Part IIA, Paper 1 Consumer and Producer Theory

1
Part IIA, Paper 1Consumer and Producer Theory

• Lecture 8
• Profit Maximisation and Supply Functions
• Flavio Toxvaerd

2
Recap and Todays Outline

• Last lecture looked at problem of cost
  minimisation, and obtained Cost function -
  C(p,y).
• We maintained throughout the analysis that factor
  prices were unaffected by the firms employment
  decisions that is, the firm was a price taker
  in the input markets
• Today we look specifically at the problem of
  profit maximisation maintaining the assumption
  that the firm is a price taker in the factor
  markets
• Note the mathematical requirements for unique
  solutions become more complicated if the
  assumption is dropped but the basic properties
  of the cost function remain unchanged

3
Profit Maximisation
Two ways to represent profit maximisation problem
1.
with FOC
MC curve cuts MR curve from below
and SOC
2.
4
Profit Maximisation
Simplifying Assumption Firm is a price taker
in the output market thus p(y)py
Thus maximisation problem 1 becomes
and solution depends on both input and output
prices
This is the supply function for the firm
As solution obtained from FOC
have that supply function is the inverse of the
marginal cost curve
5
Profit Maximisation
Formulated the second way we have
MRMC
First order conditions
These are the same conditions as those from cost
minimisation
6
Profit Maximisation
Once again we have that
y
Graphically have isoprofit line tangental to
production function
PPS
x
7
Supply and Factor Demand Functions
Solution to maximisation problem gives
Supply Function
Factor Demand Functions
8
Properties of Supply and Factor Demand Functions

• Homogeneous degree zero.

Proof Increasing all prices by the same
proportions leaves the maximisation problem
unchanged.

• Output is increasing and factor demand decreasing
  in own price

Proof Follows from second order conditions of
maximisation prob.
9
Profit Function
Evaluating the maximisation problem at the
optimal solution gives the Profit Function
From Envelope Theorem
10
Properties of Profit Function

• Homogeneous of degree 1 in prices

Proof No change in relative prices, so no change
in optimal solutions - so profit changes by same
proportion.

• Non-decreasing in output price and non-increasing
  in input prices

Proof Envelope Theorem.

• Convex in prices

Proof If prices change, profits will increase
linearly if inputs and outputs are unchanged.
Thus any change must be to increase profits - and
so profits increases more than linearly.
11
Recovering Production Function from Profit
Function
y
y0
Generates concave production function
yF(x)
y1
x0
x1
12
Short Run Profit Function
We may want to consider the profit function when
some inputs are not variable
Clearly
with equality if and only if
13
LeChatelier Principle
The price elasticity of supply is greater in the
long run than in the short run. Alternatively,
thus the slope of the Supply Function is lower
in the long run.
Proof Let
be the optimal solutions when prices are
14
LeChatelier Principle
We know that h(py) is minimised at py
15
Alternative Specification
Throughout we have kept the vectors of inputs and
output separate. However we could also consider
inputs to be nothing more than negative outputs
and we could consider a vector of net outputs
from a production process z y - x
Profit maximisation becomes maxz p.z subject to
z ? Z , where Z is the set of feasible (net)
production plans.
16
Graphically
Isoprofit line
y
PPS
x
17
General Equilibrium
We have developed methods to generate demand
functions for individuals and supply functions
for firms GIVEN a specific vector of prices. We
have made no effort to see whether or not supply
and demand are equal at those prices if the
prices are EQUILIBRIUM prices.
18
Readings

• Varian, Intermediate Economics, chapter 19
• Varian, Microeconomic Analysis, chapters 2,3