7K0.3
Q= 1000.7×640.3
Q≈10.593 ×4.938
Q≈52.462
2. Next, calculate how much output will increase if labor increases by 10%.
New quantity of labor: 100 + 0.10 ×100 = 110
15
New output when labor increases:
Q′= 1100.7×640.3
Q′≈13.020 ×4.938
Q′≈64.213
3. Finally, calculate how much output will increase if capital increases by
20%. New quantity of capital: 64 + 0.20 ×64 = 76.8
New output when capital increases:
Q′′ = 1000.7×76.80.3
Q′′ ≈10.593 ×5.794
Q′′ ≈61.406
Therefore, if the firm increases labor by 10% and capital by 20%, the output
will increase by approximately 1.751 units.
Question 21
Question
Consider a production function given by Q= 100L0.5K0.3, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis the
quantity of capital input. Find the marginal product of labor and the marginal
product of capital.
Solution
To find the marginal product of labor (M P L) and the marginal product of
capital (MP K), we need to take the partial derivative of the production function
with respect to each input variable.
Step 1: Find the Marginal Product of Labor (M P L)
MP L =∂Q
∂L = 50L−0.5K0.3
Step 2: Find the Marginal Product of Capital (M P K)
MP K =∂Q
∂K = 30L0.5K−0.7
Therefore, the marginal product of labor is 50L−0.5K0.3and the marginal
product of capital is 30L0.5K−0.7.
16
Question 22
Question
Suppose a production function is given by Q=L3/5K2/5, where Qrepresents
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-minimizing combination of inputs needed to produce Qunits of output.
Solution
Step 1: The cost of producing the given output Qusing the inputs Land K
can be represented by the total cost function C=wL +rK.
Step 2: To minimize cost, we need to minimize C=wL +rK subject to the
production function Q=L3/5K2/5.
Step 3: Since Q=L3/5K2/5, we can rewrite the cost function as C=
wL +r(Q
L3/5)5/2.
Step 4: We can now minimize Cby taking the partial derivatives with respect
to Land Kand setting them equal to 0.
Step 5: Computing the partial derivative of Cwith respect to L, we get:
∂C
∂L =w−5
2r·Q·L−8/5= 0
Step 6: Similarly, computing the partial derivative of Cwith respect to K,
we get: ∂C
∂K =−5
2r·Q·K−3/5= 0
Step 7: Solving the above two equations simultaneously, we can find the
values of Land Kthat minimize cost.
Step 8: After solving, we find that the cost-minimizing combination of inputs
is:
L=2w
5r5/8
·Q3/8
K=2w
5r5/8
·Q2/8
Therefore, the cost-minimizing combination of inputs to produce Qunits of
output is given by L=2w
5r5/8·Q3/8and K=2w
5r5/8·Q2/8.
Question 23
Question
Consider a production function given by Q=L1
3K2
3, where Qrepresents the
output quantity, Lrepresents the quantity of labor, and Krepresents the quan-
tity of capital. Assume that the price of labor (w) is
$
10 per unit and the price
17
of capital (r) is
$
20 per unit. If the firm wants to minimize the cost of producing
100 units of output, what amount of labor and capital should the firm hire?
Solution
Step 1: The cost of producing output Qusing labor Land capital Kcan be
expressed as the total cost function: C=w·L+r·K.
Step 2: In this case, we want to minimize the cost function C= 10L+ 20K
subject to the production function Q=L1
3K2
3and the constraint Q= 100.
Step 3: We can rewrite the constraint as 100 = L1
3K2
3.
Step 4: To minimize cost, we can use the Lagrange multiplier method. Define
the Lagrangian function as:
L= 10L+ 20K+λ(100 −L1
3K2
3)
Step 5: Calculate the partial derivatives of the Lagrangian function with
respect to L,K, and λ, and set them equal to 0 to find the critical points.
Step 6: Taking the derivative of Lwith respect to L, we get:
∂L
∂L = 10 −λ
3L−2
3K2
3= 0
Step 7: Taking the derivative of Lwith respect to K, we get:
∂L
∂K = 20 −2λ
3L1
3K−1
3= 0
Step 8: Taking the derivative of Lwith respect to λ, we get:
∂L
∂λ = 100 −L1
3K2
3= 0
Step 9: Solve the system of equations formed by setting the partial deriva-
tives equal to 0 to find the values of L,K, and λ.
Step 10: Calculate the values of Land Kwhich minimize the cost function
C= 10L+ 20K, subject to the production constraint Q= 100.
Question 24
Question
Consider a production function given by Q= 2L0.5K0.5, where Qis the level of
output, Lis the amount of labor input, and Kis the amount of capital input. If
the price of labor is wand the price of capital is r, what is the cost-minimizing
combination of labor and capital inputs that will produce a given level of output
Q?
18
Solution
To find the cost-minimizing combination of labor and capital inputs, we need
to minimize the total cost function, which is given by C=wL +rK subject to
the production function Q= 2L0.5K0.5.
Step 1: Set up the Lagrangian function:
Let Λ = wL +rK +λ(Q−2L0.5K0.5), where λis the Lagrange multiplier.
Step 2: Find the first-order conditions by taking the partial derivatives of
the Lagrangian function with respect to L,K, and λand setting them equal to
zero:
∂Λ
∂L :w−λL−0.5K0.5= 0
∂Λ
∂K :r−λK−0.5L0.5= 0
∂Λ
∂λ :Q−2L0.5K0.5= 0
Step 3: Solve the system of equations to find the optimal values of L,K,
and λ:
From the first equation, we have w=λL−0.5K0.5, which implies λ=w
√LK0.5.
Substitute λinto the second equation and simplify:
r=w
√LK0.5K−0.5L0.5
r=w
√L
Solving for L, we get L=w2
r2.
Similarly, solving for K, we get K=r2
w2.
Step 4: The cost-minimizing combination of labor and capital inputs is
given by L=w2
r2and K=r2
w2. This combination minimizes the total cost while
producing the given level of output Q.
Question 25
Question
Consider a production function Q= 5L0.4K0.6, where Qrepresents the output,
Lrepresents the quantity of labor, and Krepresents the quantity of capital.
Determine the marginal product of labor (MPL) and the marginal product of
capital (MPK).
Solution
Step 1: To find the marginal product of labor (MPL), we need to take the
partial derivative of the production function Qwith respect to Lwhile holding
Kconstant.
MPL=∂Q
∂L
19
Step 2: Calculate the partial derivative of Qwith respect to L.
∂Q
∂L = 0.4×5×L−0.6×K0.6= 2K0.6L−0.6
Step 3: Simplify the expression.
MPL= 2K0.6L−0.6
Step 4: To find the marginal product of capital (MPK), we need to take the
partial derivative of the production function Qwith respect to Kwhile holding
Lconstant.
MPK=∂Q
∂K
Step 5: Calculate the partial derivative of Qwith respect to K.
∂Q
∂K = 0.6×5×L0.4×K−0.4= 3L0.4K−0.4
Step 6: Simplify the expression.
MPK= 3L0.4K−0.4
Therefore, the marginal product of labor is MPL= 2K0.6L−0.6and the
marginal product of capital is MPK= 3L0.4K−0.4.
Question 26
Question
Consider a production function given by Q= 4L0.5K0.5, where Qis the total
output, Lis the amount of labor input, and Kis the amount of capital input.
If the price of labor is w= 16 and the price of capital is r= 25, find the
cost-minimizing combination of labor and capital inputs required to produce
100 units of output.
Solution
Step 1: The cost function is defined as C=wL +rK, where wis the price of
labor and ris the price of capital.
Step 2: Given the production function Q= 4L0.5K0.5and the output level
Q= 100, we can substitute Qwith 100 in the production function:
100 = 4L0.5K0.5
Step 3: To find the cost-minimizing combination of labor and capital inputs,
we need to minimize the cost function C= 16L+ 25Ksubject to the constraint
100 = 4L0.5K0.5.
20
Step 4: First, we rewrite the constraint equation in terms of one variable
using the Lagrange multiplier method:
L0.5K0.5=100
4= 25
Step 5: Set up the Lagrange function J= 16L+ 25K+λ(25 −L0.5K0.5).
Step 6: Calculate the first partial derivatives of Jwith respect to L,K, and
λ, and set them equal to 0:
∂J
∂L = 16 −λ
2√K= 0
∂J
∂K = 25 −λ
2√L= 0
∂J
∂λ = 25 −L0.5K0.5= 0
Step 7: Solve the system of equations to find the values of Land K.
Step 8: Substitute the values of Land Kinto the cost function C= 16L+
25Kto find the cost-minimizing combination of labor and capital inputs.
Question 27
Question
Let Q=L1/3K2/3represent a production function where Qis the quantity of
output produced, Lis the quantity of labor input, and Kis the quantity of
capital input. If the wage rate is w= 6 and the rental rate of capital is r= 4,
find the minimum cost of producing 64 units of output.
Solution
Step 1: To minimize cost, we need to minimize the total cost T C =wL +rK
subject to the constraint Q= 64. We can rewrite this constraint using the
production function as 64 = L1/3K2/3.
Step 2: We can now form the Lagrangian function L=wL +rK +λ(64 −
L1/3K2/3) where λis the Lagrange multiplier.
Step 3: Taking the partial derivatives of Lwith respect to L, K, and λand
setting them equal to 0, we get:
∂L
∂L =w−1
3λL−2/3K2/3= 0
∂L
∂K =r−2
3λL1/3K−1/3= 0
∂L
∂λ = 64 −L1/3K2/3= 0
21
Step 4: Solving these equations simultaneously, we get:
w=1
3λL−2/3K2/3
r=2
3λL1/3K−1/3
64 = L1/3K2/3
Step 5: From the first equation, we have λ= 3wL2/3K−2/3= 18L2/3K−2/3.
Substitute into the second equation, we get r= 12L1/3K−1/3.
Step 6: Substituting r= 4 and w= 6 into the rand wequations, we get
L= 27 and K= 8.
Therefore, the minimum cost of producing 64 units of output is T C =wL +
rK = 6(27) + 4(8) = 162 + 32 = 194.
Question 28
Question
A production function is given by Q= 5L0.5K0.5, where Qis the level of output,
Lis the amount of labor, and Kis the amount of capital. Given the total amount
of capital is fixed at K= 25, determine the marginal product of labor when
L= 9.
Solution
Step 1: Calculate the total product of labor.
Total Product of Labor (TP) = 5L0.5K0.5= 5(9)0.5(25)0.5= 5(3)(5) = 75
Step 2: Calculate the total product of labor when L= 9 + ∆L.
Total Product of Labor (TP’) = 5(9+∆L)0.5(25)0.5= 5(3+∆L)(5) = 25+25∆L
Step 3: Calculate the marginal product of labor (MPL).
MPL = ∆TP
∆L=75 −25
9−0=50
9≈5.56
Therefore, the marginal product of labor when L= 9 is approximately 5.56
units of output.
Question 29
Question
Let f(K, L)=3K0.4L0.6be a production function where Kis capital and Lis
labor. Find the average product of labor.
22
Solution
Step 1: Calculate the total product of labor by plugging L= 1 into the produc-
tion function.
f(K, 1) = 3K0.4·10.6= 3K0.4
Step 2: Define the average product of labor as the total product of labor
divided by the quantity of labor used.
APL=f(K, L)
L
Step 3: Substitute f(K, 1) = 3K0.4into the average product of labor for-
mula.
APL=3K0.4
1= 3K0.4
Therefore, the average product of labor is 3K0.4.
Question 30
Question
Consider a production function given by Q= 4L0.5K0.3, where Qrepresents
the total output, Lrepresents the quantity of labor input, and Krepresents the
quantity of capital input. If the wage rate is w= 10 and the rental rate for
capital is r= 20, determine the minimum cost of producing 100 units of output.
Solution
Step 1: The cost function for producing Qunits of output is given by C=
wL +rK.
Step 2: We need to find the values of Land Kthat minimize the cost
function Csubject to the production function constraint 4L0.5K0.3=Q= 100.
Step 3: We can rewrite the production function as L=Q
4K0.32.
Step 4: Substitute the expression for Linto the cost function: C= 10 100
4K0.32+
20K.
Step 5: By taking the derivative of the cost function with respect to Kand
setting it equal to 0, we can find the value of Kthat minimizes the cost.
Step 6: Differentiate the cost function Cwith respect to K:
dC
dK = 10 ×2×100
4K0.3×(−0.3) ×K−0.3−1+ 20
Step 7: Simplify the derivative expression:
−60 ×100
4K0.3×K−1.3+ 20
23
Step 8: Set the derivative equal to 0 and solve for K:
−60 ×100
4K0.3×K−1.3+ 20 = 0
Step 9: After solving for K, substitute the value of Kback into the produc-
tion function L=Q
4K0.32to find the value of L.
Step 10: Finally, calculate the cost C= 10L+ 20Kusing the values of L
and Kobtained. This minimum cost will be the cost of producing 100 units of
output.
Question 31
Question
Given a production function Q= 10L3/5K2/5, where Qrepresents output, L
represents labor input, and Krepresents capital input, find the level of output
when L= 8 and K= 32.
Solution
Step 1: Substitute the given values of Land Kinto the production function.
Q= 10(8)3/5(32)2/5
= 10(83/5)(322/5)
= 10(2.2974)(4)
= 91.89
Therefore, the level of output when L= 8 and K= 32 is 91.89 units.
Question 32
Question
Let f(K, L) = K0.3L0.7be a production function where Krepresents capital
and Lrepresents labor. Determine the marginal product of labor, M PL, at the
point (10,20).
Solution
Step 1: To find the marginal product of labor M PL, we need to take the partial
derivative of the production function f(K, L) with respect to labor L.
Step 1: ∂f
∂L = 0.7K0.3L−0.3
24
Step 2: Next, evaluate the expression at the point (10,20).
Step 2: ∂f
∂L
(10,20)
= 0.7×100.3×20−0.3
Step 3: Calculate the marginal product of labor MPL.
Step 3: MPL= 0.7×100.3×20−0.3= 1.2942
Therefore, the marginal product of labor M PLat the point (10,20) is 1.2942.
Question 33
Question
Suppose a production function is given by Q(K, L) = 4K0.5L0.5. Find the
marginal product of labor.
Solution
Step 1: To find the marginal product of labor (MPL), we need to take the
partial derivative of the production function Q(K, L) with respect to labor (L).
Step 2: Differentiate Q(K, L) = 4K0.5L0.5with respect to L:
∂Q
∂L = 4 ·0.5·K0.5·L0.5−1
Step 3: Simplify the derivative:
∂Q
∂L = 2K0.5L−0.5
Step 4: Therefore, the marginal product of labor is MPL = 2K0.5L−0.5.
Question 34
Question
Let Q=f(L, K)=4L0.5K0.5represent a production function where Land K
are inputs of labor and capital, respectively. Suppose the price of labor is w= 5
and the price of capital is r= 10. Find the cost minimizing input combination
to produce 100 units of output.
25
Solution
Step 1: The cost function is given by C=wL +rK.
Step 2: We want to minimize cost, subject to the constraint Q= 100.
Step 3: Substitute the given production function into the cost function to
get C= 5L+ 10K.
Step 4: We can re-write the production function as Q= 4√L·K.
Step 5: Use the constraint Q= 100 to find the relationship between Land
K: 4√LK = 100 ⇒√LK = 25 ⇒LK = 625.
Step 6: We want to minimize C= 5L+ 10Ksubject to LK = 625.
Step 7: Rewrite the cost function as C(L)=5L+ 10(625/L).
Step 8: Take the derivative of Cwith respect to Land set it equal to zero
to find the minimum: C′(L)=5−6250/L2= 0.
Step 9: Solve for L: 5L2= 6250 ⇒L2= 1250 ⇒L=√1250 = 25√2.
Step 10: Find the corresponding value of Kusing LK = 625: K=625
25√2=
25√2.
Step 11: Therefore, the cost-minimizing input combination to produce 100
units of output is L=K= 25√2.
Question 35
Question
Let Q=f(L, K) be a production function representing the output Qproduced
by using labor Land capital K. Suppose the production function is given by
Q= 4L0.5K0.5. Determine 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 take the
partial derivative of the production function with respect to labor L:
∂Q
∂L = 2L−0.5K0.5
Step 2: Simplify the expression:
∂Q
∂L = 2K0.5
L0.5= 2K0.5
√L
Therefore, the marginal product of labor is MPL= 2K0.5
√L.
Step 3: 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 = 2L0.5K−0.5
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Step 4: Simplify the expression:
∂Q
∂K = 2 L0.5
K0.5= 2L0.5
√K
Therefore, the marginal product of capital is MPK= 2L0.5
√K.
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