MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Firm Optimisation Useful but optional Firm Demand and Supply Prerequisites July 2015 1 Note the detail in slides marked can only be seen if you run the slideshow ID: 284192
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Slide1
The Multi-Output Firm
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Firm: OptimisationUseful, but optionalFirm: Demand and Supply
Prerequisites
April 2018
1Slide2
Introduction
This presentation focuses on analysis of firm producing more than one goodmodelling issuesproduction function
profit maximisationFor the single-output firm, some things are obvious: the direction of productionreturns to scalemarginal productsBut what of multi-product processes?Some rethinking required...?
nature of inputs and outputs?tradeoffs between outputs?counterpart to cost function?
April 2018
2Slide3
Profit maximisation
Overview
Net outputs
Production possibilities
The Multi-Output Firm
A fundamental concept
April 2018
3Slide4
Multi-product firm: issues
“Direction” of productionNeed a more general notationAmbiguity of some commoditiesIs paper an input or an output?Aggregation over processesHow do we add firm 1’s inputs and firm 2’s outputs?
April 2018
4Slide5
Net output
Net output, written as qi if positive denotes the amount of good i produced as output
if negative denotes the amount of good i used up as outputKey concepttreat outputs and inputs symmetricallyoffers a representation that is consistentProvides consistencyin aggregation
in “direction” of productionApril 2018
5
We just need some reinterpretationSlide6
Approaches to outputs and inputs
–z
1
–z
2
...
–z
m
+
q
=
q
1
q
2
...
q
n
-1
q
n
OUTPUT
q
INPUTS
z
1
z
2
...
z
m
NET OUTPUTS
q
1
q
2
q
n
-1
q
n
...
Right-hand boxes:
standard “accounting” approach
Left-hand boxes: approach
using “net outputs”
Bottom-left: how
the two are related
Outputs:
+
net additions to the
stock of a good
Inputs:
reductions in the
stock of a good
Bottom-right:
simple sign convention
April 2018
6Slide7
Aggregation
Consider an industry with two firmsLet qif
be net output for firm f of good i, f = 1,2Let qi be net output for whole industry of good i
How is total related to quantities for individual firms?Just add upqi = qi1
+ qi2 Example 1: both firms produce
i
as output
q
i
1
= 100,
q
i
2
= 100
q
i
= 200Example 2: both firms use i as input
qi1 = − 100, q
i2 = − 100 q
i = − 200Example 3: firm 1 produces i that is used by firm 2 as input
qi1 = 100, qi
2 = − 100 q
i
= 0
April 2018
7Slide8
Net output: summary
Sign convention is common senseIf i is an outputaddition to overall supply of i
so sign is positiveIf i is an inputnet reduction in overall supply of i so sign is negativeIf i is a pure intermediate good
no change in overall supply of i so assign it a zero in aggregateApril 2018
8Slide9
Profit maximisation
Overview
Net outputs
Production possibilities
The Multi-Output Firm
A production function with many outputs, many inputs…
April 2018
9Slide10
Rewriting the production function…
Reconsider single-output firm example given earliergoods 1,…,m are inputs
good m+1 is outputn = m + 1Conventional way of writing feasibility condition:q £
f (z1, z2, ...., zm )
where f is the production functionExpress this in net-output notation and rearrange:
q
n
£
f
(−
q
1
,
−
q
2
, ....,
−
qn-1 )q
n − f (−q1
, −q2, ...., −qn
-1 ) £ 0Rewrite this relationship as
F (q1, q
2, ...., qn-1, q
n
)
£
0
where
F
is the implicit production function
Properties of
F
are implied by those of
f
April 2018
10Slide11
The production function F
Recall equivalence for single output firm: q
n − f (−q1, −q2, ...., −qn
-1 ) £ 0F (q1
, q2, ...., qn-1, q
n
)
£
0
So, for this case:
F
is increasing in
q
1
, q
2
, ....,
qnif
f is homogeneous of degree 1, F is homogeneous of degree 0if f
is differentiable so is Ffor any i
, j = 1,2,…, n − 1, MRTSij =
Fj(q)/Fi(
q)It makes sense to generalise these…
April 2018
11Slide12
The production function
F (more)For a vector q of net outputs
q is feasible if F(q) £ 0q is technically efficient if F(
q) = 0q is infeasible if F(q)
> 0For all feasible q: F
(
q
) is increasing in
q
1
, q
2
, ....,
q
n
if there is CRTS then
F
is homogeneous of degree 0
if
f is differentiable so is Ffor inputs
i, j, MRTSij = Fj
(q)/Fi(q)
for outputs i, j, marginal rate of transformation of i into
j is MRTij = Fj(q
)/Fi(q)Illustrate the last concept using the
transformation curve
April 2018
12Slide13
Firm’s transformation curve
q
2
q
1
F
1
(
q
°
)/
F
2
(
q
°
)
q
°
F
(
q
)
=
0
F
(
q
)
0
Goods 1 and 2 are outputs
Green area: feasible
outputs
Orange boundary: technically
efficient outputs
Pink line:
MRT
at
q
o
April 2018
13Slide14
An example with five goods
Goods 1 and 2 are outputsGoods 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
April 2018
14Slide15
The three input constraints
q
2
q
1
blue:
Constraint
3
purple
:
Constraint 5
brown:
Constraint 4
points satisfying
q
1
a
13
+ q
2
a
23
£
−q
3
points satisfying
q
1
a
14
+ q
2
a
24
£
−q
4
points satisfying
q
1
a
15
+ q
2
a
25
£
−q
5
Intersection is
the feasible set for the two outputs
April 2018
15Slide16
The resulting feasible set
q
2q1
how this responds to changes in available inputs
The transformation curve
April 2018
16Slide17
*Changing
quantities of inputs
q
2
q
1
q
1
a
13
+ q
2
a
23
=
−q
3
Green area:
feasible set
as
before
Shift blue line: more
of input 3
Shift brown line:
less of input 4
q
1
a
14
+ q
2
a
24
=
−
q
4
+
d
q
4
q
1
a
13
+ q
2
a
23
=
−q
3
−
d
q
3
April 2018
17Slide18
Profit maximisation
Overview
Net outputs
Production possibilities
The Multi-Output Firm
Integrated approach to optimisation
April 2018
18Slide19
Profits
The basic concept is (of course) the sameRevenue CostsBut we use the concept of net output
this simplifies the expressionexploits symmetry of inputs and outputsConsider an “accounting” presentationApril 2018
19Slide20
Accounting with net outputs
Costs
Suppose goods 1,...,m are inputs and goods
m+1 to n are outputs
Revenue
n
å
p
i
q
i
i
=m
+1
= Profits
m
å
p
i
[
q
i
]
i =
1
n
å
p
i
q
i
i =
1
–
2
nd
line: Cost
of inputs
(
1, ..., m)
1st line: Revenue from outputs ( m+1, ..., n
) Subtract cost from revenue to get profits
April 201820Slide21
Iso-profit lines...
q
2
q1`
increasing
profit
pink-line: net-output
vectors
for
a given
P
0
higher lines
for higher profit levels.
p
1
q
1
+ p
2
q
2
=
P
0
p
1
q
1
+ p
2
q
2
=
constant
use this to represent profit-maximisation
April 2018
21Slide22
Profit maximisation: multi-product firm (1)
q
2
q
1`
q
*
Green area: Feasible
outputs
pink
line:
Isoprofit
increasing
profit
move
isoprofit
out to max profit
q*
: Profit-maximising
output
q
*
is technically efficient
slope of
isoprofit
: MRTS
Slope at
q
*
equals price ratio
Here
q
1
*
> 0
and
q
2
*
> 0
April 2018
22Slide23
Profit maximisation: multi-product firm (2)
q
2
q
1`
q
*
q
*
is technically efficient
Slope at
q*
≤
price ratio
increasing
profit
Here
q
1
*
> 0
but
q
2
*
= 0
April 2018
23
Green area: Feasible
outputs
pink
line:
Isoprofit
q*
: Profit-maximising
output
slope of
isoprofit
: MRTS
move
isoprofit
out to max profitSlide24
Maximising profits
FOC for an interior maximum is pi
lFi(q) = 0
n å
pi q
i
lF
(
q
)
i =
1
n
å
p
i
q
i
subject to
F
(
q
) ≤ 0
i =
1
Problem is to choose
q
so as to maximise
Lagrangean is
April 2018
24Slide25
Maximised profits
Introduce the
profit function the solution function for the profit maximisation problem n
n P(
p) = max å p
i
q
i
=
å
p
i
q
i
*
{
F
(
q
) ≤ 0}
i
=
1
i = 1
Works like other solution functions:
non-decreasing
homogeneous of degree 1
continuous
convex
Take derivative with respect to
p
i
:
P
i
(
p
) =
q
i
*
write
q
i
*
as net supply function
qi* = q
i(p)
April 2018
25Slide26
Summary
Three key concepts Net outputsimplifies analysiskey to modelling multi-output firm
easy to rewrite production function in terms of net outputsTransformation curve summarises tradeoffs between outputsProfit functioncounterpart of cost functionApril 2018
26