and Technology Profit
Maximization
Instructor: Dr. Arghya Ghosh
([e-mail])
Office Hours: Wed, 1pm 4pm, ASB 406
Weeks: 5 8, 11-12
Theme: Production, Firms, Market Structure
Chapter 18: Technology
Chapter 19: Profit Maximization
Technology
Technology
Technology describes how inputs are converted
output.
Production Functions
Isoquants
Marginal Product
Returns to Scale
Technical rate of substitution
Long run and short run
Lecture is being produced by using computer,
projector, software, labour
PRODUCTION FUNCTION: The maximum
amount of output (y) one can produce using
inputs x1, x2 ,
y = f(x1, x2 , )
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One input, one output
y = f(x) is the production
function.
y = f(x)
y = f(x) is the maximal
output level obtainable
from x input units.
x
Input Level
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Production Function with more than
one input
Production Function
y
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Labour and Capital used for
manufacturing production
Labour and Land for agricultural
production
How are different factors/inputs
combined to produce output?
ISOQUANTS : combination of
different inputs that produce same
level of output
x
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Fixed Proportion Production Fn
Fixed Proportion Isoquants
Y = Digging holes , X1 = labourers, X2 =
shovels
Y = min (X1, X2) ,
X1 = 1, X2 = 1; Y = min (1,1) = 1
X1 = 2, X2 = 1; Y = min (2,1) = 1
X1 = 1, X2 = 2; Y = min (1,2) = 1
(1,1), (1,2), (2,1) . all these combinations
produce 1 unit of output.
Two inputs need to be used in fixed
proportion. Adding more of one without
having more of the other does not help
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Perfect Substitutes
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Perfect Substitutes Isoquants
Y = X1 + X2 ,
X1 = 1, X2 = 1;
Y =2
Y =2
X1 = 0, X2 = 2;
Y =2
X1 = 2, X2 = 0;
(1,1), (0,2), (2,0) . all these
combinations produce 2 units of
output.
Two inputs are perfect substitutes
Grading multiple choice exams:
labour and capital (machine) are
perfect substitutes
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Cobb-Douglas Production Function
b
y = Ax1 x2
a
Let a = , b = and A =1. Find input
combinations such that y = 2.
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Cobb-Douglas Isoquants
Different combinations of
(x1, x2) that yield y = 2
x2
x'2
1
2 = x1 x2 / 2
1/ 2
x1 = 1, x2 = 4;
x1 = 4, x2 = 1;
x1 = 2, x2 = 2;
Note if x1 or x2 = 0, y = 0. Thus both
inputs are essential
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x"
2
y= 2
x'
1
x"
1
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2
Marginal Product
Computing Marginal Product (MP)
The marginal product of any input i is the rate of change
of the output level as the level of input i changes.
1
f (L, K) = L / 3K2/ 3
Production function: y = f(L,K)
y
Marginal product of labour (MPL) is L =
Marginal product of capital (MPK) is
f ( L, K )
L
y f ( L, K )
=
K
K
Properties (that typically hold)
Marginal product of L depends on K and similarly
marginal product of K depends on L
Diminishing Marginal product --- MPL declines as L
increases. MPK declines as K increases
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Returns to Scale
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Returns to scale: an example
Consider
Note, for Cobb-Douglas technology
MPL increases as K increases
MPL diminishes as L increases
2
MP f (L, K ) 2 f ( L, K )
L
=
= L5/ 3 K 2/ 3 < 0
=
L2
9
L L L
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Returns to Scale (contd.)
Marginal product describe the change
in output level as a single input level
changes
Returns-to-scale describes how the
output level changes as all input
levels change in direct proportion
(e.g. all input levels or
halved).
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f (L, K) 1 2 / 3 2 / 3
= L K >0
L
3
MPL is
f ( L, K ) = La K b
Double L and K. Then
Consider the production function
y = F(L,K)
Increase t fold where t > 1
Constant returns to scale (CRS) if
F(tL, tK) = tF(L,K)
Increasing returns to scale (IRS) if
F(tL, tK) > tF(L,K)
Decreasing returns to scale (DRS) if
F(tL, tK) < tF(L,K)
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Technical Rate of Substitution
At what rate can a firm substitute one
input for another without changing its
output level
F(2L,2K)= ( 2L)a( 2K)b = 2a+b La K b = 2a+b F(L, K)
Constant returns to scale if a + b = 1
Increasing returns to scale if a + b > 1
Decreasing returns to scale if a + b <1
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Technical rate of substitution(contd.)
To compute TRS output level must remain
unchanged. Thus dy = 0
This implies
Technical rate of substitution
The slope is the rate at which
input 2 must be given up as
input 1s level is increased so as
not to change the output level.
The slope of an isoquant is its
technical rate-of-substitution.
x2
f ( L, K )
f ( L, K )
dL +
dK = 0
L
K
Rearranging we get the TRS
TRS =
f ( L, K )
dK
MPL
= L
=
f ( L, K )
dL
MPK
K
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Technical rate of substitution
Compute TRS at K = 1, L = 4 for a CobbDouglas production function
f(L,K) L / 3K 2 / 3
=1
MPL =
MPK =
1
f(L,K)
= L 2 / 3 K
3
L
2
f(L,K)
= L1 / 3 K
3
K
2/3
1 / 3
TRS = - (MPL/MPK) = - (K/2L); At L = 4, K
= 2, the TRS is 1/8
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Long run and the short run (contd.)
Suppose y = f(x1, x2)
Consider a short run where
x1 is variable
x2 is fixed.
Note, depending on the level of x2,
the short run production function
changes.
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x'2
y
x'
1
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x1
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Long Run and the Short run
The long-run is the circumstance in
which a firm is unrestricted in its
choice of all input levels.
A short-run is a circumstance in
which a firm is restricted in some way
in its choice of at least one input
level.
In a manufacturing firm that employs
capital and labour, capital (K) might
be fixed in the short run but variable
in the long run
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Short run production functions
y = f ( x1 , x2 ) = x1/ 3 x2 / 3
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Profit maximization
Preliminaries
Short run profit maximization
Long run profit maximization
Factor demand curves
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Short run profit-maximization
Suppose x2 is fixed in the short run
What is the level of x1 that maximize
profits?
Example: A manufacturing firm uses
labour (L) and capital (K) to produce
its own output. Suppose capital is
fixed in the short run. How much
labour should it employ?
To answer that first we need to find
out the benefit and cost from
employing an extra unit of labour.
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Preliminaries
Output level y = f(x1,x2)
Product price p
Revenue = py = pf(x1,x2)
Inputs: x1,x2.
Input prices: w1,w2.
Costs = w1x1 + w2x2
Profits = Revenue Costs
= pf(x1,x2) - w1x1 - w2x2
Taking p, w1 and w2 as given each
firm chooses x1,x2.to maximize
Profits
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Thinking clearly
Production function
y = f(L,K)
where K is fixed in the short run.
As a owner you are contemplating
whether to employ a bit more L. Let
w denotes the wage rate.
Increase in output: dy = (MPL) X dL
Increase in revenue:
pdy = p X MPL X dL
Increase in cost: w X dL
Optimum: p X MPL = w
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Next
Profit maximization (contd.)
Pages: 339 345, Chapter 19,
Varian
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