110 CHAPTER 5. THE CONSUMER AND THE MARKET
Figure 5.5: Consumption in the household-production model
j + 1 is
w
j
b
2j+1
 w
j+1
b
2j
w
j
b
1j+1
 w
j+1
b
1j
: (5.22)
We can think of this as ratio of notional prices 
1
=
2
. So it is clear that a simple
increase in the budget y (from a larger resource endowment) j ust “in‡ates”the
attainable set –look at the way each vertex (5.21) changes with y –without
altering the relative slopes of di¤erent parts of the frontier (5.22). However
changes in input prices or productivities will change the shape of the frontier.
As illustrated the household would consume at x


utility directly. The household’s new c onsu mption point is at x

.
The fact that some commodities are purchased by households n ot for direct
consumption but as inputs to produce other goods within the household enables
us to understand a number of phenomena that are di¢ cult to reconcile in the
simple consumer-choice model of section 4.5 (chapter 4):
 If m > n, some market goods may not be purchased. By contrast, in
the model of chapter 4, if all indi¤erence curves are strictly convex to the
origin, all go ods must be consumed in positive amounts.
 If the market price of a good falls, or indeed if there is a technical im-
provement in some input this may lead to no change in the consumer’s
equilibrium.
 Even though each x
i
may be a “normal”good, certain purchased market
goods may appear “inferior”if preferences are non-homothetic.
11
 The demand for inputs purchased in the market may exhibit jumps: as
the price of an input drops to a critical level we may get a sudden switch
from one facet to another in the optimal consumption plan.
11
Provide an intuitive argument why this may occur.
112 CHAPTER 5. THE CONSUMER AND THE MARKET
5.5 Aggregation over goods
If we were to try to use any of the consumer models in an empirical study we
would encounter a number of practical di¢ culties. If we want to capture the …ne
detail of consumer choice, distinguishing not just broad categories of consumer
expenditure (food, clothing, housing ) but individual product types within
those categories (olive oil, peanut oil) almost certainly th is would require that

2
+ p
3
x
3
 y is equivalent to maximising a function U(x
1
; x) subject
to p
1
x
1
+ px  y where p := p
2
+ p
3
, x := x
2
+ [1  ]x
3
,  := p
2
=p.
An extension of this result can be made from three to an arbitrary number of
commodities,
12
so e¤ectively resolving the p roblem of aggregation over groups
of goods. The implication of Theorem 5.1 is that if the relative prices of a group
of commodities remain unchanged we can treat this group as though it were a
single commodity.

 D
hi
is the corresponding demand function.
We also write n
h
for the number of households.
The issues that we need to address are: (a) How is aggregate (market) de-
mand for commodity i related to the demand f or i by each individual household
h ? (b) What additional conditions, if any, need to be imposed on preferences
in order to get sensib le results from the aggregation? Let u s do this in three
steps.
Adding up the goods
Suppose we know exactly the amount that is being consumed by each household
of a particular good:
x
h
i
; h = 1; :::; n
h
: (5.23)
To get the total amount of i that is being consumed in the economy, it might
seem that we should just stick a summation sign in front of (5.23) so as to get
n
h
X
h=1
x
h
i
: (5.24)

hi

p; y
h

(5.25)
The idea of equation (5.25) is depicted in the two-person case in Figure 5.7
and it seems that the elementary process is similar to that of aggregating the
supply of output by …rms, depicted in Figures 3.1 and 3.2. There are similar
caveats on aggregation and market equilibrium as for the …rm
15
–see pages 51
to 53 for a reminder –but in the case of aggregating over consumers there is a
more subtle problem.
Will the entity in (5.25) behave like a “proper” demand function? The
problem is that a demand function typically is de…ned on prices and some simple
measure of income –but clearly the right-hand side of (5.25) could be sensitive
to the distribution of income amongst households, not just its total. One way
of addressing this issue is to consider the problem as that of characterising the
behaviour of a representative consumer. This could be done by focusing on the
person with average income
16
y :=
1
n
h
n
h
X
h=1

i
=

D
i
(p; y) (5.26)
where each

D
i
behaves like a conventional demand curve. If such a relationship
exists, then we may write:

D
i
(p; y) =
1
n
h
n
h
X
h=1
D
hi

p; y
h

(5.27)

(p)). This is illustrated in Figure 5.8. Of course
imposing this requirement on the demand function also imposes a correspond-
ing condition on the class of utility functions that allow one to characterise the
behaviour of the market as though it were that of a representative consumer.
Market demand and WA RP
What happens if this regularity condition is not satis…ed? Aggregate demand
may b e have very oddly indeed. There is an even deeper problem than just
the possibility that market demand may depend on income distribution. This is
illustrated in Figure 5.9 which allows for the possibility that incomes are endoge-
nously determined by prices as in (5.1). Alf and Bill each have conventionally
shaped utility functions: although clearly they di¤er markedly in terms of their
income e¤ects: in neither case is there a “Gi¤en good”. The original prices are
shown by the budget sets in the …rst two panels: Alf’s demands are at point
x
a
and Bill’s at x
b
. Prices then change so that good 1 is cheaper (the budget
constraint is now the ‡atter line): Alf’s and Bill’s demands are now at points
x
a0
and x
b0
respectively; clearly their individual demands satisfy th e Weak Ax-
iom of Revealed Preference. However, look now at the combined result of their
behaviour (third panel): the average demand shifts from x to x
0
. It is clear that
this change in average demand could not be made consistent with the behaviour
of some imaginary “representative consumer”–it does not even satisfy WARP!

5.9 Exercises
5.1 A peasant consumer has the utility function
a log (x
1
) + [1  a] log (x
2
 k)
where good 1 is rice and good 2 is a “basket” of all other commodities, and
0 < a < 1, k  0.
1. Brie‡y interpret the parameters a and k.
2. Assume that the peasant is endowed with …xed amounts (R
1
; R
2
) of the two
goods and that market prices for the two goods are known. Under what
circumstances will the peasant wish to supply rice to the market? Will the
supply of rice increase with the price of rice?
3. What would be the e¤ect of imposing a quota ration on the consumption
of good 2?
118 CHAPTER 5. THE CONSUMER AND THE MARKET
5.2 Take the model of Exercise 5.1. Suppose that it is possible for the peasant
to invest in rice production. Sacri…cing an amount z of commodity 2 would
yield additional rice output of
b

1  e
z

where b > 0 is a productivity parameter.

is consumption in period t, and , k are parameters such that 0 <  < 1
and k  0. The consumer is endowed with an exogenous income stream (y
1
; y
2
)
and he can lend on the capital market at a …xed interest rate r, but is not allowed
to borrow.
1. Interpret the parameters of the utility function.
2. Assume that y
1
 y where
y := k 

1  

y
2
 k
1 + r

Find the individual’s optimal consumption in each period.
3. If y
1
 y what is the impact on period 1 consumption of
(a) an increase in the interest rate?
5.9. EXERCISES 119
(b) an increase in y
1
?

is the amount of leisure enjoyed by person 2, and x
0
is the amount of the
single, composite consumption good enjoyed by the household. The two members
of the household have, respectively (T
1
; T
2
) hours which can either be enjoyed as
leisure or spent in paid work. The hourly wage rates for the two individuals are
w
1
, w
2
respectively, they jointly have non-wage income of y, and the price of
the composite consumption good is unity.
1. Write down the budget constraint for the household.
2. If the utility function U takes the form
U(x
0
; x
1
; x
2
) =
2
X
i=0

i

1
T
1
+ w
2
T
2
]  [
0
+ w
1

1
+ w
2

2
]]
3. Write down the labour supply function for the tw o individuals.
4. What is the response of an individual’s labour supply to an increase in
(a) his/her own wage,
(b) the other person’s wage, and
(c) the non-wage income?
120 CHAPTER 5. THE CONSUMER AND THE MARKET
5.7 Let the demand by household 1 for good 1 be given by
x
1
1
=
8

of households 1 and 2 for good 1 and show that at p
1
= a there are now
three possible values of
1
2
[x
1
1
+ x
2
1
].
3. Extend the argument to n
h
identical consumers. Show that n
h
! 1 the
possible values of the consumption of good 1 per household becomes the
entire segment

y
4a
;
y
2a

.
Chapter 6
A Simple Economy

from an isolated enterprise to an entire economy. It is useful to be able to
think about a collection of production processes that deal with di¤erent parts
of the economy and their relationship to one another. Fortunately there is a
comparatively easy way of doing this.
6.2.1 Processes and net outputs
In order to describe the technological possibilities it is useful to use the concept
brie‡y introduced in chapter 2:
De…nition 6.1 The net output vector q is a list of all potential inputs to and
outputs from a production process, using the convention that outputs are mea-
sured positively and inputs negatively.
We can apply this concept at the level of a particular production process or
to the economy as a whole. At each level of operation if more of commodity i is
being produced than is b eing used as input, then q
i
is positive, whereas if more
of i is used up than is produced as output, then q
i
is negative. To illustrate this
6.2. ANOTHER LOOK AT PRODUCTION 123
usage and its application to multiple production processes, consider Figure 6.1
which illustrates three processes in which labour, land, pigs and potatoes are
used as inputs, and pigs, potatoes and sausages are obtained as outputs.
We could represent process 1 in vector form as
q
1
=
2
6
6
6

7
7
7
7
5
(6.2)
q
3
=
2
6
6
6
6
4
+1000 [sausage s]
0 [potatoes]
20 [pigs]
10 [la bour]
0 [land]
3
7
7
7
7
5
(6.3)
Expressions (6.1) to (6.3) give a succinct description of each of the processes.
But we could also imagine a simpli…ed economy in which these …ve commodities
were the only economic goods and q

7
5
(6.4)
So, viewed from the point of view of the economy as a whole, our three processes
produce sausages and potatoes as outputs using labour and labour as inputs;
pigs are a pure intermediate good.
In sum, we have a simple method of deriving the production process in the
economy as a whole, q, from its constituent parts. But this leaves open a number
of issues: How do we handle multiple techniques in each process? What is the
relationship of this approach to the production function introduced in section
2.5? Is the simple adding-up procedure always valid?
124 CHAPTER 6. A SIMPLE ECONOMY
6.2.2 The technology
The vectors in (6.1) to (6.3) or their combination (6.4) describe one possible list
of production activities. It is useful to be able to describe the “state of the art”,
the set of all available processes for transforming inputs into outputs –i.e. the
technology. We shall accordingly refer to Q, a subset of Euclidean n-space, R
n
,
as the technology set (also known as the production set.) If we write q 2 Q we
mean simply that the list of inputs and outputs given by the vector is technically
feasible. We assume the set Q is exogenously given –a preordained collection
of blueprints of the production process. Our immediate task is to consider the
possible structure of the set Q: the characteristics of the set that incorporate
the properties of the technology.
We approach this task by imposing on Q a set of axioms which seem to
provide a plausible description of the technology available to the community.
These axioms will then form a basis of almost all our subsequent discussion of
the production side of the economy, although sometimes one or other of them
may be relaxed. We shall proceed by …rst providing a formal statement of the

2 Q then tq

2 Q.
Let us see the implications of all six axioms by using a d iagram. Accordingly
take Process III in Figure 6.1 and consider the technology of turning pigs (good
3) and labour (good 4) into sausages (good 1). In Figure 6.2 the vector q

q

= (1800; 0; 18; 20; 0) (6.5)
represents one speci…c technique in terms of the list of the two inputs and the
output they produce;
q
0
= (500; 0; 10; 5; 0) (6.6)
6.2. ANOTHER LOOK AT PRODUCTION 125
Figure 6.2: Labour and pigs produ ce sausages
represents another, less labour-intensive technique producing less output. The
three unlabelled vectors represent other techniques for combining the two inputs
to produce sausages: note that all …ve points lie in the (+; ; ; ; ) orthant
indicating that sausages are the output (+), pigs and labour the inputs () –
the two “”symbols are there just to remind us that goods 2 (potatoes) and 5
(land) are irrelevant in this production process.
These axioms can be used to build up a picture of the technology set in Figure
6.3. Axiom 6.1 simply states that the origin 0 must belong to the technology
set – no pigs, no labour: no sausages. Axiom 6.2 rules out there being any
points in the (; +; ; +; +; ) orthant –you cannot have a technique that produces
sausages and pigs and labour time to be enjoyed as leisure. Axiom 6.3 …xes the
“direction”of pro d uction in that the sausage machine d oes not have a reverse
gear – if q is technically possible, then there is no feasible vector q lying in

q

=(900; 0; 9; 10; 0) is also technologically feasible;
hence also the entire cone shape in Figure 6.3 must belong to Q.
Axioms 6.1–6.3 are fairly unexceptionable, and it is not easy to imagine
circumstances under which one would want to relax them. The free disposal
axiom 6.4 is almost innocuous: perhaps only the case of noxious wastes and the
like need to be excluded. However, we should think some more about Axioms
6.5 and 6.6 before moving on.
The additivity axiom rules out the possibility of decreasing returns to scale
– de…ned analogously to the way we did it for the case of a single output on
page 16. As long as every single output is correctly identi…ed and accounted for
this axiom seems reasonable: if, say, land were also required for sausage making
then it might well be the case that multiplying the vector q
0
by 2000 would
produce less than a million sausages, because the sausage makers might get in
each other’s way – but this is clearly a problem of incomplete speci…cation of
the model, not the inappropriateness of the axiom. However, at the level of
the individual …rm (rather than across th e whole economy) apparent violations
of additivity may be relevant. If certain essential features of the …rm are non-
expandable, then decreasing returns may apply within the …rm; in the whole
economy additivity might still apply if “clones”of individual …rms could be set
up.
6.2. ANOTHER LOOK AT PRODUCTION 127
Figure 6.4: The potato-sausage tradeo¤
The divisibility axiom rules out increasing returns (since this implies that any
net output vector can be “scaled down”to any arbitrary extent) and is perhaps
the most suspect. Clearly some processes do involve indivisibilities, and whilst
it is reasonable to speak of single pigs or quarter pigs in process III, there is

for all q
1
 0g
If both technologies were available at the same time, what would be the combined technology
set?
128 CHAPTER 6. A SIMPLE ECONOMY
6.2.3 The production function again
The extended example in section 6.2.2 dealt with one production process; but
all the principles discussed there apply to the combined processes for the whole
economy. Naturally there is the di¢ culty of trying to visualise things in …ve
dimensions –so, to get a feel for the nature of the technology set Q it is useful
to look at particular sections of the set. One particularly useful ins tance of
this is illustrated in Figure 6.4 that illustrates the technological possibilities for
producing the two outputs in the …ve-good economy (sausages and potatoes),
for given values of the three other goods. The kinks in the boundary of the set
correspond to the speci…c techniques of production that were discussed earlier.
3
In the case where there are lots of basic processes, this view of the technology
set, giving the production possibilities for the two outputs, will look like Figure
6.5. Clearly we have recreated Figure 2.17 that we introduced rather abruptly
in chapter 2’s discussion of the single multiproduct …rm.
This connection of ideas suggests a further step. Using the idea of the
technology set Q we can then write the production function for the economy as
a whole. This speci…es the set of net output vectors (in other words the set of
input-output combinations) that are feasible given the technology available to
the economy. In other words this is a function  such that
4
(q)  0 (6.7)
if and only if q 2 Q. The particular feature of the production function high-
lighted in Figures 6.4 and 6.5 is of course the transformation curve –the implicit

explicit production function q  (z
1
; z
2
); rewrite this production function in implicit form
usin g  no tation. Ske tch the set of technologically feas ible net output vectors.
5
Use Theorem A.6 (page 499) to provide a 1-line proof of this.
6.3. THE ROBINSON CRUSOE ECONOMY 129
Figure 6.5: Smooth potato-sausage tradeo¤
Theorem 6.1 (Convexity in aggregation) If each the technology set or …rm
is convex and if there are no production externalities then the technology set for
the economy is also convex.
If, to the contrary, there were externalities then it is possible that the aggre-
gate technology set is nonconvex. Clearly the independence implied by the ab-
sence of externalities considerably simpli…es the step of moving from the analysis
of the individual …rm or process to the analysis of the whole economy.
6.3 The Robinson Crusoe economy
Now that we have a formal description of the production side of our simple econ-
omy we need to build this into a complete model. The model will incorporate
both production and consumption sectors and will take into account the natural
resource constraints of the economy. To take this step we turn to a well-known
story that contains an appropriately simple account of economic organisation –
the tale of Robinson Crusoe.
To set the scene we are on the sunny shores of a desert island which is cut
o¤ from the rest of the world so that:
 there is no trade with world markets,
 we have a single economic agent (Robinson Crusoe),
130 CHAPTER 6. A SIMPLE ECONOMY
x consumption goods

1
; R
2
; :::; R
n
, of each of the n commodities, where each R
i
must be positive or
zero. Then we can write down the materials balance condition for commodity
i:
x
i
 q
i
+ R
i
(6.9)
which simply states that the amount consumed of commod ity i must not exceed
the total production of commodity i plus preexisting stocks of i. Technology
and resources enable us to specify the attainable set for consumption in this
model, sometimes known as the production-possibility set.
6
This f ollows from
6
Use the production model of Exercise 2.10. If Crusoe has stocks of three resources
R
3
; R
4
; R

Figure 6.6 (for the case where R
1
= R
2
= 0) and a standard set of indi¤er-
ence curves has been introduced to represent Crusoe’s preferen ces . Clearly the
maximum will be at the point where
7

i
(q)

j
(q)
=
U
i
(x)
U
j
(x)
(6.11)
for any pair of goods that are produced in non-zero quantities, and consumed in
positive quantities at the optimum. This condition is illustrated in Figure 6.7
where the highlighted solution p oint (representing both optimal consumption
x

and optimal net outputs q

= x

+ 
2
q
2
+ ::: + 
n
q
n
(6.12)
where 
1
; 
2
; ; 
n
are some notional prices. (I have used a di¤erent symbol
for prices here because at the moment there is no market, and therefore there
8
Use your answer to the exercise in note 6 to illus trate the e¤ect of an increase in the
stock R
4
.
9
Use the diagram in the text to show the e¤ect of a technological improvement that enables
Crusoe to produce more of commodity i fo r ever y input combination.
6.4. DECENTRALISATION AND TRADE 133
are no “prices” in the usual meaning of the word (if we want to invent a story
for this let us suppose that Robinson Crusoe does some accounting as a spare-
time activity). If we were to draw the projection of (6.12) on the diagram
for di¤erent values of the sum  we would generate a family of isopro…t lines

; :::; 
n
are set equal to these announced
10
Use the de…nition of net outputs to expla in how to rewrite the expression for pro…ts in
(6. 12) the more conventio nal format of “Revenue - Cost”.
134 CHAPTER 6. A SIMPLE ECONOMY
MRS values then a simple geometrical experiment con…rms that the pro…t-
maximising net outputs q chosen by Friday (left-hand panel of Figure 6.6) lead
to a vector of commodities available for consumption q + R (right-hand panel
of Figure 6.6) that exactly corresponds to the optimal x vector (Figure 6.7).
In the light of this story we can interpret the numbers 
1
; :::; 
n
as shadow
prices –the imputed values of commodities given Crusoe’s tastes. The notional
“shadow pro…ts” made by the desert island at any net output vector q will be
given by (6.12). So the notional valuation of the whole island at these shadow
prices is just
 := 
1
[q
1
+ R
1
] + 
2
[q
2

U
i
(x)
U
j
(x)
=

i

j
: (6.15)
11
How wo uld this sort of problem change if Crusoe could not thoroughly monitor Friday’s
actions ?