ECON 350 - CLASSICAL
ECONOMICS - Production Functions
Question Bank - Set 4
Liberty University
Question 1
Question
Let Q=L2/3K1/3be the production function where Qis the output, Lis the
quantity of labor, and Kis the quantity of capital. Find the marginal product
of labor.
Solution
Step 1: To find the marginal product of labor, we need to differentiate the
production function Q=L2/3K1/3with respect to labor Lwhile keeping capital
Kconstant. ∂Q
∂L =2
3L−1/3K1/3
Step 2: Simplify the expression for the marginal product of labor.
∂Q
∂L =2
3·K1/3
L1/3=2
3K
L1/3
Therefore, the marginal product of labor is 2
3K
L1/3.
Question 2
Question
Consider a production function given by Q= 4L0.5K0.5, where Qrepresents the
quantity of output, Lrepresents the quantity of labor input, and Krepresents
the quantity of capital input. If the price of labor is wand the price of capital is
r, determine the optimal combination of labor and capital inputs that minimizes
the cost of producing 100 units of output.
Solution
Step 1: The cost function can be expressed as C=wL +rK.
Step 2: We are given the production function Q= 4L0.5K0.5= 100. This
can be rewritten as L0.5K0.5= 25.
Step 3: To minimize the cost of producing 100 units of output, we need to
solve the following minimization problem:
Minimize wL +rK subject to L0.5K0.5= 25
Step 4: We can rewrite the minimization problem in terms of a single variable
by substituting K= 25/L0.5into the cost function:
C=wL +r25
L0.5=wL +25r
L0.5
Step 5: To find the minimum cost, we differentiate the cost function with
respect to Land set the derivative equal to 0:
dC
dL =w−25r
2L1.5= 0
Step 6: Solving for L, we get L=q50r
w.
Step 7: Substitute the value of Lback into the production function L0.5K0.5=
25 to find K:r50r
w×K0.5= 25
K=252w
50r
Step 8: Therefore, the optimal combination of labor and capital inputs that
minimizes the cost of producing 100 units of output is L=q50r
wand K=252w
50r.
Question 3
Question
Consider a production function given by Q= 2K0.5L0.5, where Qrepresents the
total quantity produced, Kis the amount of capital input, and Lis the amount
of labor input. If the price of capital (PK) is 3 and the price of labor (PL) is
4, find the optimal combination of capital and labor inputs that minimizes the
total cost of production.
2
Solution
Step 1: The total cost (C) of production can be expressed as a function of the
quantities of capital and labor used, along with their respective prices. There-
fore, C=PK·K+PL·L.
Step 2: Substituting the expression for Qinto the cost function, we get
C= 3K+ 4L.
Step 3: To minimize total cost subject to the production constraint Q=
2K0.5L0.5, we can substitute the expression for Cinto the production function
and solve for Kin terms of L.
Step 4: Substituting Cinto the production function gives 2K0.5L0.5= 3K+
4L.
Step 5: Simplifying the equation, we get K0.5=4L
2−3.
Step 6: Squaring both sides gives K=16L2
4, which simplifies to K= 4L2.
Step 7: Now, we substitute the expression for Kback into the cost function
to get C= 3(4L2)+4L.
Step 8: Simplifying further, we have C= 12L2+ 4L.
Step 9: To find the minimum cost, we need to differentiate the cost function
with respect to Land set it equal to zero. This gives dC
dL = 24L+ 4 = 0.
Step 10: Solving the above equation for L, we find L=−1
6.
Step 11: Since the amount of labor cannot be negative, this negative value
is not valid. Therefore, the critical point is at L= 0.
Step 12: Substituting L= 0 back into the production function, we find that
the corresponding value of Kis also 0.
Therefore, the optimal combination of capital and labor inputs that mini-
mizes the total cost of production is K= 0 and L= 0.
Question 4
Question
Let Q= 2L0.5K0.5be the production function of a company, where Qis the
total output, Lis the amount of labor input, and Kis the amount of capital
input. If the wage rate is w= 4 and the rental rate of capital is r= 9, find the
cost-minimizing combination of labor and capital inputs that produce 100 units
of output.
Solution
Step 1: The cost function is given by C=wL +rK, where Cis the total cost.
We need to minimize the cost function subject to the constraint of producing
100 units of output.
Step 2: From the production function, we have Q= 2L0.5K0.5. Since we
are producing 100 units of output, we can substitute this into the production
function to get:
100 = 2L0.5K0.5
3
Step 3: To minimize the cost function, we’ll use the Lagrange multiplier
method. Let J=wL +rK −λ(2L0.5K0.5−100), where λis the Lagrange
multiplier.
Step 4: Taking the partial derivatives of Jwith respect to L,K, and λand
setting them equal to zero, we get the following equations:
∂J
∂L =w−λL−0.5K0.5= 0
∂J
∂K =r−λL0.5K−0.5= 0
∂J
∂λ = 2L0.5K0.5−100 = 0
Step 5: Solving the system of equations, we find:
L= 25 and K= 25
Therefore, the cost-minimizing combination of labor and capital inputs that
produce 100 units of output is L= 25 and K= 25.
Question 5
Question
Consider a production function in the form of Cobb-Douglas function:
Q(K, L) = K0.4L0.6
where Qis the total output, Kis the quantity of capital, and Lis the quantity
of labor.
Assume that the price of capital (PK) is 10 and the price of labor (PL) is 5.
Determine the total cost function (T C) in terms of Q.
Solution
Step 1: The total cost function (T C) can be calculated by multiplying the
quantity of capital (K) by the price of capital (PK) and the quantity of labor
(L) by the price of labor (PL).
T C =K·PK+L·PL
Step 2: Since Kand Lare related to the total output (Q) by the Cobb-
Douglas function, we can express them in terms of Q.
K=Q0.4
L=Q0.6
4
Step 3: Substitute the expressions for Kand Lin terms of Qinto the total
cost function:
T C =Q0.4·PK+Q0.6·PL
Step 4: Replace the given prices for capital (PK) and labor (PL) to get the
total cost function in terms of Q:
T C =Q0.4·10 + Q0.6·5
Therefore, the total cost function in terms of Qis T C = 10Q0.4+ 5Q0.6.
Question 6
Question
Consider a production function for a firm given by Q=L0.3K0.7, where Q
represents the quantity of output, Lrepresents the quantity of labor, and K
represents the quantity of capital. Determine the minimum cost combination of
labor and capital needed to produce 100 units of output, given that the cost of
labor is
$
5 per unit and the cost of capital is
$
10 per unit.
Solution
Step 1: We are given the production function Q=L0.3K0.7. Since we want to
produce 100 units of output, we can set Q= 100 in the production function to
get the relationship between Land K.
Step 2: Substituting Q= 100 into the production function, we have 100 =
L0.3K0.7.
Step 3: To minimize the cost of production, we need to minimize the total
cost Cwhich is given by C= 5L+10K, subject to the constraint 100 = L0.3K0.7.
Step 4: We can rewrite the constraint in terms of Kto substitute it into the
cost function. Taking the natural logarithm of both sides of the constraint gives
ln(100) = lnL0.3K0.7. Simplifying, we get 2.3026 = 0.3 ln(L)+0.7 ln(K).
Step 5: Rearranging the constraint equation, we can express Kin terms of
Las K=100
L0.31/0.7.
Step 6: Substituting the expression for Kinto the cost function, we have
C= 5L+ 10 100
L0.31/0.7.
Step 7: To find the minimum cost combination of labor and capital, we
need to minimize C. This involves finding the critical points of Cby taking its
derivative with respect to L, setting it equal to 0, and solving for L.
Step 8: After finding the value of L, we can substitute it back into the
constraint equation to find the corresponding value of K.
5
Question 7
Question
Consider a production function given by Q=K0.3L0.7, where Qis the quantity
produced, Kis the amount of capital used, and Lis the amount of labor used.
Suppose the price of capital (PK) is
$
10 per unit and the price of labor (PL) is
$
5 per unit. If the firm’s budget is
$
200, how many units of capital and labor
should the firm hire in order to maximize output?
Solution
Step 1: Set up the firm’s cost constraint. Given that the firm’s budget is
$
200,
the cost constraint can be written as:
10K+ 5L= 200
Step 2: Rewrite the production function. Rewrite the production function
as follows:
Q= 100.3L0.7
Step 3: Express production in terms of one variable. Solve the cost constraint
for Kin terms of L:
K= 20 −2L
Step 4: Express output as a function of one variable. Substitute the expres-
sion for Kinto the production function:
Q= (20 −2L)0.3L0.7
Step 5: Find the first derivative of the production function with respect to
L. Calculate the first derivative of Qwith respect to L:
dQ
dL = 0.7(20 −2L)0.3L−0.3−0.3(20 −2L)0.3L0.7
Step 6: Set the derivative equal to zero and solve for L. Setting the derivative
equal to zero:
0.7(20 −2L)0.3L−0.3−0.3(20 −2L)0.3L0.7= 0
Solving for Lyields L= 10.
Step 7: Calculate the amount of capital and labor for maximizing output.
Use the solved value of Lto find K:
K= 20 −2(10) = 0
Therefore, the firm should hire 0 units of capital and 10 units of labor to maxi-
mize output.
6
Question 8
Question
Consider a production function Q=L1/3K2/3, where Qrepresents the output
quantity, Lrepresents the quantity of labor input, and Krepresents the quantity
of capital input. If the wage rate is wand the rental rate of capital is r, find
the equations for the isoquants associated with levels of output Q1< Q2.
Solution
To find the equations for the isoquants associated with levels of output Q1<
Q2in this production function, we begin by setting up the isoquant equation
Q=L1/3K2/3.
Step 1: Find the equation for the isoquant Q1Given that Q1=
L1/3
1K2/3
1, the isoquant equation becomes:
Q1=L1/3
1K2/3
1
Step 2: Find the equation for the isoquant Q2Given that Q2=
L1/3
2K2/3
2, the isoquant equation becomes:
Q2=L1/3
2K2/3
2
Step 3: Equate the two isoquant equations Since we want to find the
isoquants associated with levels of output Q1< Q2, we set Q1=Q2to find the
equation for the isoquants:
L1/3
1K2/3
1=L1/3
2K2/3
2
This equation represents the isoquants associated with levels of output Q1<
Q2.
Question 9
Question
Consider a production function given by Q= 2L0.6K0.3, where Qrepresents
output, Lis labor input, and Kis capital input. If the firm currently employs
100 units of labor and 64 units of capital, calculate the marginal product of
labor at this point.
Solution
Step 1: Compute the total product of labor (T PL) by substituting L= 100 into
the production function:
T PL= 2(100)0.6(64)0.3
7
T PL= 2(1000.6)(640.3)
T PL= 2(39.81)(4)
T PL= 318.48
Step 2: Calculate the total product of labor when the firm employs one
additional unit of labor (T PL+1) by substituting L= 101 into the production
function:
T PL+1 = 2(101)0.6(64)0.3
T PL+1 = 2(1010.6)(640.3)
T PL+1 = 2(40.01)(4)
T PL+1 = 320.17
Step 3: Determine the marginal product of labor (MPL) as the change in
total product of labor when an additional unit of labor is employed:
MP L =T PL+1 −T PL
MP L = 320.17 −318.48
MP L1.69
Therefore, the marginal product of labor when the firm employs 100 units
of labor and 64 units of capital is approximately 1.69.
Question 10
Question
Let Q=L3/4K1/4be the production function where Lrepresents labor and
Krepresents capital. Determine the level of marginal product of labor at the
point (8,4).
Solution
Step 1: To find the marginal product of labor, we need to take the partial deriva-
tive of the production function Qwith respect to labor L. Step 2: Compute the
partial derivative of Qwith respect to L. Step 3: Substitute L= 8 and K= 4
into the partial derivative to find the marginal product of labor.
Step 1: The marginal product of labor is given by
MPL=∂Q
∂L
8
Step 2: Compute the partial derivative of Qwith respect to Lby differenti-
ating Q=L3/4K1/4with respect to L.
∂Q
∂L =3
4L−1/4K1/4
Step 3: Substitute L= 8 and K= 4 into the partial derivative to find the
marginal product of labor.
MPL(8,4) = 3
4(8)−1/4(4)1/4
MPL(8,4) = 3
4(2−1)(2)
MPL(8,4) = 3
4
Therefore, the level of marginal product of labor at the point (8,4) is 3
4.
Question 11
Question
Consider a production function given by Q= 2L1/2K1/2, where Qrepresents
the quantity of output, Lis the quantity of labor input, and Kis the quantity
of capital input. Suppose the wage rate is w= 4 and the rental rate for capital
is r= 9. If the firm wants to produce 64 units of output at the minimum cost,
how many units of labor and capital should be employed?
Solution
Step 1: To minimize the cost of production subject to the output target, we
need to minimize the total cost function C=w·L+r·Kwhile producing 64
units of output.
Step 2: Substituting the cost and output functions into the total cost func-
tion, we have
C= 4L+ 9K
Step 3: We are given that Q= 64, so substituting Q= 64 into the production
function:
64 = 2L1/2K1/2
Step 4: Rearranging the production function:
32 = L1/2K1/2
Step 5: Squaring both sides to get rid of the fractional exponents:
1024 = LK
9
Step 6: We now have two equations: 32 = L1/2K1/2and 1024 = LK. We
can solve these two equations simultaneously.
Step 7: From 1024 = LK, we can express Kin terms of Las K=1024
L.
Step 8: Substituting K=1024
Linto 32 = L1/2K1/2:
32 = L1/21024
L1/2
Step 9: Simplifying the above equation gives:
32 = √1024
32 = 32
Step 10: Therefore, the firm should employ L= 1024 units of labor and
K= 1 unit of capital to produce 64 units of output at the minimum cost.
Question 12
Question
Consider a production function given by Q=LαKβ, where Qrepresents the
output level, Lis the quantity of labor input, and Kis the quantity of capital
input. If the firm has a fixed budget for inputs, derive the cost-minimizing input
combination to produce a given level of output Q.
Solution
Step 1: We first need to set up the Lagrangian function:
L(L, K, λ) = wL +rK +λ(Q−LαKβ)
where wis the wage rate for labor, ris the rental rate for capital, and λis the
Lagrange multiplier.
Step 2: Next, we take the partial derivatives of Lwith respect to L,K, and
λand set them equal to zero:
∂L
∂L =w−λαLα−1Kβ= 0
∂L
∂K =r−λβLαKβ−1= 0
∂L
∂λ =Q−LαKβ= 0
Step 3: Solve the first equation for λand substitute it into the second
equation:
λ=w
αLα−1Kβ=r
βLαKβ−1
10
Step 4: From the equations above, we can find the optimal input combination
(L∗, K∗) that minimizes the cost for producing output Q.
Step 5: The cost-minimizing input combination can be found using the op-
timal values of Land Kderived from the Lagrangian approach.
Question 13
Question
Consider a production function f(x, y) = x2y3. Determine the average product
of labor when x= 2 and y= 3.
Solution
Let the average product of labor be denoted by AP L. The formula for average
product of labor is given by:
AP L =f(x, y)
x
Step 1: Calculate the production function value at x= 2 and y= 3.
f(2,3) = 22·33= 4 ·27 = 108
Step 2: Substitute x= 2 and y= 3 into the formula for average product of
labor.
AP L =f(2,3)
2=108
2= 54
Therefore, the average product of labor when x= 2 and y= 3 is 54.
Question 14
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the level of
output, Lis the quantity of labor input, and Kis the quantity of capital input.
If the wage rate is w= 10 and the rental rate of capital is r= 20, find the
minimum cost of producing 400 units of output.
Solution
1. The cost function for producing Qunits of output can be expressed as
C(w, r, Q) = wL +rK.
2. Given the production function Q=L0.5K0.5, we need to find the quanti-
ties of labor (L) and capital (K) that minimize the cost of producing 400 units
of output.
3. Since Q= 400, we have 400 = L0.5K0.5.
11
4. To minimize the cost, we need to minimize C(w, r, Q) = wL +rK subject
to the constraint 400 = L0.5K0.5.
5. By the Lagrange Multiplier method, we construct the Lagrangian as
follows:
L(L, K, λ) = wL +rK −λ(400 −L0.5K0.5)
6. Taking partial derivatives with respect to L,K, and λand setting them
equal to zero, we have:
∂L
∂L =w−0.5λL−0.5K0.5= 0
∂L
∂K =r−0.5λL0.5K−0.5= 0
∂L
∂λ = 400 −L0.5K0.5= 0
7. Solving the system of equations, we find L= 100 and K= 400.
8. Substituting L= 100 and K= 400 back into the cost function C(w, r, Q) =
wL +rK, we get:
C(10,20,400) = 10(100) + 20(400) = 1000 + 8000 = 9000
Therefore, the minimum cost of producing 400 units of output is 9000.
Question 15
Question
Consider a production function given by Q=L0.5K0.5, where Qrepresents the
total output, Lis the amount of labor input, and Kis the amount of capital
input. If the wage rate is w= 10 and the rental rate for capital is r= 20,
find the least-cost combination of labor and capital that produces 100 units of
output.
Solution
Step 1: The cost minimization problem can be formulated as follows: Minimize
C=wL +rK subject to the production function Q=L0.5K0.5and output
level Q= 100.
Step 2: Substitute the production function into the cost function: C=
10L+ 20K.
Step 3: Also, substitute the output level Q= 100 into the production func-
tion: 100 = L0.5K0.5.
Step 4: Rewrite the production function in terms of one variable using the
output level constraint: K=100
L2.
Step 5: Substitute Kinto the cost function to get cost in terms of one
variable: C= 10L+ 20 100
L2.
Step 6: To minimize cost, differentiate Cwith respect to Land set it equal
to zero: dC
dL = 10 −40 100
L3= 0.
12
Step 7: Solve for L: 10L3−40000 = 0. L=3
√4000 = 20.
Step 8: Calculate the corresponding value of Kusing the production func-
tion: K=100
20 2= 25.
Step 9: Therefore, the least-cost combination of labor and capital to produce
100 units of output is L= 20 units and K= 25 units.
Question 16
Question
Let Q=L0.3K0.7represent a production function, where Qis the total output,
Lis the quantity of labor, and Kis the quantity of capital. Determine the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function Qwith respect to L.
∂Q
∂L = 0.3L−0.7K0.7
Step 2: Simplify the expression to find the marginal product of labor.
∂Q
∂L = 0.3K0.7
L0.7= 0.3K
L0.7
Step 3: Similarly, to find the marginal product of capital, differentiate the
production function Qwith respect to K.
∂Q
∂K = 0.7L0.3K−0.3
Step 4: Simplify the expression to find the marginal product of capital.
∂Q
∂K = 0.7L0.3
K0.3= 0.7L
K0.3
Therefore, the marginal product of labor is 0.3K
L0.7and the marginal
product of capital is 0.7L
K0.3.
Question 17
Question
Suppose a firm has a production function given by Q=L0.5K0.5, where Q
represents the quantity of output, Lrepresents the quantity of labor, and K
represents the quantity of capital. If the firm has 100 units of capital and wants
to produce 400 units of output, what is the minimum amount of labor the firm
must use?
13
Solution
We are given the production function Q=L0.5K0.5. We are also given that
K= 100 and Q= 400. We need to find the minimum amount of labor L
required to produce 400 units of output.
Step 1: Substitute the given values into the production function.
400 = L0.5×1000.5
Step 2: Simplify the expression.
400 = L0.5×10
Step 3: Divide both sides by 10 to isolate L.
40 = L0.5
Step 4: Square both sides to solve for L.
L= 402
Step 5: Calculate the minimum amount of labor required.
L= 1600
Therefore, the firm must use a minimum of 1600 units of labor to produce
400 units of output when given 100 units of capital.
Question 18
Question
Given a production function Q= 10L0.5K0.5, where Qrepresents the total
output, Lis the quantity of labor, and Kis the quantity of capital, find the
marginal product of labor (MPL) and the marginal product of capital (M PK).
Solution
To find the marginal product of labor (MPL), we need to take the derivative of
the production function with respect to labor (L), while holding the quantity of
capital (K) constant. Similarly, to find the marginal product of capital (M PK),
we need to take the derivative of the production function with respect to capital
(K), while holding the quantity of labor constant.
14
Step 1: Find MPL
MPL=∂Q
∂L
=∂
∂L(10L0.5K0.5)
= 5K0.5∂
∂LL0.5
= 5K0.5·0.5L−0.5
= 2.5K0.5L−0.5
Hence, the marginal product of labor (MPL) is 2.5K0.5L−0.5.
Step 2: Find MPK
MPK=∂Q
∂K
=∂
∂K (10L0.5K0.5)
= 5L0.5∂
∂K K0.5
= 5L0.5·0.5K−0.5
= 2.5L0.5K−0.5
Therefore, the marginal product of capital (MPK) is 2.5L0.5K−0.5.
Question 19
Question
Consider a production function given by Q= 4L0.5K0.5, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the wage rate is
$
10 per unit of labor and the
rental rate for capital is
$
20 per unit of capital, determine the cost-minimizing
combination of labor and capital that the firm should use to produce 100 units
of output.
Solution
Step 1: Write the cost-minimization problem. The cost-minimization problem
for the firm can be written as:
Minimize 10L+ 20Ksubject to 4L0.5K0.5= 100
Step 2: Solve for one of the variables in the constraint. Since the objective
function contains Land Kseparately, solve for one of the variables in the
constraint. We can solve for Kin terms of L:
4L0.5K0.5= 100 ⇒K=100
4L0.5=25
L0.5
15
Step 3: Substitute the expression for Kinto the objective function. Sub-
stitute the expression for Kinto the cost function to obtain a single-variable
function in terms of L:
Minimize 10L+ 20 25
L0.5
Step 4: Simplify the expression.
Minimize 10L+500
L0.5
Step 5: Find the minimum value. To minimize the cost function, differentiate
it with respect to Land set it equal to 0:
d
dL 10L+500
L0.5= 10 −250
L1.5= 0
Solving for L, we get:
L=250
10
2
3
= 25
Step 6: Determine the corresponding value of K. Using the expression
derived earlier for Kin terms of L:
K=25
L0.5=25
5= 5
Therefore, the firm should use 25 units of labor and 5 units of capital to
produce 100 units of output at minimum cost.
Question 20
Question
Consider a production function given by Q=KαLβ, where Qis the quantity
of output, Kis the quantity of capital input, Lis the quantity of labor input,
and α, β > 0. Show that the marginal product of labor can be expressed as
∂Q/∂L
Q=βL/Q.
Solution
Step 1: Calculate the marginal product of labor, ∂Q
∂L .
Step 1: ∂Q
∂L =βKαLβ−1
Step 2: Calculate the total product, Q.
Q=KαLβ
16
Step 3: Express the marginal product of labor as a proportion of the total
product.
Step 3: ∂Q/∂L
Q=βKαLβ−1
KαLβ
Step 4: Simplify the expression.
βKαLβ−1
KαLβ=βL
L=βL/Q
Therefore, the marginal product of labor can be expressed as ∂Q/∂L
Q=βL/Q.
Question 21
Question
Suppose a firm’s production function is given by Q= 10L1/2K1/2, where Q
represents output, Lrepresents labor input, and Krepresents capital input. If
the firm hires labor at a wage rate of w= 4 and rents capital at a rate of r= 5,
determine the minimum cost of producing 100 units of output.
Solution
Step 1: Calculate the total cost function using the given production function.
Total Cost (TC) = Total Labor Cost + Total Capital Cost
Step 2: Calculate the labor input required to produce 100 units of output.
Q= 10L1/2K1/2=⇒100 = 10L1/2K1/2
Step 3: Substitute the value of Kfrom the production function into the cost
function and differentiate to find the minimum cost.
Minimize T C =wL +rK = 4L+ 5 100
10L1/2
Step 4: Differentiate to find the minimum.
dT C
dL = 4 −50
L3/2=⇒4 = 50
L3/2=⇒L3/2=50
4= 12.5 =⇒L≈7.90
Step 5: Calculate the minimum cost.
K=100
10(7.90)1/2≈15.85
Minimum Cost = 4(7.90) + 5(15.85) ≈97.75
Therefore, the minimum cost of producing 100 units of output is approxi-
mately 97.75.
17
Question 22
Question
Consider a production function Q= 2K1
3L2
3, where Qrepresents the total
output, Kis the amount of capital input, and Lis the amount of labor input.
At a certain point, the amount of capital is fixed at 64 units. If each unit of
labor costs
$
4 and each unit of capital costs
$
6, how many units of labor should
be hired to minimize the cost of producing 60 units of output?
Solution
Step 1: To minimize the cost of producing 60 units of output, we need to
minimize the total cost function. The total cost function is given by C=
rK +wL, where ris the cost of capital per unit and wis the cost of labor per
unit.
Step 2: Given that each unit of labor costs
$
4 and each unit of capital costs
$
6, the total cost function becomes C= 6(64) + 4L.
Step 3: We are given that the total output Qis 60 units. Substituting
Q= 60 into the production function, we have 60 = 2(64) 1
3L2
3.
Step 4: Simplifying the production function gives 30 = 4L2
3, which further
simplifies to L2
3=30
4.
Step 5: Solving for L, we get L=30
4
3
2=303
43
1
2=27000
64
1
2.
Step 6: Calculating the value inside the square root gives 27000
64 = 421.875.
Thus, L=√421.875 ≈20.54.
Step 7: Since the number of units of labor must be a whole number, we hire
21 units of labor to minimize the cost of producing 60 units of output.
Question 23
Question
Let Q=f(L, K)=4L0.5K0.5be a production function, where Qis the quantity
of output, Lis the quantity of labor, and Kis the quantity of capital. Determine
the marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function f(L, K) with respect to Lwhile holding Kconstant.
∂f
∂L = 2L−0.5K0.5
Step 2: Simplifying the expression, we find the marginal product of labor:
MPL= 2√K/L
18
Step 3: To find the marginal product of capital, we differentiate the produc-
tion function f(L, K) with respect to Kwhile holding Lconstant.
∂f
∂K = 2L0.5K−0.5
Step 4: Simplifying the expression, we find the marginal product of capital:
MPK= 2√L/K
Therefore, the marginal product of labor is 2√K/L and the marginal prod-
uct of capital is 2√L/K.
Question 24
Question
Suppose a production function is given by Q=L1/2K1/4, where Qis the quan-
tity produced, Lis labor input, and Kis capital input. If the wage rate is w= 4
and the rental rate for capital is r= 9, what is the cost-minimizing combination
of labor and capital inputs to produce 100 units of output?
Solution
Step 1: To find the cost-minimizing combination of inputs, we need to minimize
the firm’s cost function, C=wL +rK, subject to the production constraint
Q=L1/2K1/4and the given output level of 100 units.
Step 2: The Lagrangian for this problem is:
L=wL +rK +λ(100 −L1/2K1/4)
Step 3: Taking the first-order conditions with respect to L,K, and λ, we
get the following three equations:
∂L
∂L =w−λ
2L1/2K1/4= 0
∂L
∂K =r−λ
4L1/2K−3/4= 0
∂L
∂λ = 100 −L1/2K1/4= 0
Step 4: Solving the first two equations for λand setting them equal to each
other, we get: w
2L1/2K1/4=r
4L1/2K3/4
Step 5: Simplifying the equation above, we find:
2rK = 4w
19
Step 6: Substituting r= 9 and w= 4 into the equation above, we get
2(9)K= 4(4). Solving for K, we find K= 8.
Step 7: Substituting K= 8 into the production constraint, we have 100 =
L1/2(8)1/4. Solving for L, we find L= 25.
Therefore, the cost-minimizing combination of labor and capital inputs to
produce 100 units of output is L= 25 and K= 8.
Question 25
Question
Consider a production function f(x, y) = 3x2y−2xy2. Find the marginal
product of labor (MP L) and the marginal product of capital (M P K) at the
point (4,2).
Solution
To find the marginal product of labor (M P L), we need to calculate the partial
derivative of the production function f(x, y) with respect to xwhen yis held
constant. Similarly, to find the marginal product of capital (M P K), we need
to calculate the partial derivative of the production function with respect to y
when xis held constant.
Step 1: Calculate M P L
MP L =∂f
∂x =∂(3x2y−2xy2)
∂x = 6xy −2y2
Step 2: Evaluate MP L at (4,2)
MP L(4,2) = 6(4)(2) −2(2)2= 48 −8 = 40
Therefore, the marginal product of labor at (4,2) is 40.
Step 3: Calculate M P K
MP K =∂f
∂y =∂(3x2y−2xy2)
∂y = 3x2−4xy
Step 4: Evaluate MP K at (4,2)
MP K(4,2) = 3(4)2−4(4)(2) = 48 −32 = 16
Therefore, the marginal product of capital at (4,2) is 16.
Question 26
Question
Suppose a production function is given by Q=L0.6K0.4. If the wage rate is
w= 2 and the rental rate of capital is r= 3, what is the minimum cost of
producing 100 units of output given these factor prices?
20
Solution
Step 1: The cost minimization problem can be formulated as: Minimize C=
wL +rK Subject to the production function Q=L0.6K0.4and the output
target Q= 100.
Step 2: We can substitute the production function into the cost function to
get: C=wL +rK = 2L+ 3K
Step 3: Now, we can substitute the output target into the production func-
tion to get: 100 = L0.6K0.4
Step 4: From the cost function, we have C= 2L+ 3Kand from the pro-
duction function, we have 100 = L0.6K0.4. We can use the Lagrange method to
find the minimum cost efficiently.
Step 5: Setting up the Lagrangian function: L= 2L+3K+λ(100−L0.6K0.4)
Step 6: Taking the partial derivatives of Lwith respect to L,K, and λand
setting them equal to zero gives us the first-order conditions.
Step 7: Solving the first-order conditions, we find the values of Land K
that minimize the cost.
Step 8: Finally, substitute the values of Land Kback into the cost function
C= 2L+ 3Kto find the minimum cost of producing 100 units of output.
Question 27
Question
Let Q=L0.5K0.5represent a production function where Qis the quantity
produced, Lis the amount of labor input, and Kis the amount of capital input.
Determine the total product of labor when capital is fixed at 16 units, and find
the average product of labor when 64 units of output are produced.
Solution
Step 1: To find the total product of labor, we need to substitute K= 16 into
the production function Q=L0.5K0.5.
Total Product of Labor (TPL) = Q=L0.5×160.5= 4L0.5
Step 2: To find the average product of labor, we first need to find the amount
of labor input that produces 64 units of output. Given Q= 64, we substitute
Q= 64 into the production function Q=L0.5K0.5.
64 = L0.5×160.5= 4L0.5
16 = L0.5
L= 162= 256
21
Step 3: Now that we know the amount of labor input (L= 256), we can
find the average product of labor by substituting L= 256 into the production
function.
Average Product of Labor (APL) = Q
L=64
256 = 0.25
Therefore, the total product of labor when capital is fixed at 16 units is 4L0.5
and the average product of labor when 64 units of output are produced is 0.25
units.
Question 28
Question
Consider a production function given by Q=K0.3L0.7, where Qis the level
of output, Kis the amount of capital, and Lis the amount of labor. If the
marginal product of capital is given by MPK= 0.3K−0.7L0.7, and the marginal
product of labor is given by MPL= 0.7K0.3L−0.3, find the equation of the
isoquant that passes through the point (16,4).
Solution
Step 1: The equation of an isoquant represents all combinations of capital and
labor that can produce a given level of output. To find the equation of the
isoquant passing through the point (16,4), we need to substitute these values
of capital and labor into the production function Q=K0.3L0.7.
Step 2: Substituting K= 16 and L= 4 into the production function:
Q= 160.3·40.7
Q= 2.2974 ·5.1538
Q≈11.8396
Step 3: The equation of the isoquant passing through the point (16,4) is:
11.8396 = K0.3L0.7
Step 4: To simplify the equation above, the isoquant can be written in the
following form:
K0.3L0.7= 11.8396
Therefore, the equation of the isoquant passing through the point (16,4) is
K0.3L0.7= 11.8396.
22
Question 29
Question
Consider the production function Q= 4L0.5K0.5, where Qrepresents the quan-
tity of output produced, Lrepresents labor input, and Krepresents capital
input.
If the price of labor is w= 2 and the price of capital is r= 3, calculate the
cost minimizing quantities of labor and capital required to produce 36 units of
output.
Solution
Step 1: Write the cost minimization problem. Given the production function
Q= 4L0.5K0.5, the cost minimization problem can be expressed as:
minimize wL +rK
subject to the production constraint Q= 36.
Step 2: Substitute the given production function into the cost function.
Substitute Q= 36 into the production function Q= 4L0.5K0.5to get:
36 = 4L0.5K0.5
Step 3: Express the cost function in terms of one variable. Since w= 2 and
r= 3, the cost function becomes:
C= 2L+ 3K
Step 4: Express the production constraint in terms of one variable. Solve
the production constraint 36 = 4L0.5K0.5for K:
K=36
4L0.5= 9L−0.5
Step 5: Substitute the production constraint into the cost function to get
a single variable cost function. Substitute K= 9L−0.5into the cost function
C= 2L+ 3K:
C= 2L+ 3(9L−0.5)=2L+ 27L−0.5
Step 6: Find the minimum cost. To find the minimum cost, differentiate the
cost function C= 2L+ 27L−0.5with respect to Land set it equal to 0:
dC
dL = 2 −27
2L−1.5= 0
Step 7: Solve for the optimal value of L. Solving 27
2L−1.5= 2 gives L=
(27
4)2
3= 9.
Step 8: Find the optimal value of K. Substitute L= 9 back into K= 9L−0.5:
K= 9(9)−0.5= 3
Therefore, the cost-minimizing quantities of labor and capital required to
produce 36 units of output are L= 9 and K= 3, respectively.
23
Question 30
Question
Consider a production function Q= 2K0.5L0.5where Qrepresents the total
output, Kis the amount of capital input, and Lis the amount of labor input.
Calculate the marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor (M PL), we differentiate the
production function with respect to L, holding Kconstant.
MPL=∂Q
∂L =∂(2K0.5L0.5)
∂L
Step 2: Applying the power rule of differentiation, we get
MPL= 2 ·0.5·K0.5·L0.5−1=K0.5L−0.5
Step 3: Similarly, to find the marginal product of capital (M PK), we differ-
entiate the production function with respect to K, holding Lconstant.
MPK=∂Q
∂K =∂(2K0.5L0.5)
∂K
Step 4: Applying the power rule of differentiation, we get
MPK= 2 ·0.5·L0.5·K0.5−1=L0.5K−0.5
Therefore, the marginal product of labor is K0.5L−0.5and the marginal
product of capital is L0.5K−0.5.
Question 31
Question
Consider a production function with two inputs, labor (L) and capital (K),
given by the function Q= 2L0.5K0.5. Determine the marginal product of labor
and the marginal product of capital.
Solution
To find the marginal product of labor (MPL) and the marginal product of
capital (MPK), we need to take the partial derivative of the production function
with respect to each input while holding the other input constant.
Step 1: Find the Marginal Product of Labor (M PL)To find M PL,
we take the partial derivative of Qwith respect to L:
MPL=∂Q
∂L = 0.5·2K0.5·L−0.5=K0.5·L−0.5
24
Step 2: Find the Marginal Product of Capital (M PK)To find MPK,
we take the partial derivative of Qwith respect to K:
MPK=∂Q
∂K = 0.5·2L0.5·K−0.5=L0.5·K−0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=L0.5
K0.5.
Question 32
Question
Consider a production function given by Q= 2LK, where Qis the total output,
Lis the amount of labor, and Kis the amount of capital. Determine the
marginal product of labor and the marginal product of capital.
Solution
To find the marginal product of labor (MPL), we differentiate the production
function with respect to L, holding Kconstant. To find the marginal product
of capital (MPK), we differentiate the production function with respect to K,
holding Lconstant.
Step 1: Find the marginal product of labor (M PL)
Q= 2LK
MPL=∂Q
∂L = 2K
Step 2: Find the marginal product of capital (MPK)
Q= 2LK
MPK=∂Q
∂K = 2L
Therefore, the marginal product of labor is MPL= 2Kand the marginal
product of capital is MPK= 2L.
Question 33
Question
Let Qbe the total output, Lbe the quantity of labor, and Kbe the quantity
of capital in a production process. A production function is defined by Q=
2L0.5K0.5. Determine the marginal product of labor (M PL) and the marginal
product of capital (MPK).
25
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, while holding the
quantity of capital Kconstant.
MPL=∂Q
∂L
Given Q= 2L0.5K0.5, we calculate:
MPL=∂
∂L(2L0.5K0.5) = K0.5∂
∂L(2L0.5)
MPL=K0.5·1L−0.5=K0.5
L0.5
Step 2: Now, to find the marginal product of capital (M PK), we need to
take the partial derivative of the production function with respect to capital K,
while holding the quantity of labor Lconstant.
MPK=∂Q
∂K
Given Q= 2L0.5K0.5, we calculate:
MPK=∂
∂K (2L0.5K0.5) = L0.5∂
∂K (2K0.5)
MPK=L0.5·1·2K−0.5=2L0.5
K0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=2L0.5
K0.5.
Question 34
Question
Suppose a firm has the production function given by Q=LαKβ, where Q
represents the quantity of output, Lis the quantity of labor input, and Kis the
quantity of capital input. The firm wants to minimize its total cost by choosing
optimal quantities of labor and capital inputs. If the wage rate is wand the
rental rate of capital is r, find the firm’s cost-minimizing combination of Land
Kin terms of w,r,α, and β.
26
Solution
Step 1: Write the firm’s cost minimization problem. The firm wants to minimize
its total cost C, given by C=wL +rK, subject to the production constraint
Q=LαKβ.
Step 2: Form the Lagrangian function. We can set up the Lagrangian func-
tion as:
L(L, K, λ) = wL +rK +λ(Q−LαKβ)
Step 3: Find the first-order conditions. Taking the derivative of Lwith
respect to L,K, and λ, and setting them equal to zero, we get the following
three equations:
∂L
∂L =w−αλLα−1Kβ= 0
∂L
∂K =r−βλLαKβ−1= 0
∂L
∂λ =Q−LαKβ= 0
Step 4: Solve the system of equations. From the first two equations, we
have: w
αLα−1Kβ=r
βLαKβ−1
Solving for L
K, we get:
L
K=wβ
rα
1
β−α
Step 5: Substitute L
Kback into the production function. Substitute L
Kback
into Q=LαKβ, we get:
Q=wβ
β−αrα
β−α1/α+β
Therefore, the firm’s cost-minimizing combination of Land Kis L=wβ
rα
1
β−α
and K=rα
wβ
1
β−α.
Question 35
Question
Let Q= 3L0.5K0.5be a production function where Qis the level of output, Lis
the amount of labor input, and Kis the amount of capital input. The current
amount of labor input is 16 units and the current amount of capital input is 25
units. Calculate the marginal product of labor at the current input levels.
27
Solution
Step 1: The cost function can be expressed as C=wL +rK.
Step 2: We are given the production function Q= 4L0.5K0.5= 100. This
can be rewritten as L0.5K0.5= 25.
Step 3: To minimize the cost of producing 100 units of output, we need to
solve the following minimization problem:
Minimize wL +rK subject to L0.5K0.5= 25
Step 4: We can rewrite the minimization problem in terms of a single variable
by substituting K= 25/L0.5into the cost function:
C=wL +r25
L0.5=wL +25r
L0.5
Step 5: To find the minimum cost, we differentiate the cost function with
respect to Land set the derivative equal to 0:
dC
dL =w−25r
2L1.5= 0
Step 6: Solving for L, we get L=q50r
w.
Step 7: Substitute the value of Lback into the production function L0.5K0.5=
25 to find K:r50r
w×K0.5= 25
K=252w
50r
Step 8: Therefore, the optimal combination of labor and capital inputs that
minimizes the cost of producing 100 units of output is L=q50r
wand K=252w
50r.
Question 3
Question
Consider a production function given by Q= 2K0.5L0.5, where Qrepresents the
total quantity produced, Kis the amount of capital input, and Lis the amount
of labor input. If the price of capital (PK) is 3 and the price of labor (PL) is
4, find the optimal combination of capital and labor inputs that minimizes the
total cost of production.
2
Solution
Step 1: The total cost (C) of production can be expressed as a function of the
quantities of capital and labor used, along with their respective prices. There-
fore, C=PK·K+PL·L.
Step 2: Substituting the expression for Qinto the cost function, we get
C= 3K+ 4L.
Step 3: To minimize total cost subject to the production constraint Q=
2K0.5L0.5, we can substitute the expression for Cinto the production function
and solve for Kin terms of L.
Step 4: Substituting Cinto the production function gives 2K0.5L0.5= 3K+
4L.
Step 5: Simplifying the equation, we get K0.5=4L
2−3.
Step 6: Squaring both sides gives K=16L2
4, which simplifies to K= 4L2.
Step 7: Now, we substitute the expression for Kback into the cost function
to get C= 3(4L2)+4L.
Step 8: Simplifying further, we have C= 12L2+ 4L.
Step 9: To find the minimum cost, we need to differentiate the cost function
with respect to Land set it equal to zero. This gives dC
dL = 24L+ 4 = 0.
Step 10: Solving the above equation for L, we find L=−1
6.
Step 11: Since the amount of labor cannot be negative, this negative value
is not valid. Therefore, the critical point is at L= 0.
Step 12: Substituting L= 0 back into the production function, we find that
the corresponding value of Kis also 0.
Therefore, the optimal combination of capital and labor inputs that mini-
mizes the total cost of production is K= 0 and L= 0.
Question 4
Question
Let Q= 2L0.5K0.5be the production function of a company, where Qis the
total output, Lis the amount of labor input, and Kis the amount of capital
input. If the wage rate is w= 4 and the rental rate of capital is r= 9, find the
cost-minimizing combination of labor and capital inputs that produce 100 units
of output.
Solution
Step 1: The cost function is given by C=wL +rK, where Cis the total cost.
We need to minimize the cost function subject to the constraint of producing
100 units of output.
Step 2: From the production function, we have Q= 2L0.5K0.5. Since we
are producing 100 units of output, we can substitute this into the production
function to get:
100 = 2L0.5K0.5
3
Step 3: To minimize the cost function, we’ll use the Lagrange multiplier
method. Let J=wL +rK −λ(2L0.5K0.5−100), where λis the Lagrange
multiplier.
Step 4: Taking the partial derivatives of Jwith respect to L,K, and λand
setting them equal to zero, we get the following equations:
∂J
∂L =w−λL−0.5K0.5= 0
∂J
∂K =r−λL0.5K−0.5= 0
∂J
∂λ = 2L0.5K0.5−100 = 0
Step 5: Solving the system of equations, we find:
L= 25 and K= 25
Therefore, the cost-minimizing combination of labor and capital inputs that
produce 100 units of output is L= 25 and K= 25.
Question 5
Question
Consider a production function in the form of Cobb-Douglas function:
Q(K, L) = K0.4L0.6
where Qis the total output, Kis the quantity of capital, and Lis the quantity
of labor.
Assume that the price of capital (PK) is 10 and the price of labor (PL) is 5.
Determine the total cost function (T C) in terms of Q.
Solution
Step 1: The total cost function (T C) can be calculated by multiplying the
quantity of capital (K) by the price of capital (PK) and the quantity of labor
(L) by the price of labor (PL).
T C =K·PK+L·PL
Step 2: Since Kand Lare related to the total output (Q) by the Cobb-
Douglas function, we can express them in terms of Q.
K=Q0.4
L=Q0.6
4
Step 3: Substitute the expressions for Kand Lin terms of Qinto the total
cost function:
T C =Q0.4·PK+Q0.6·PL
Step 4: Replace the given prices for capital (PK) and labor (PL) to get the
total cost function in terms of Q:
T C =Q0.4·10 + Q0.6·5
Therefore, the total cost function in terms of Qis T C = 10Q0.4+ 5Q0.6.
Question 6
Question
Consider a production function for a firm given by Q=L0.3K0.7, where Q
represents the quantity of output, Lrepresents the quantity of labor, and K
represents the quantity of capital. Determine the minimum cost combination of
labor and capital needed to produce 100 units of output, given that the cost of
labor is
$
5 per unit and the cost of capital is
$
10 per unit.
Solution
Step 1: We are given the production function Q=L0.3K0.7. Since we want to
produce 100 units of output, we can set Q= 100 in the production function to
get the relationship between Land K.
Step 2: Substituting Q= 100 into the production function, we have 100 =
L0.3K0.7.
Step 3: To minimize the cost of production, we need to minimize the total
cost Cwhich is given by C= 5L+10K, subject to the constraint 100 = L0.3K0.7.
Step 4: We can rewrite the constraint in terms of Kto substitute it into the
cost function. Taking the natural logarithm of both sides of the constraint gives
ln(100) = lnL0.3K0.7. Simplifying, we get 2.3026 = 0.3 ln(L)+0.7 ln(K).
Step 5: Rearranging the constraint equation, we can express Kin terms of
Las K=100
L0.31/0.7.
Step 6: Substituting the expression for Kinto the cost function, we have
C= 5L+ 10 100
L0.31/0.7.
Step 7: To find the minimum cost combination of labor and capital, we
need to minimize C. This involves finding the critical points of Cby taking its
derivative with respect to L, setting it equal to 0, and solving for L.
Step 8: After finding the value of L, we can substitute it back into the
constraint equation to find the corresponding value of K.
5
Question 7
Question
Consider a production function given by Q=K0.3L0.7, where Qis the quantity
produced, Kis the amount of capital used, and Lis the amount of labor used.
Suppose the price of capital (PK) is
$
10 per unit and the price of labor (PL) is
$
5 per unit. If the firm’s budget is
$
200, how many units of capital and labor
should the firm hire in order to maximize output?
Solution
Step 1: Set up the firm’s cost constraint. Given that the firm’s budget is
$
200,
the cost constraint can be written as:
10K+ 5L= 200
Step 2: Rewrite the production function. Rewrite the production function
as follows:
Q= 100.3L0.7
Step 3: Express production in terms of one variable. Solve the cost constraint
for Kin terms of L:
K= 20 −2L
Step 4: Express output as a function of one variable. Substitute the expres-
sion for Kinto the production function:
Q= (20 −2L)0.3L0.7
Step 5: Find the first derivative of the production function with respect to
L. Calculate the first derivative of Qwith respect to L:
dQ
dL = 0.7(20 −2L)0.3L−0.3−0.3(20 −2L)0.3L0.7
Step 6: Set the derivative equal to zero and solve for L. Setting the derivative
equal to zero:
0.7(20 −2L)0.3L−0.3−0.3(20 −2L)0.3L0.7= 0
Solving for Lyields L= 10.
Step 7: Calculate the amount of capital and labor for maximizing output.
Use the solved value of Lto find K:
K= 20 −2(10) = 0
Therefore, the firm should hire 0 units of capital and 10 units of labor to maxi-
mize output.
6
Question 8
Question
Consider a production function Q=L1/3K2/3, where Qrepresents the output
quantity, Lrepresents the quantity of labor input, and Krepresents the quantity
of capital input. If the wage rate is wand the rental rate of capital is r, find
the equations for the isoquants associated with levels of output Q1< Q2.
Solution
To find the equations for the isoquants associated with levels of output Q1<
Q2in this production function, we begin by setting up the isoquant equation
Q=L1/3K2/3.
Step 1: Find the equation for the isoquant Q1Given that Q1=
L1/3
1K2/3
1, the isoquant equation becomes:
Q1=L1/3
1K2/3
1
Step 2: Find the equation for the isoquant Q2Given that Q2=
L1/3
2K2/3
2, the isoquant equation becomes:
Q2=L1/3
2K2/3
2
Step 3: Equate the two isoquant equations Since we want to find the
isoquants associated with levels of output Q1< Q2, we set Q1=Q2to find the
equation for the isoquants:
L1/3
1K2/3
1=L1/3
2K2/3
2
This equation represents the isoquants associated with levels of output Q1<
Q2.
Question 9
Question
Consider a production function given by Q= 2L0.6K0.3, where Qrepresents
output, Lis labor input, and Kis capital input. If the firm currently employs
100 units of labor and 64 units of capital, calculate the marginal product of
labor at this point.
Solution
Step 1: Compute the total product of labor (T PL) by substituting L= 100 into
the production function:
T PL= 2(100)0.6(64)0.3
7
T PL= 2(1000.6)(640.3)
T PL= 2(39.81)(4)
T PL= 318.48
Step 2: Calculate the total product of labor when the firm employs one
additional unit of labor (T PL+1) by substituting L= 101 into the production
function:
T PL+1 = 2(101)0.6(64)0.3
T PL+1 = 2(1010.6)(640.3)
T PL+1 = 2(40.01)(4)
T PL+1 = 320.17
Step 3: Determine the marginal product of labor (MPL) as the change in
total product of labor when an additional unit of labor is employed:
MP L =T PL+1 −T PL
MP L = 320.17 −318.48
MP L1.69
Therefore, the marginal product of labor when the firm employs 100 units
of labor and 64 units of capital is approximately 1.69.
Question 10
Question
Let Q=L3/4K1/4be the production function where Lrepresents labor and
Krepresents capital. Determine the level of marginal product of labor at the
point (8,4).
Solution
Step 1: To find the marginal product of labor, we need to take the partial deriva-
tive of the production function Qwith respect to labor L. Step 2: Compute the
partial derivative of Qwith respect to L. Step 3: Substitute L= 8 and K= 4
into the partial derivative to find the marginal product of labor.
Step 1: The marginal product of labor is given by
MPL=∂Q
∂L
8
Step 2: Compute the partial derivative of Qwith respect to Lby differenti-
ating Q=L3/4K1/4with respect to L.
∂Q
∂L =3
4L−1/4K1/4
Step 3: Substitute L= 8 and K= 4 into the partial derivative to find the
marginal product of labor.
MPL(8,4) = 3
4(8)−1/4(4)1/4
MPL(8,4) = 3
4(2−1)(2)
MPL(8,4) = 3
4
Therefore, the level of marginal product of labor at the point (8,4) is 3
4.
Question 11
Question
Consider a production function given by Q= 2L1/2K1/2, where Qrepresents
the quantity of output, Lis the quantity of labor input, and Kis the quantity
of capital input. Suppose the wage rate is w= 4 and the rental rate for capital
is r= 9. If the firm wants to produce 64 units of output at the minimum cost,
how many units of labor and capital should be employed?
Solution
Step 1: To minimize the cost of production subject to the output target, we
need to minimize the total cost function C=w·L+r·Kwhile producing 64
units of output.
Step 2: Substituting the cost and output functions into the total cost func-
tion, we have
C= 4L+ 9K
Step 3: We are given that Q= 64, so substituting Q= 64 into the production
function:
64 = 2L1/2K1/2
Step 4: Rearranging the production function:
32 = L1/2K1/2
Step 5: Squaring both sides to get rid of the fractional exponents:
1024 = LK
9
Step 6: We now have two equations: 32 = L1/2K1/2and 1024 = LK. We
can solve these two equations simultaneously.
Step 7: From 1024 = LK, we can express Kin terms of Las K=1024
L.
Step 8: Substituting K=1024
Linto 32 = L1/2K1/2:
32 = L1/21024
L1/2
Step 9: Simplifying the above equation gives:
32 = √1024
32 = 32
Step 10: Therefore, the firm should employ L= 1024 units of labor and
K= 1 unit of capital to produce 64 units of output at the minimum cost.
Question 12
Question
Consider a production function given by Q=LαKβ, where Qrepresents the
output level, Lis the quantity of labor input, and Kis the quantity of capital
input. If the firm has a fixed budget for inputs, derive the cost-minimizing input
combination to produce a given level of output Q.
Solution
Step 1: We first need to set up the Lagrangian function:
L(L, K, λ) = wL +rK +λ(Q−LαKβ)
where wis the wage rate for labor, ris the rental rate for capital, and λis the
Lagrange multiplier.
Step 2: Next, we take the partial derivatives of Lwith respect to L,K, and
λand set them equal to zero:
∂L
∂L =w−λαLα−1Kβ= 0
∂L
∂K =r−λβLαKβ−1= 0
∂L
∂λ =Q−LαKβ= 0
Step 3: Solve the first equation for λand substitute it into the second
equation:
λ=w
αLα−1Kβ=r
βLαKβ−1
10
Step 4: From the equations above, we can find the optimal input combination
(L∗, K∗) that minimizes the cost for producing output Q.
Step 5: The cost-minimizing input combination can be found using the op-
timal values of Land Kderived from the Lagrangian approach.
Question 13
Question
Consider a production function f(x, y) = x2y3. Determine the average product
of labor when x= 2 and y= 3.
Solution
Let the average product of labor be denoted by AP L. The formula for average
product of labor is given by:
AP L =f(x, y)
x
Step 1: Calculate the production function value at x= 2 and y= 3.
f(2,3) = 22·33= 4 ·27 = 108
Step 2: Substitute x= 2 and y= 3 into the formula for average product of
labor.
AP L =f(2,3)
2=108
2= 54
Therefore, the average product of labor when x= 2 and y= 3 is 54.
Question 14
Question
Consider a production function Q=L0.5K0.5, where Qrepresents the level of
output, Lis the quantity of labor input, and Kis the quantity of capital input.
If the wage rate is w= 10 and the rental rate of capital is r= 20, find the
minimum cost of producing 400 units of output.
Solution
1. The cost function for producing Qunits of output can be expressed as
C(w, r, Q) = wL +rK.
2. Given the production function Q=L0.5K0.5, we need to find the quanti-
ties of labor (L) and capital (K) that minimize the cost of producing 400 units
of output.
3. Since Q= 400, we have 400 = L0.5K0.5.
11
4. To minimize the cost, we need to minimize C(w, r, Q) = wL +rK subject
to the constraint 400 = L0.5K0.5.
5. By the Lagrange Multiplier method, we construct the Lagrangian as
follows:
L(L, K, λ) = wL +rK −λ(400 −L0.5K0.5)
6. Taking partial derivatives with respect to L,K, and λand setting them
equal to zero, we have:
∂L
∂L =w−0.5λL−0.5K0.5= 0
∂L
∂K =r−0.5λL0.5K−0.5= 0
∂L
∂λ = 400 −L0.5K0.5= 0
7. Solving the system of equations, we find L= 100 and K= 400.
8. Substituting L= 100 and K= 400 back into the cost function C(w, r, Q) =
wL +rK, we get:
C(10,20,400) = 10(100) + 20(400) = 1000 + 8000 = 9000
Therefore, the minimum cost of producing 400 units of output is 9000.
Question 15
Question
Consider a production function given by Q=L0.5K0.5, where Qrepresents the
total output, Lis the amount of labor input, and Kis the amount of capital
input. If the wage rate is w= 10 and the rental rate for capital is r= 20,
find the least-cost combination of labor and capital that produces 100 units of
output.
Solution
Step 1: The cost minimization problem can be formulated as follows: Minimize
C=wL +rK subject to the production function Q=L0.5K0.5and output
level Q= 100.
Step 2: Substitute the production function into the cost function: C=
10L+ 20K.
Step 3: Also, substitute the output level Q= 100 into the production func-
tion: 100 = L0.5K0.5.
Step 4: Rewrite the production function in terms of one variable using the
output level constraint: K=100
L2.
Step 5: Substitute Kinto the cost function to get cost in terms of one
variable: C= 10L+ 20 100
L2.
Step 6: To minimize cost, differentiate Cwith respect to Land set it equal
to zero: dC
dL = 10 −40 100
L3= 0.
12
Step 7: Solve for L: 10L3−40000 = 0. L=3
√4000 = 20.
Step 8: Calculate the corresponding value of Kusing the production func-
tion: K=100
20 2= 25.
Step 9: Therefore, the least-cost combination of labor and capital to produce
100 units of output is L= 20 units and K= 25 units.
Question 16
Question
Let Q=L0.3K0.7represent a production function, where Qis the total output,
Lis the quantity of labor, and Kis the quantity of capital. Determine the
marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function Qwith respect to L.
∂Q
∂L = 0.3L−0.7K0.7
Step 2: Simplify the expression to find the marginal product of labor.
∂Q
∂L = 0.3K0.7
L0.7= 0.3K
L0.7
Step 3: Similarly, to find the marginal product of capital, differentiate the
production function Qwith respect to K.
∂Q
∂K = 0.7L0.3K−0.3
Step 4: Simplify the expression to find the marginal product of capital.
∂Q
∂K = 0.7L0.3
K0.3= 0.7L
K0.3
Therefore, the marginal product of labor is 0.3K
L0.7and the marginal
product of capital is 0.7L
K0.3.
Question 17
Question
Suppose a firm has a production function given by Q=L0.5K0.5, where Q
represents the quantity of output, Lrepresents the quantity of labor, and K
represents the quantity of capital. If the firm has 100 units of capital and wants
to produce 400 units of output, what is the minimum amount of labor the firm
must use?
13
Solution
We are given the production function Q=L0.5K0.5. We are also given that
K= 100 and Q= 400. We need to find the minimum amount of labor L
required to produce 400 units of output.
Step 1: Substitute the given values into the production function.
400 = L0.5×1000.5
Step 2: Simplify the expression.
400 = L0.5×10
Step 3: Divide both sides by 10 to isolate L.
40 = L0.5
Step 4: Square both sides to solve for L.
L= 402
Step 5: Calculate the minimum amount of labor required.
L= 1600
Therefore, the firm must use a minimum of 1600 units of labor to produce
400 units of output when given 100 units of capital.
Question 18
Question
Given a production function Q= 10L0.5K0.5, where Qrepresents the total
output, Lis the quantity of labor, and Kis the quantity of capital, find the
marginal product of labor (MPL) and the marginal product of capital (M PK).
Solution
To find the marginal product of labor (MPL), we need to take the derivative of
the production function with respect to labor (L), while holding the quantity of
capital (K) constant. Similarly, to find the marginal product of capital (M PK),
we need to take the derivative of the production function with respect to capital
(K), while holding the quantity of labor constant.
14
Step 1: Find MPL
MPL=∂Q
∂L
=∂
∂L(10L0.5K0.5)
= 5K0.5∂
∂LL0.5
= 5K0.5·0.5L−0.5
= 2.5K0.5L−0.5
Hence, the marginal product of labor (MPL) is 2.5K0.5L−0.5.
Step 2: Find MPK
MPK=∂Q
∂K
=∂
∂K (10L0.5K0.5)
= 5L0.5∂
∂K K0.5
= 5L0.5·0.5K−0.5
= 2.5L0.5K−0.5
Therefore, the marginal product of capital (MPK) is 2.5L0.5K−0.5.
Question 19
Question
Consider a production function given by Q= 4L0.5K0.5, where Qrepresents
the quantity of output produced, Lis the quantity of labor input, and Kis
the quantity of capital input. If the wage rate is
$
10 per unit of labor and the
rental rate for capital is
$
20 per unit of capital, determine the cost-minimizing
combination of labor and capital that the firm should use to produce 100 units
of output.
Solution
Step 1: Write the cost-minimization problem. The cost-minimization problem
for the firm can be written as:
Minimize 10L+ 20Ksubject to 4L0.5K0.5= 100
Step 2: Solve for one of the variables in the constraint. Since the objective
function contains Land Kseparately, solve for one of the variables in the
constraint. We can solve for Kin terms of L:
4L0.5K0.5= 100 ⇒K=100
4L0.5=25
L0.5
15
Step 3: Substitute the expression for Kinto the objective function. Sub-
stitute the expression for Kinto the cost function to obtain a single-variable
function in terms of L:
Minimize 10L+ 20 25
L0.5
Step 4: Simplify the expression.
Minimize 10L+500
L0.5
Step 5: Find the minimum value. To minimize the cost function, differentiate
it with respect to Land set it equal to 0:
d
dL 10L+500
L0.5= 10 −250
L1.5= 0
Solving for L, we get:
L=250
10
2
3
= 25
Step 6: Determine the corresponding value of K. Using the expression
derived earlier for Kin terms of L:
K=25
L0.5=25
5= 5
Therefore, the firm should use 25 units of labor and 5 units of capital to
produce 100 units of output at minimum cost.
Question 20
Question
Consider a production function given by Q=KαLβ, where Qis the quantity
of output, Kis the quantity of capital input, Lis the quantity of labor input,
and α, β > 0. Show that the marginal product of labor can be expressed as
∂Q/∂L
Q=βL/Q.
Solution
Step 1: Calculate the marginal product of labor, ∂Q
∂L .
Step 1: ∂Q
∂L =βKαLβ−1
Step 2: Calculate the total product, Q.
Q=KαLβ
16
Step 3: Express the marginal product of labor as a proportion of the total
product.
Step 3: ∂Q/∂L
Q=βKαLβ−1
KαLβ
Step 4: Simplify the expression.
βKαLβ−1
KαLβ=βL
L=βL/Q
Therefore, the marginal product of labor can be expressed as ∂Q/∂L
Q=βL/Q.
Question 21
Question
Suppose a firm’s production function is given by Q= 10L1/2K1/2, where Q
represents output, Lrepresents labor input, and Krepresents capital input. If
the firm hires labor at a wage rate of w= 4 and rents capital at a rate of r= 5,
determine the minimum cost of producing 100 units of output.
Solution
Step 1: Calculate the total cost function using the given production function.
Total Cost (TC) = Total Labor Cost + Total Capital Cost
Step 2: Calculate the labor input required to produce 100 units of output.
Q= 10L1/2K1/2=⇒100 = 10L1/2K1/2
Step 3: Substitute the value of Kfrom the production function into the cost
function and differentiate to find the minimum cost.
Minimize T C =wL +rK = 4L+ 5 100
10L1/2
Step 4: Differentiate to find the minimum.
dT C
dL = 4 −50
L3/2=⇒4 = 50
L3/2=⇒L3/2=50
4= 12.5 =⇒L≈7.90
Step 5: Calculate the minimum cost.
K=100
10(7.90)1/2≈15.85
Minimum Cost = 4(7.90) + 5(15.85) ≈97.75
Therefore, the minimum cost of producing 100 units of output is approxi-
mately 97.75.
17
Question 22
Question
Consider a production function Q= 2K1
3L2
3, where Qrepresents the total
output, Kis the amount of capital input, and Lis the amount of labor input.
At a certain point, the amount of capital is fixed at 64 units. If each unit of
labor costs
$
4 and each unit of capital costs
$
6, how many units of labor should
be hired to minimize the cost of producing 60 units of output?
Solution
Step 1: To minimize the cost of producing 60 units of output, we need to
minimize the total cost function. The total cost function is given by C=
rK +wL, where ris the cost of capital per unit and wis the cost of labor per
unit.
Step 2: Given that each unit of labor costs
$
4 and each unit of capital costs
$
6, the total cost function becomes C= 6(64) + 4L.
Step 3: We are given that the total output Qis 60 units. Substituting
Q= 60 into the production function, we have 60 = 2(64) 1
3L2
3.
Step 4: Simplifying the production function gives 30 = 4L2
3, which further
simplifies to L2
3=30
4.
Step 5: Solving for L, we get L=30
4
3
2=303
43
1
2=27000
64
1
2.
Step 6: Calculating the value inside the square root gives 27000
64 = 421.875.
Thus, L=√421.875 ≈20.54.
Step 7: Since the number of units of labor must be a whole number, we hire
21 units of labor to minimize the cost of producing 60 units of output.
Question 23
Question
Let Q=f(L, K)=4L0.5K0.5be a production function, where Qis the quantity
of output, Lis the quantity of labor, and Kis the quantity of capital. Determine
the marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor, we differentiate the production
function f(L, K) with respect to Lwhile holding Kconstant.
∂f
∂L = 2L−0.5K0.5
Step 2: Simplifying the expression, we find the marginal product of labor:
MPL= 2√K/L
18
Step 3: To find the marginal product of capital, we differentiate the produc-
tion function f(L, K) with respect to Kwhile holding Lconstant.
∂f
∂K = 2L0.5K−0.5
Step 4: Simplifying the expression, we find the marginal product of capital:
MPK= 2√L/K
Therefore, the marginal product of labor is 2√K/L and the marginal prod-
uct of capital is 2√L/K.
Question 24
Question
Suppose a production function is given by Q=L1/2K1/4, where Qis the quan-
tity produced, Lis labor input, and Kis capital input. If the wage rate is w= 4
and the rental rate for capital is r= 9, what is the cost-minimizing combination
of labor and capital inputs to produce 100 units of output?
Solution
Step 1: To find the cost-minimizing combination of inputs, we need to minimize
the firm’s cost function, C=wL +rK, subject to the production constraint
Q=L1/2K1/4and the given output level of 100 units.
Step 2: The Lagrangian for this problem is:
L=wL +rK +λ(100 −L1/2K1/4)
Step 3: Taking the first-order conditions with respect to L,K, and λ, we
get the following three equations:
∂L
∂L =w−λ
2L1/2K1/4= 0
∂L
∂K =r−λ
4L1/2K−3/4= 0
∂L
∂λ = 100 −L1/2K1/4= 0
Step 4: Solving the first two equations for λand setting them equal to each
other, we get: w
2L1/2K1/4=r
4L1/2K3/4
Step 5: Simplifying the equation above, we find:
2rK = 4w
19
Step 6: Substituting r= 9 and w= 4 into the equation above, we get
2(9)K= 4(4). Solving for K, we find K= 8.
Step 7: Substituting K= 8 into the production constraint, we have 100 =
L1/2(8)1/4. Solving for L, we find L= 25.
Therefore, the cost-minimizing combination of labor and capital inputs to
produce 100 units of output is L= 25 and K= 8.
Question 25
Question
Consider a production function f(x, y) = 3x2y−2xy2. Find the marginal
product of labor (MP L) and the marginal product of capital (M P K) at the
point (4,2).
Solution
To find the marginal product of labor (M P L), we need to calculate the partial
derivative of the production function f(x, y) with respect to xwhen yis held
constant. Similarly, to find the marginal product of capital (M P K), we need
to calculate the partial derivative of the production function with respect to y
when xis held constant.
Step 1: Calculate M P L
MP L =∂f
∂x =∂(3x2y−2xy2)
∂x = 6xy −2y2
Step 2: Evaluate MP L at (4,2)
MP L(4,2) = 6(4)(2) −2(2)2= 48 −8 = 40
Therefore, the marginal product of labor at (4,2) is 40.
Step 3: Calculate M P K
MP K =∂f
∂y =∂(3x2y−2xy2)
∂y = 3x2−4xy
Step 4: Evaluate MP K at (4,2)
MP K(4,2) = 3(4)2−4(4)(2) = 48 −32 = 16
Therefore, the marginal product of capital at (4,2) is 16.
Question 26
Question
Suppose a production function is given by Q=L0.6K0.4. If the wage rate is
w= 2 and the rental rate of capital is r= 3, what is the minimum cost of
producing 100 units of output given these factor prices?
20
Solution
Step 1: The cost minimization problem can be formulated as: Minimize C=
wL +rK Subject to the production function Q=L0.6K0.4and the output
target Q= 100.
Step 2: We can substitute the production function into the cost function to
get: C=wL +rK = 2L+ 3K
Step 3: Now, we can substitute the output target into the production func-
tion to get: 100 = L0.6K0.4
Step 4: From the cost function, we have C= 2L+ 3Kand from the pro-
duction function, we have 100 = L0.6K0.4. We can use the Lagrange method to
find the minimum cost efficiently.
Step 5: Setting up the Lagrangian function: L= 2L+3K+λ(100−L0.6K0.4)
Step 6: Taking the partial derivatives of Lwith respect to L,K, and λand
setting them equal to zero gives us the first-order conditions.
Step 7: Solving the first-order conditions, we find the values of Land K
that minimize the cost.
Step 8: Finally, substitute the values of Land Kback into the cost function
C= 2L+ 3Kto find the minimum cost of producing 100 units of output.
Question 27
Question
Let Q=L0.5K0.5represent a production function where Qis the quantity
produced, Lis the amount of labor input, and Kis the amount of capital input.
Determine the total product of labor when capital is fixed at 16 units, and find
the average product of labor when 64 units of output are produced.
Solution
Step 1: To find the total product of labor, we need to substitute K= 16 into
the production function Q=L0.5K0.5.
Total Product of Labor (TPL) = Q=L0.5×160.5= 4L0.5
Step 2: To find the average product of labor, we first need to find the amount
of labor input that produces 64 units of output. Given Q= 64, we substitute
Q= 64 into the production function Q=L0.5K0.5.
64 = L0.5×160.5= 4L0.5
16 = L0.5
L= 162= 256
21
Step 3: Now that we know the amount of labor input (L= 256), we can
find the average product of labor by substituting L= 256 into the production
function.
Average Product of Labor (APL) = Q
L=64
256 = 0.25
Therefore, the total product of labor when capital is fixed at 16 units is 4L0.5
and the average product of labor when 64 units of output are produced is 0.25
units.
Question 28
Question
Consider a production function given by Q=K0.3L0.7, where Qis the level
of output, Kis the amount of capital, and Lis the amount of labor. If the
marginal product of capital is given by MPK= 0.3K−0.7L0.7, and the marginal
product of labor is given by MPL= 0.7K0.3L−0.3, find the equation of the
isoquant that passes through the point (16,4).
Solution
Step 1: The equation of an isoquant represents all combinations of capital and
labor that can produce a given level of output. To find the equation of the
isoquant passing through the point (16,4), we need to substitute these values
of capital and labor into the production function Q=K0.3L0.7.
Step 2: Substituting K= 16 and L= 4 into the production function:
Q= 160.3·40.7
Q= 2.2974 ·5.1538
Q≈11.8396
Step 3: The equation of the isoquant passing through the point (16,4) is:
11.8396 = K0.3L0.7
Step 4: To simplify the equation above, the isoquant can be written in the
following form:
K0.3L0.7= 11.8396
Therefore, the equation of the isoquant passing through the point (16,4) is
K0.3L0.7= 11.8396.
22
Question 29
Question
Consider the production function Q= 4L0.5K0.5, where Qrepresents the quan-
tity of output produced, Lrepresents labor input, and Krepresents capital
input.
If the price of labor is w= 2 and the price of capital is r= 3, calculate the
cost minimizing quantities of labor and capital required to produce 36 units of
output.
Solution
Step 1: Write the cost minimization problem. Given the production function
Q= 4L0.5K0.5, the cost minimization problem can be expressed as:
minimize wL +rK
subject to the production constraint Q= 36.
Step 2: Substitute the given production function into the cost function.
Substitute Q= 36 into the production function Q= 4L0.5K0.5to get:
36 = 4L0.5K0.5
Step 3: Express the cost function in terms of one variable. Since w= 2 and
r= 3, the cost function becomes:
C= 2L+ 3K
Step 4: Express the production constraint in terms of one variable. Solve
the production constraint 36 = 4L0.5K0.5for K:
K=36
4L0.5= 9L−0.5
Step 5: Substitute the production constraint into the cost function to get
a single variable cost function. Substitute K= 9L−0.5into the cost function
C= 2L+ 3K:
C= 2L+ 3(9L−0.5)=2L+ 27L−0.5
Step 6: Find the minimum cost. To find the minimum cost, differentiate the
cost function C= 2L+ 27L−0.5with respect to Land set it equal to 0:
dC
dL = 2 −27
2L−1.5= 0
Step 7: Solve for the optimal value of L. Solving 27
2L−1.5= 2 gives L=
(27
4)2
3= 9.
Step 8: Find the optimal value of K. Substitute L= 9 back into K= 9L−0.5:
K= 9(9)−0.5= 3
Therefore, the cost-minimizing quantities of labor and capital required to
produce 36 units of output are L= 9 and K= 3, respectively.
23
Question 30
Question
Consider a production function Q= 2K0.5L0.5where Qrepresents the total
output, Kis the amount of capital input, and Lis the amount of labor input.
Calculate the marginal product of labor and the marginal product of capital.
Solution
Step 1: To find the marginal product of labor (M PL), we differentiate the
production function with respect to L, holding Kconstant.
MPL=∂Q
∂L =∂(2K0.5L0.5)
∂L
Step 2: Applying the power rule of differentiation, we get
MPL= 2 ·0.5·K0.5·L0.5−1=K0.5L−0.5
Step 3: Similarly, to find the marginal product of capital (M PK), we differ-
entiate the production function with respect to K, holding Lconstant.
MPK=∂Q
∂K =∂(2K0.5L0.5)
∂K
Step 4: Applying the power rule of differentiation, we get
MPK= 2 ·0.5·L0.5·K0.5−1=L0.5K−0.5
Therefore, the marginal product of labor is K0.5L−0.5and the marginal
product of capital is L0.5K−0.5.
Question 31
Question
Consider a production function with two inputs, labor (L) and capital (K),
given by the function Q= 2L0.5K0.5. Determine the marginal product of labor
and the marginal product of capital.
Solution
To find the marginal product of labor (MPL) and the marginal product of
capital (MPK), we need to take the partial derivative of the production function
with respect to each input while holding the other input constant.
Step 1: Find the Marginal Product of Labor (M PL)To find M PL,
we take the partial derivative of Qwith respect to L:
MPL=∂Q
∂L = 0.5·2K0.5·L−0.5=K0.5·L−0.5
24
Step 2: Find the Marginal Product of Capital (M PK)To find MPK,
we take the partial derivative of Qwith respect to K:
MPK=∂Q
∂K = 0.5·2L0.5·K−0.5=L0.5·K−0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=L0.5
K0.5.
Question 32
Question
Consider a production function given by Q= 2LK, where Qis the total output,
Lis the amount of labor, and Kis the amount of capital. Determine the
marginal product of labor and the marginal product of capital.
Solution
To find the marginal product of labor (MPL), we differentiate the production
function with respect to L, holding Kconstant. To find the marginal product
of capital (MPK), we differentiate the production function with respect to K,
holding Lconstant.
Step 1: Find the marginal product of labor (M PL)
Q= 2LK
MPL=∂Q
∂L = 2K
Step 2: Find the marginal product of capital (MPK)
Q= 2LK
MPK=∂Q
∂K = 2L
Therefore, the marginal product of labor is MPL= 2Kand the marginal
product of capital is MPK= 2L.
Question 33
Question
Let Qbe the total output, Lbe the quantity of labor, and Kbe the quantity
of capital in a production process. A production function is defined by Q=
2L0.5K0.5. Determine the marginal product of labor (M PL) and the marginal
product of capital (MPK).
25
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, while holding the
quantity of capital Kconstant.
MPL=∂Q
∂L
Given Q= 2L0.5K0.5, we calculate:
MPL=∂
∂L(2L0.5K0.5) = K0.5∂
∂L(2L0.5)
MPL=K0.5·1L−0.5=K0.5
L0.5
Step 2: Now, to find the marginal product of capital (M PK), we need to
take the partial derivative of the production function with respect to capital K,
while holding the quantity of labor Lconstant.
MPK=∂Q
∂K
Given Q= 2L0.5K0.5, we calculate:
MPK=∂
∂K (2L0.5K0.5) = L0.5∂
∂K (2K0.5)
MPK=L0.5·1·2K−0.5=2L0.5
K0.5
Therefore, the marginal product of labor is MPL=K0.5
L0.5and the marginal
product of capital is MPK=2L0.5
K0.5.
Question 34
Question
Suppose a firm has the production function given by Q=LαKβ, where Q
represents the quantity of output, Lis the quantity of labor input, and Kis the
quantity of capital input. The firm wants to minimize its total cost by choosing
optimal quantities of labor and capital inputs. If the wage rate is wand the
rental rate of capital is r, find the firm’s cost-minimizing combination of Land
Kin terms of w,r,α, and β.
26
Solution
Step 1: Write the firm’s cost minimization problem. The firm wants to minimize
its total cost C, given by C=wL +rK, subject to the production constraint
Q=LαKβ.
Step 2: Form the Lagrangian function. We can set up the Lagrangian func-
tion as:
L(L, K, λ) = wL +rK +λ(Q−LαKβ)
Step 3: Find the first-order conditions. Taking the derivative of Lwith
respect to L,K, and λ, and setting them equal to zero, we get the following
three equations:
∂L
∂L =w−αλLα−1Kβ= 0
∂L
∂K =r−βλLαKβ−1= 0
∂L
∂λ =Q−LαKβ= 0
Step 4: Solve the system of equations. From the first two equations, we
have: w
αLα−1Kβ=r
βLαKβ−1
Solving for L
K, we get:
L
K=wβ
rα
1
β−α
Step 5: Substitute L
Kback into the production function. Substitute L
Kback
into Q=LαKβ, we get:
Q=wβ
β−αrα
β−α1/α+β
Therefore, the firm’s cost-minimizing combination of Land Kis L=wβ
rα
1
β−α
and K=rα
wβ
1
β−α.
Question 35
Question
Let Q= 3L0.5K0.5be a production function where Qis the level of output, Lis
the amount of labor input, and Kis the amount of capital input. The current
amount of labor input is 16 units and the current amount of capital input is 25
units. Calculate the marginal product of labor at the current input levels.
27
Solution
Step 1: Calculate the total product of labor.
We calculate the total product of labor by plugging in the current level of capital
input into the production function.
Q= 3(16)0.5(25)0.5
Q= 3(4)(5)
Q= 60
Step 2: Calculate the total product of labor when the labor input is increased
by 1 unit. To calculate the total product of labor when labor is increased by 1
unit, we need to plug in L+ 1 into the production function.
QL+1 = 3(16 + 1)0.5(25)0.5
QL+1 = 3(17)0.5(25)0.5
QL+1 = 3(4.123)(5)
QL+1 = 61.845
Step 3: Calculate the marginal product of labor. The marginal product of
labor is the change in total product when labor input is increased by 1 unit.
MPL=QL+1 −Q
MPL= 61.845 −60
MPL= 1.845
Therefore, the marginal product of labor at the current input levels is 1.845
units.
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