Download Q = 100K 0.5 L 0.5

Economic Analysis
for Business
Session XVII: Production Function
and Factor Markets
Instructor
Sandeep Basnyat
9841892281
[email protected]
Recall: Profit Maximization
As stated before,
 Firms will hire extra labour and capital until:
MRPL = w and MRPK = r
PxMPL = w and PxMPK = r
Dividing,
MPL / MPK = w / r
MPL / w = MPK / r ……………..(i)
Equation (i) is known as efficiency condition. Or
Least cost combination input.
Meaning: if a firm is maximizing in the above condition,
then it is efficiently operating.
Numerical Problems
1) Consider the Production function:
Q = 3LK + 2K
Price per unit of K and L are 76 and 6
respectively. Per unit selling price for
output is 2. Find:
a) Amount of factors demanded
b) Amount of product produced
c) Amount of profit or Loss generated
Solution Numerical Prob. (1)
Q = 3LK + 2K
MPK = 3L+2
MPL = 3K
Profit Maximizing condition,
P x MPL = w and
P x MPK = r
2(3K) = 6
and
2 (3L+2) = 76
K=1
and
L = 12
Q = 3(12)(1) + 2(1) = 38
Profit = TR –TC = 2(38) –[6(12) + 76(1)] = - 72 (Loss)
Numerical 2
Assume the production function:
Q = 100K0.5L0.5
a) If the price of labour and capital are 4 and 2
respectively, find the efficient input
combination (least cost combination input) for
producing 1000 units of output.
b) How would the input mix change if the price
of capital increased to r = 4 and the firm still
wanted to produce 1000 units of output?
c) Interpret the result of (b).
Solution to Numerical 2
a) Q = 100K0.5L0.5
MPK = 50(L0.5 / K0.5)
MPL = 50(K0.5 / L0.5)
Efficiency condition,
MPL / MPK = w / r
K = (w/r)L …………….(i)
Substituting the value of K in Production function,
1000 = 100 ((w/r)L )0.5L0.5 = 100 L (4/2) 0.5
L = 7.07. Substituting the value of L in eq. (i), K = 14.14
b) When k = 4,
1000 = 100 L (4/4) 0.5 = 100 L (4/4) 0.5 ; L = 10 and K = 10
c) The firm responded to the higher price of capital by
substituting labour for capital
Numerical 3

Suppose that a firm’s production function is given by
Q = K² L. Further suppose that w = $10 and r = $20
a) Suppose the firm wants to produce 27,000 units of output.
What is the most efficient combination of labor and capital?
Solution:
MPL = K², MPK = 2KL
MRTS = MPL / MPK = K / 2L
K / 2L = 10 / 20...therefore K=L
To produce 27000 units
K²L = 27000....therefore K³ = 27000, so K=30, L=30
Numerical 3

Suppose that a firm’s production function is given by
Q = K² L. Further suppose that w = $10 and r = $20
b) Suppose that the firm wants to produce 27,000 units of
output in the most efficient way possible. How much does
the firm spend?
Solution:
Budget constraint is
wL + rK = (10)(30) + (20)(30) = $900
Numerical 3

Suppose that a firm’s production function is given by
Q = K² L. Further suppose that w = $10 and r = $20
c) Suppose that the firm wants to produce 27,000 units of
output and has exactly 10 units of capital in hand. In this
situation, how many labor has to be employed?
Solution:
27000 = 10²L, so L = 270 units of labour
Numerical 3

Suppose that a firm’s production function is given by
Q = K² L. Further suppose that w = $10 and r = $20
d)Suppose that the firm wants to spend exactly $1,200. What
is the most efficient combination of labor and capital ?
Solution:
wL + rK = 1200
We know that L=K, w=10, r=20, so
(10)(L) + (20)(L) = 1200, therefore L=40, K=40
Numerical 3

Suppose that a firm’s production function is given by
Q = K² L. Further suppose that w = $10 and r = $20
e) Suppose that the firm spends exactly $1200 in the most
efficient way possible. How much output can the firm
produce?
Solution:
Substitute K=L=40 into production function
(40)²(40) = 64000 units
Summary of Factor Market
Factor market
Derived demand – derived from a firm’s
decision to supply a good in another market.
 Production function- provides relationship
 MRPL/MRPK (VMPL/VMPK) curves demand curve for factor market: determine
additional labour or capital hired
 Profit maximizing or efficiency condition

◦ VMPL = P x MPL = MR (or MC) x MPL = W
◦ VMPK = P x MPK = MR (or MC) x MPK = r
MPL and VMPL Example
The VMPL curve
$6,000
5,000
4,000
3,000
$2,500
2,000
1,000
0
0
1
2
3
4
L (number of workers)
5
Supply Curve for factor market-eg. Labour
W
S1
W2
W1
L1 L2
L
Equilibrium in the Factor Market
W
S
W1
D
L1
L
Linkages Among the Factors of Production
In most cases, factors of production are
used together in a way that makes each
factor’s productivity dependent on the
quantities of the other factors.
 Example: an increase in the quantity of
capital

◦ The marginal product and rental price of
capital fall.
◦ Having more capital makes workers more
productive, MPL and W rise.
Thank you