Transcript Slide 1
Prerequisites
Almost essential
Firm: Optimisation
Frank Cowell: Microeconomics
Useful, but optional
Firm: Demand and Supply
October 2006
The Multi-Output Firm
MICROECONOMICS
Principles and Analysis
Frank Cowell
Introduction
Frank Cowell: Microeconomics
This presentation focuses on analysis of firm producing
more than one good
For the single-output firm, some things are obvious:
modelling issues
production function
profit maximisation
the direction of production
returns to scale
marginal products
But what of multi-product processes?
Some rethinking required...?
nature of inputs and outputs?
tradeoffs between outputs?
counterpart to cost function?
Overview...
The Multi-Output
Firm
Frank Cowell: Microeconomics
Net outputs
A fundamental
concept
Production
possibilities
Profit
maximisation
Multi-product firm: issues
Frank Cowell: Microeconomics
“Direction” of production
Ambiguity of some commodities
Need a more general notation
Is paper an input or an output?
Aggregation over processes
How do we add firm 1’s inputs and firm 2’s
outputs?
Net output
Frank Cowell: Microeconomics
Net output, written as qi,
Key concept
if positive denotes the amount of good i produced as
output
if negative denotes the amount of good i used up as
output
treat outputs and inputs symmetrically
offers a representation that is consistent
Provides consistency
in aggregation
in “direction” of production
We just need some
reinterpretation
Approaches to outputs and inputs
Frank Cowell: Microeconomics
NET
OUTPUTS
OUTPUT
INPUTS
q1
z1
q2
z2
...
...
qn-1
zm
qn
A standard “accounting” approach
An approach using “net outputs”
How the two are related
A simple sign convention
q
q1
–z1
q2
–z2
...
= ...
qn-1
–zm
qn
+q
Outputs:
Inputs:
+
net additions to the
stock of a good
reductions in the
stock of a good
Aggregation
Frank Cowell: Microeconomics
Consider an industry with two firms
How is total related to quantities for individual firms?
qi1 = 100, qi2 = 100
qi = 200
Example 2: both firms use i as input
Just add up
qi = qi1 + qi2
Example 1: both firms produce i as output
Let qif be net output for firm f of good i, f = 1,2
Let qi be net output for whole industry of good i
qi1 = − 100, qi2 = − 100
qi = − 200
Example 3: firm 1 produces i that is used by firm 2 as input
qi1 = 100, qi2 = − 100
qi = 0
Net output: summary
Frank Cowell: Microeconomics
Sign convention is common sense
If i is an output…
addition to overall supply of i
so sign is positive
If i is an inputs
net reduction in overall supply of i
so sign is negative
If i is a pure intermediate good
no change in overall supply of i
so assign it a zero in aggregate
Overview...
The Multi-Output
Firm
Frank Cowell: Microeconomics
Net outputs
A production
function with
many outputs,
many inputs…
Production
possibilities
Profit
maximisation
Rewriting the production function…
Frank Cowell: Microeconomics
Reconsider single-output firm example given earlier
Conventional way of writing feasibility condition:
qn f(−q1, −q2, ...., −qn-1 )
qn − f(−q1, −q2, ...., −qn-1 )
Rewrite this relationship as
q f(z1, z2, ...., zm )
where f is the production function
Express this in net-output notation and rearrange:
goods 1,…,m are inputs
good m+1 is output
n=m+1
F (q1, q2, ...., qn-1, qn ) 0
where F is the implicit production function
Properties of F are implied by those of f…
The production function F
Frank Cowell: Microeconomics
Recall equivalence for single output firm:
So, for this case:
qn − f(−q1, −q2, ...., −qn-1 )
F (q1, q2, ...., qn-1, qn ) 0
F is increasing in q1, q2, ...., qn
if f is homogeneous of degree 1, Fis homogeneous of
degree 0
if f is differentiable so is F
for any i, j = 1,2,…, n−1 MRTSij = Fj(q)/Fi(q)
It makes sense to generalise these…
The production function F (more)
Frank Cowell: Microeconomics
For a vector q of net outputs
For all feasible q:
q is feasible if F(q) 0
q is technically efficient if F(q) = 0
q is infeasible if F(q) > 0
F(q) is increasing in q1, q2, ...., qn
if there is CRTS then Fis homogeneous of degree 0
if f is differentiable so is F
for any two inputs i, j, MRTSij = Fj(q)/Fi(q)
for any two outputs i, j, the marginal rate of transformation of i into
j is MRTij = Fj(q)/Fi(q)
Illustrate the last concept using the transformation curve…
Firm’s transformation curve
Frank Cowell: Microeconomics
Goods 1 and 2 are outputs
Feasible outputs
q2
Technically efficient outputs
MRT at qo
q°
F(q) 0
F1(q°)/F2(q°)
F(q)=0
q1
An example with five goods
Frank Cowell: Microeconomics
Goods 1 and 2 are outputs
Goods 3, 4, 5 are inputs
A linear technology
fixed proportions of each input needed for the production of each
output:
q1 a1i + q2 a2i −qi
where aji is a constant i = 3,4,5, j = 1,2
given the sign convention −qi > 0
Take the case where inputs are fixed at some arbitrary
values…
The three input constraints
Frank Cowell: Microeconomics
q1
points satisfying
q1a13 + q2a23 −q3
Draw the feasible set for the
two outputs:
input Constraint 3
Add Constraint 4
Add Constraint 5
points satisfying
q1a14 + q2a24 −q4
Intersection
is the feasible
set for the two
outputs
points satisfying
q1a15 + q2a25 −q5
q2
The resulting feasible set
Frank Cowell: Microeconomics
q1
The transformation
curve
how this responds
to changes in
available inputs
q2
Changing quantities of inputs
Frank Cowell: Microeconomics
q1
points satisfying
q1a13 + q2a23 −q3
The feasible set for the two
consumption goods as before:
Suppose there were more of
input 3
Suppose there were less of
input 4
points satisfying
q1a13 + q2a23 −q3 −dq3
points satisfying
q1a14 + q2a24 −q4 + dq4
q2
Overview...
The Multi-Output
Firm
Frank Cowell: Microeconomics
Net outputs
Integrated
approach to
optimisation
Production
possibilities
Profit
maximisation
Profits
Frank Cowell: Microeconomics
The basic concept is (of course) the same
But we use the concept of net output
Revenue Costs
this simplifies the expression
exploits symmetry of inputs and outputs
Consider an “accounting” presentation…
Accounting with net outputs
Frank Cowell: Microeconomics
Suppose goods 1,...,m are inputs
and goods m+1 to n are outputs
n
i=m+1
pi qi
Revenue
m
pi [ qi]
– Costs
i=1
n
pi qi
i=1
= Profits
Cost of inputs (goods 1,...,m)
Revenue from outputs (goods
m+1,...,n)
Subtract cost from revenue to
get profits
Iso-profit lines...
Frank Cowell: Microeconomics
Net-output vectors yielding a
given P0.
Iso-profit lines for higher profit
levels.
q2
p1q1+ p2q2 = constant
p1q1+ p2q2 = P
use this to represent
profit-maximisation
q1`
Frank Cowell: Microeconomics
Profit maximisation: multiproduct firm (1)
Feasible outputs
q2
Isoprofit line
Maximise profits
Profit-maximising output
MRTS at profit-maximising
output
Here q1*>0
and q2*>0
*
q
q* is
technically
efficient
q1`
Slope at q*
equals price
ratio
Frank Cowell: Microeconomics
Profit maximisation: multiproduct firm (2)
Feasible outputs
q2
Isoprofit line
Maximise profits
Profit-maximising output
MRTS at profit-maximising
output
Here q1*>0
but q2* = 0
q* is
technically
efficient
q*
q1`
Slope at q* ≤
price ratio
Maximising profits
Frank Cowell: Microeconomics
Problem is to choose q so as to maximise
n
pi qi
subject to F(q) ≤ 0
i=1
Lagrangean is
n
pi qi
lF(q)
i=1
FOC for an interior maximum is
pi lFi(q) = 0
Maximised profits
Frank Cowell: Microeconomics
Introduce the profit function
the solution function for the profit maximisation problem
P(p) = max
n
pi qi
{F(q) ≤ 0} i = 1
= pi qi*
i=1
Works like other solution functions:
n
non-decreasing
homogeneous of degree 1
continuous
convex
Take derivative with respect to pi :
Pi(p) = qi*
write qi* as net supply function
qi* = qi(p)
Summary
Frank Cowell: Microeconomics
Three key concepts
Net output
Transformation curve
simplifies analysis
key to modelling multi-output firm
easy to rewrite production function in terms of net
outputs
summarises tradeoffs between outputs
Profit function
counterpart of cost function