Theory of the Firm
• Firms want to maximize profit
• This implies minimizing cost
• Need to identify underlying technological
relationships between inputs and outputs.
Factors of Production
Inputs
• Broadly – labor, land, raw materials,
capital
• Motion picture studio:
– producers, directors, actors, lots, sound
stages, equipment, film.
• Electricity Generator
– Managers, technicians, coal, generation
equipment, (clean air)
Specification of Technology –
one output
• Production function: f(x) = {y such that y is
the maximum output associated with –x}
– Cobb Douglas f(x1,x2) = x1ax2(1-a)
– Leontiff
f(x1,x2) = min(ax1,bx2)
– CES
f(x1,x2) = [ax1r + bx2r]1/r
– Linear
f(x1,x2) = x1 + x2
Sketch the Cobb Douglas, Leontiff, and Linear
Isoquants
Specification of Technology –
one output
5
Input 2
4
Q(y1)
3
Q(y2)
2
Q(y3)
1
0
0
0.5
1
Input 1
Cobb Douglas Isoquants
f(x1,x2) = x1ax2(1-a)
1.5
2
Specification of Technology –
one output
input 2
slope a/b
Q(y2)
Q(y1)
input 1
Leontiff isoquant
f(x1,x2) = min(ax1,bx2)
Technical Rate of Substitution
• The technical rate of substitution is the
amount that you need to adjust one input
in order to keep output constant for a small
change in another output. This is
equivalent to the slope of the isoquant.
If the production function is f(x1,x2) what is
the technical rate of substitution?
Elasticity of substitution
• The elasticity of substitution measures the
curvature of the isoquant:
s = d(x1/x2)/(x1/x2)
d(TRS)/TRS
The higher s, the less curved the isoquant
is, and easier it is to substitute between
inputs.
Elasticity of substitution
s=∞
Input 2
s = 1 (Cobb Douglas)
s=-∞
Input 1
Constant Elasticity of Substitution
f(x1,x2) = [ax1r + bx2r]1/r
s = 1/(1-r)
Example
a  -a
1 2
• Let f(x1,x2) = x x
• Find the TRS
• Then, try to find the elasticity of
substitution, using the ln formulation.
x y  ln x
=
y x  ln y
• Elasticity of substitution is a good measure
of flexibility.
Returns to Scale
• Constant returns to scale: a doubling of
inputs will result in a doubling of output.
– f(tx) = tf(x) the production function is
homogeneous of degree 1.
• Increasing returns to scale
– f(tx) > tf(x)
– example: fixed costs
• Decreasing returns to scale
– f(tx) < tf(x)
– example: a fixed input
Returns to Scale: Examples
• Bausch and Lomb. A 12-oz bottle of saline
solution costs $2.79. A 1-oz bottle of eye drops
costs $5.65.
• People’s Express. Very successful, attributed
partly to management practices (minimal
hierachy, training, profit sharing, performance
pay). Grew from 300 to 5000 employees. Active
involvement became difficult, more hierarch
necessary, output increased less than input.
• Oil shippers: unlimited liability in case of a spill.
Therefore a small firm with only one ship is
preferable.
Short run vs. Long run
• In the short run some inputs may be fixed.
• In the long run, we generally consider all
inputs to be variable.
• Example: Capacity
Profit Maximization
• Basic assumption of Economics. Is it
right?
• The firm takes actions that will maximize
the total revenues – costs.
Max R(a) – C(a)
F.O.C. R’(a) = C’(a)
MR = MC
Profit Maximization - examples
• The level of output should be chosen so
that the cost of producing the last unit of
output is equal to the revenue from that
unit.
• In 1962 Continental Airlines filled only 50%
of certain flights. It considered dropping
some of these flights, but each flight had a
(marginal) cost of $2000 and revenue of
$3100.
Profit Maximization
• Revenue = the price of what is sold X the
amount that is sold
• Cost = the price of the inputs X the amount
of input used.
• Technological Constraints
• Market Constraints
• Assume that the firm is a price-taker
(competitive firm)
Profit Maximization- price taker
• Max pf(x) –wx
• foc characterizes profit-maximizing
behavior.
• “value marginal product = its price”
• What is the requirement on f(x) for a
maximum to exist? What is the SOC?
Profit, supply, and demand
functions
• demand function
x(p,w) = argmax pf(x) – wx
• Supply function
y(p,w) = f(x(p,w))
• Profit Function
p(p,w) = py(p,w) – wx(p,w)
Profit, supply, and demand
functions
• A function is homogeneous of degree n if
f(tx)=tnx
• If a function is homogneous of degree 0,
then doubling all its inputs doesn’t change
the output.
• If a function is homogenous of degree 1,
then doubling all inputs doubles the
outputs.
Profit, supply, and demand
functions
• Consider the demand function x(p,w).
What happens if you double all prices?
• Consider the profit function, p(p,w). What
happens if you double all prices?
• What about the supply function?
• What does this say about behavior under
inflation?
Properties of the Profit Function
p(p,w) = py(p,w) – wx(p,w)
• Increasing in output prices; decreasing in
input prices
• Homogeneous of degree 1 in prices.
• Convex in p (!)
These properties follow only from the
assumption of profit maximization.
The profit function is convex
p(p,w) = Max pf(x) –wx
Prove it using the envelope theorem
Alternative proof that the profit
function is convex
Let y max profits at p, y’ at p’, and y’’ at p’’,
where p’’ = tp + (1-t)p’
Then
p(p’’) = p’’y’’ = (tp + (1-t)p’)y’’
= tpy’’ + (1-t)p’y’’ < tpy + (1-t)p’y’
= tp(p) + (1-t)p(p’)
Why is this true?
The profit function is convex
profits
p(p)
py*-w*x*
p(p*)
p*
output price
Profit function
Suppose the price of output is randomly
fluctuating. Is it desirable to stabilize this
price?
Jensen’s inequality:
p(E[p]) < E[p(p)] for any convex p.
Thus, firms do better when price fluctuates!
Because they change their production plan
accordingly.
When might this not be true?
Hotelling’s Lemma
• Use the envelope theorem to find
p
=
p
p
=
wi
Hotelling’s Lemma
• Use the envelope theorem to find
p
= y*
p
p
*
= - xi
wi
Example
• Consider a firm with 3 inputs – x, ec,enc
with prices w,pc,pnc
• Carbon emissions are equal to ec
• Carbon is taxed at level t
• Thus total price of ec is pc+t
• Now consider technical change that
reduces the carbon intensity.
• Let technical change a reduce the carbon
intesity from 1 to (1-a)
Example-continued
• After technical change the total price for ec
would be pc+(1-a)t
• The profit function is p(w, pc+(1-a)t,pnc)
max - g (a   p (w, pc  (1 - a t , pnc 
x
p
*
- g ' (a  = -t
= -tec
pc
Cost Minimization
• In order to maximize profit, a firm must be
minimizing the cost of producing its output.
• Cost minimization is an alternate
characterization of price-taking firms.
Max py – c(w,y) Where
c(w,y) = min wx
s.t. y = f(x)
• Cost minimization is equally valid for other
types of firms
Cost Minimization
Min wx
s.t. f(x) = y
Lagrangian
L(l,x) = wx – l(f(x) – y)
F.O.C.
wi-l df(x*)/dxi = 0
f(x*) = y
Cost Minimization
Divide the ith constraint by the jth
df(x*)/dxi = wi
df(x*)/dxj wj
Economic rate of substitution = TRS
Cost Minimization
Input 2
f(x1,x2) = y
Input 1
isocost line. slope = -w1/w2
Cost Minimization
Input 2
f(x1,x2) = y
Input 1
isocost line. slope = -w1/w2
How does this differ from the utility
maximization problem?
Rice Milling
Capital
80
70
60
50
40
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20
10
0
0
5
10
15
20
Labor
25
30
35
Rice Milling
Capital
80
70
60
50
40
30
20
10
0
0
5
10
15
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Labor
25
30
35
Rice Milling
Capital
80
70
60
50
40
30
20
10
0
0
5
10
15
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Labor
25
30
35
Capital
Rice Milling - high tech
80
70
60
50
40
30
20
10
0
0
5
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Labor
25
30
35
Capital
Rice Milling - high tech
80
70
60
50
40
30
20
10
0
0
5
10
15
20
Labor
25
30
35
Example
• Graph the demand for factor x1 as a
function of its price w1 for the 3 production
functions below.
x2
x2
x1
x2
x1
x1
Kuhn-Tucker Theorem
• In order to solve constrained optimization
problems when there might be a corner
solution, we need to use the Kuhn-Tucker
Theorem.
Cost Minimization
df(x*)/dxi = df(x*)/dxj
wi
wj
Marginal product per $
One dollar invested in xi increases output by
df(x*)/dxi
wi
To minimize cost you must equalize the
rates of return on each input.
Cost minimization - examples
• parking lots: tall versus expansive
Properties of the Cost Function
• Increasing or decreasing in w?
• Homogeneous of degree ? in w?
• Convex or concave?
Elasticity of substitution as a
measure of flexibilty
s=∞
Input 2
s = 1 (Cobb Douglas)
s=-∞
Input 1