MICROECONOMICS Principles and Analysis Frank Cowell Almost essential Consumer Optimisation Useful but optional Firm Optimisation Prerequisites July 2015 1 Note the detail in slides marked can only be seen if you run the slideshow ID: 135807
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Slide1
A Simple Economy
MICROECONOMICSPrinciples and Analysis Frank Cowell
Almost essential Consumer: OptimisationUseful, but optionalFirm: Optimisation
Prerequisites
April 2018
1Slide2
The setting
A closed economyPrices determined internallyA collection of natural resourcesDetermines incomesA variety of techniques of productionAlso determines incomesA single economic agent
Robinson Crusoe
Needs new notation, new concepts..
April 2018
2Slide3
Notation and concepts
x = (
x1, x2, ..., xn)consumption
q = (q1, q
2, ..., qn
)
net outputs
R
= (
R
1
, R
2
, ..., R
n
)
resources
just the same
as before
more on this soon...
available for consumption or production
April 2018
3Slide4
Overview
Structure of production
The Robinson Crusoe problem
Decentralisation
Markets and trade
A Simple Economy
Production in a multi-output, multi-process world
April 2018
4Slide5
Net output clears up problems
“Direction” of productionGet a more general notationAmbiguity of some commoditiesIs paper an input or an output?Aggregation over processesHow do we add my inputs and your outputs?We need to reinterpret production concepts
April 20185Slide6
Approaches to outputs and inputs
–
z
1–z2
...
–
z
m
+
q
=
q
1
q
2
...
q
n
-1
q
n
INPUTS
z
1
z
2
...
z
m
NET OUTPUTS
q
1
q
2
q
n
-1
q
n
...
A standard “accounting” approach
An approach using “net outputs”
How the two are related
Outputs:
+
net additions to the
stock of a good
Inputs:
reductions in the
stock of a good
Intermediate goods:
0
your output and my input cancel each other out
A simple sign convention
April 2018
6
OUTPUTS
qSlide7
Multistage production
1000
sausages
10 hrs labour 20 pigs
22
pigs
90 potatoes
10 hrs labour
2 pigs
1000
potatoes
1 unit land
10 potatoes
10
hrs
labour
Process 2 is for a pure intermediate good
Add process 3 to get 3 interrelated processes
1
Process 1 produces a consumption good / input
2
3
April 2018
7Slide8
Combining the three processes
sausages
potatoespigslabourland
0
+990
0
–
10
–
1
Process 1
+
0
–
90
20
–
10
0
Process 2
+
+1000
0
–
20
–
10
0
Process 3
=
+1000
+900
0
–
30
–
1
Economy’s net output vector
22 pigs
1000 potatoes
1000 sausages
30 hrs labour
1 unit land
22 pigs
100 potatoes
April 2018
8Slide9
More about the potato-pig-sausage story
We have described just one techniqueWhat if more were available?What would be the concept of the isoquant?What would be the marginal product?What would be the trade-off between outputs?
April 20189Slide10
An axiomatic approach
Let Q be the set of all technically feasible net output vectorsThe technology set “q
Î Q” means “q is technologically do-able”The shape of Q describes the nature of production possibilities in the economyWe build it up using some standard production axiomsApril 2018
10Slide11
Standard production axioms
Possibility of Inaction0 Q
No Free LunchQ +n = {0}IrreversibilityQ (– Q ) = {0
}Free DisposalIf q Q and
q' q
then
q
'
Q
Additivity
If
q
Q and q' Q then q + q
' QDivisibilityIf
q Q and 0 t 1 then t
q Q
A graphical interpretation
April 2018
11Slide12
All these points
The technology set
Q
pigs
labour
sausages
q
°
q
1
q
4
q
3
½
q
°
2
q
'
q
'+
q
"
0
q
"
q
'
Possibility of inaction
“No free lunch”
No points here!
Irreversibility
No points here!
Free disposal
Additivity
Some basic techniques
Divisibility
•
Implications of additivity + divisibility
Additivity+Divisibility
imply constant returns to scale
In this case
Q
is a cone
Let’s derive some familiar production concepts
April 2018
12
* detail on slide can only be seen if you run the slideshowSlide13
A “horizontal” slice through Q
pigs
labour
sausages
q
1
q
4
q
3
0
Q
April 2018
13Slide14
... to get the tradeoff
in inputs
(500 sausages)
(750 sausages)
labour
q
4
q
3
pigs
April 2018
14Slide15
…flip these to give isoquants
(500 sausages)
(750 sausages)
labour
q
4
q
3
pigs
April 2018
15Slide16
A “vertical” slice through Q
pigs
labour
sausages
q
1
q
4
q
3
0
Q
April 2018
16Slide17
The pig-sausage relationship
sausages
pigs
(with 10 units of labour)
q
1
q
3
(with 5 units of labour)
April 2018
17Slide18
(sausages)
(potatoes)
high input
q
2
q
1
The potato-sausage tradeoff
Again
take slices through
Q
Heavy shade:
low level of inputs
low input
Light
shade:
level of inputs
Rays illustrate CRTS
April 2018
18Slide19
Now rework a concept we know
In earlier presentations we used a simple production functionA way of characterising technological feasibility in the 1-output caseNow we have defined technological feasibility in the many-input many-output caseusing the set Q
Use this to define a production function for the general multi-output versionit includes the single-output case that we studied earlier
…let’s see.
April 2018
19Slide20
Technology set and production function
The technology set Q and the production function F are two ways of representing the same relationship:q
Î Q Û F(q) £ 0Properties of F inherited from properties of Q F(q1, q2,…, qn) is nondecreasing in each net output
qiIf Q is a convex set then F is a concave functionAs a convention F(q)
= 0 for efficiency, so...F(q) £ 0 for feasibility
April 2018
20Slide21
q
2
q1The set Q and the production function F
A view of set
Q
: production possibilities of two outputs
If frontier is smooth (many basic techniques)
Feasible but inefficient points in the interior
F
(
q
) < 0
Feasible and efficient points on the boundary
F
(
q
) = 0
Infeasible points outside the boundary
F
(
q
) > 0
April 2018
21
Boundary is the
transformation curve
Slope:
marginal rate of transformation
MRT
ij
:=
F
j
(
q
) /
F
i
(
q
) Slide22
How the transformation curve is derived
Do this for given stocks of resourcesPosition of transformation curve depends on technology and resourcesChanging resources changes production possibilities of consumption goods
April 201822Slide23
Overview
Structure of production
The Robinson Crusoe problem
Decentralisation
Markets and trade
A Simple Economy
A simultaneous production-and-consumption problem
April 2018
23Slide24
Setting
A single isolated economic agentNo marketNo trade (as yet)Owns all the resource stocks R on an islandActs as a rational consumerGiven utility function
Also acts as a producer of some of the consumption goodsGiven production functionApril 201824Slide25
The Crusoe problem (1)
max
U(x) by choosing x and q subject to:
a joint consumption and production decision
x Î
X
logically feasible consumption
F
(
q
)
£
0
technical feasibility:
equivalent to
“
q
Î Q ”
x £ q + R
materials balance:
can’t consume more than is available from net output + resources
The facts of life
April 2018
25Slide26
Crusoe’s problem and solution
x
2
0
April 2018
26
x
1
Attainable set with
R
1
= R
2
= 0
Positive stock of resource
1
More of resources
3,…
,n
Crusoe’s preferences
Attainable set derived from technology and materials balance condition
increasing
preference
The FOC
MRS = MRT
:
U
1
(
x
)
F
1
(
q
)
—— = ——
U
2
(
x
)
F
2
(
q
)
x
*
The optimum
* detail on slide can only be seen if you run the slideshowSlide27
The nature of the solution
From the FOC it seems as though we have two parts:A standard consumer optimum Something that looks like a firm's optimumCan these two parts be “separated out”?
A story:Imagine that Crusoe does some accountancy in his spare timeIf there were someone else on the island he would delegate production……then use the proceeds from production to maximise his utilityTo investigate this possibility we must look at the nature of income and profitsApril 2018
27Slide28
Overview
Structure of production
The Robinson Crusoe problem
Decentralisation
Markets and trade
A Simple Economy
The role of prices in separating consumption and production decision-making
April 2018
28Slide29
The nature of income and profits
The island is a closed and the single economic actor (Crusoe) has property rights over everythingProperty rights consist of “implicit income” from resources R the surplus (profit) of the production processes
We could usethe endogenous income model of the consumerthe definition of profits of the firmBut there is no market therefore there are no prices we may have to “invent” the pricesExamine the application to profitsApril 201829Slide30
Profits and income at shadow prices
We know that there is no system of pricesInvent some “shadow prices” for accounting purposesUse these to value national income
r1 r
2 ... r
n
profits
r
1
q
1
+
r
2
q
2
+...+
r
nqn
value ofresource stocks
r1R1 + r
2R
2
+...+
r
n
R
n
value of
national income
r
1
[
q
1
+R
1
]
+...+
r
n
[
q
n
+R
n
]
April 2018
30Slide31
National income contours
r
1
[
q
1
+ R
1
]
+
r
2
[
q
2
+ R
2
] =
const
contours for progressively higher values of national income q
1+R1
q
2
+R
2
April 2018
31Slide32
“National income” of the Island
Attainable set
x
2
x
1
0
Using shadow prices
r
we’ve broken down the Crusoe problem into a two-step process
:
Profit maximisation
Utility maximisation
The Island’s “budget set”
Use this to maximise utility
Iso-profit
–
income maximisaton
F
1
(
x
)
r
1
—— = —
F
2
(
x
)
r
2
U
1
(
x
)
r
1
—— = —
U
2
(
x
)
r
2
x
*
April 2018
32Slide33
A separation result
By using “shadow prices” r :a global maximisation problemis separated into sub-problems:
An income-maximisation problemA utility maximisation problemMaximised income from 1 is used in problem 2
max
U
(
x
) subject to
x
£
q + R
F
(
q)
£ 0
max
U(x) subject to
n
S
r
i
x
i
£
y
i
=1
n
max
S
r
i
[
q
i
+
R
i
]
subj. to
i
=1
F
(
q
)
£
0
April 2018
33Slide34
The separation result
The result is central to microeconomicsalthough presented in a very simple model generalises to complex economies But it raises an important question:
can this trick always be done?depends on the structure of the components of the problem To see this let’s rework the Crusoe storyvisualise it as a simultaneous value-maximisation and value minimisationthen try to spot why the separation result worksApril 2018
34Slide35
Crusoe problem: another view
The attainable set
x
2
x
1
0
A
= {
x
:
x
q
+
R
,
F
(
q
)
0
}
The price line
The “Better-than-
x
*
” set
A
B
r
1
r
2
B
= {
x
:
U
(
x)
U
(
x
*
)
}
Decentralisation
x
*
maximises income over
A
x
*
minimises expenditure over
B
x
*
April 2018
35
Here “Better-than-
x*
” is used as shorthand for
“Better-than-or-just-as-good-as-
x*
”
* detail on slide can only be seen if you run the slideshowSlide36
The role of convexity
The “separating hyperplane” theorem is useful hereGiven two convex sets A and B in Rn with no points in common, you can pass a hyperplane between
A and BIn R2 a hyperplane is just a straight lineIn our application:A is the Attainable set Derived from production possibilities + resourcesConvexity depends on divisibility of productionB is the “Better-than” set Derived from preference mapConvexity depends on whether people prefer mixturesThe hyperplane is the price system
Let's look at another case
April 2018
36Slide37
Optimum cannot be decentralised
x
2 x1
consumer optimum here
profit maximi-sation here
A nonconvex attainable set
The consumer optimum
Maximise profits at these prices
Implied prices: MRT=MRS
Production
responses do not support the consumer optimum
In this case the price system “fails”
A
x
*
April 2018
37
x
°Slide38
Overview
Structure of production
The Robinson Crusoe problem
Decentralisation
Markets and trade
A Simple Economy
How the market simplifies the model
April 2018
38Slide39
Introducing the market
Now suppose that Crusoe has contact with the worldnot restricted to “home production”can buy/sell at world pricesThis development expands the range of choice
It also modifies the separation argument in an interesting way
Think again about the attainable set
April 2018
39Slide40
Crusoe's island trades
x
2
x
1
x**
x
1
**
q
1
**
x
2
**
q
2
**
Equilibrium on the island
The possibility of trade
Max national income at world prices
Trade enlarges the attainable set
Equilibrium with trade
x
*
is the
Autarkic equilibrium
:
x
1
*
= q
1
*
;
x
2
*
=q
2
*
World prices imply a revaluation of national income
In this equilibrium the gap between
x
**
and
q
**
is bridged by imports & exports
World prices
Domestic prices
q**
April 2018
40
x*Slide41
The nonconvex case with world trade
x
2
x
1
x
1
**
q
1
**
x
2
**
q
2
**
After opening trade
Before opening trade
A
′
A
Equilibrium on the island
World prices
x*
Again maximise income at world prices
The equilibrium with trade
Attainable set before and after trade
Trade “convexifies” the attainable set
April 2018
41
x**
q**Slide42
“Convexification”
There is nothing magic about thisWhen you write down a conventional budget set you are describing a convex set S pi x
i ≤ y, xi ≥ 0When you “open up” the model to trade you changefrom a world where F(·) determines the constraintto a world where a budget set determines the constraintIn the new situation you can apply the separation theoremApril 2018
42Slide43
The Robinson Crusoe economy
The global maximum is simpleIt can be split up into two separate parts:Profit (national income) maximisation Utility maximisationRelies on the fundamental decentralisation result for the price systemFollows from the separating hyperplane result“You can always separate two eggs with a single sheet of paper”
April 2018
43