Unformatted text preview: Econ 121.
Intermediate Microeconomics. Eduardo Faingold
Yale University Lecture 17 Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
IV. Market failure Outline of the course
I. Introduction
II. Individual choice
III. Competitive markets
Partial equilibrium (Ch. 16)
Exchange economies (Ch. 31)
Production (Ch. 18–23)
Efﬁciency and equilibrium with production (Ch. 32) IV. Market failure A simple economy with production
2 consumers, A and B
2 consumption goods, 1 and 2, which A and B care about
J 1 ﬁrms, labeled j D 1; : : : ; J N input goods, n D 1; : : : ; N , which the ﬁrms can use to produce
consumption goods 1 and 2 Consumption goods are in zero net supply
Inputs goods are in positive net supply:
N
N
N
I1 > 0; I2 > 0; : : : ; IN > 0 A simple economy with production
Preferences
B
B
A
A
uA.x1 ; x2 / and uB .x1 ; x2 / Technology constraints of ﬁrm j :
j j j;1 j;1 j j j;2 j;2 y1 D f1 .I1 ; : : : ; IN /
y2 D f2 .I1 ; : : : ; IN /
where
j yi j;i
In D # of units of consumption good i produced by ﬁrm j
D # of units of input n used by ﬁrm j in the production of cons. good i Thus, ﬁrms are heterogeneous in their technologies. Resource constraints
Consumption good 1: A
B
1
2
J
x1 C x1 D y1 C y1 C : : : C y1 D J
X j y1 j D1 Consumption good 2: J
2
1
B
A
x2 C x2 D y2 C y2 C : : : C y2 D J
X
j D1 Input n D 1; : : : ; N : J
X j;2
j;1
N
.In C In / D In j D1 Thus, we have 2 C N resource constraints. j y2 Allocations
An allocation is a list .x A; x B ; I 1;1 ; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J /
comprising: a consumption bundle for each consumer:
A
A
x A D .x1 ; x2 /; B
B
x B D .x1 ; x2 / an output bundle for each ﬁrm j D 1; : : : ; J :
j j y j D .y1 ; y2 /
an input bundle for each ﬁrm j D 1; : : : ; J and each consumption good
i D 1; 2:
j;i
j;i
I j;i D .I1 ; : : : ; IN / Pareto efﬁciency
An allocation .x A; x B ; I 1;1; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J / is feasible if
it satisﬁes the technology constraints and the resource constraints.
A feasible allocation .x A; x B ; I 1;1; : : : ; I J;1 ; I 1;2; : : : ; I J;2; y 1; : : : ; y J / is
Pareto efﬁcient if there does not exist another feasible allocation
.x 0A; x 0B ; I 1;1; : : : ; I 0J;1 ; I 01;2; : : : ; I 0J;2 ; y 01; : : : ; y 0J / such that uA.x 0A/ uA.x A/ uB .x 0B / uB .x B /; with at least one strict inequality.
Note that the ﬁrm’s proﬁts do not enter the deﬁnition, but that is OK. This
model implicitly assumes that the ﬁrms are owned by the consumers, who
get shares of their proﬁts. Inputoutput economy
This simple model assumes a complete separation between consumption /
output goods and input goods.
This seems a bit farfetched: think of realworld complicated supply chains.
Even for natural resources this separation is problematic: think of crude oil
vs. reﬁned oil.
Nevertheless, we will go along with this assumption for now. (Next class,
we will remove this assumption.) Production possibilities frontier
Set of .y1; y2/ such that
NN 1
J
y1 C : : : C y1 y1 D max
N subject to
1
J
y2 C : : : C y2
j j;1 j;1 j j;2 j;2 f1 .I1 ; : : : ; IN /
f2 .I1 ; : : : ; IN /
J
X
j D1 j;1
j;2
In C In D y2
N
j D y1 ; all j D y2 ; all j D N
In; all n j Production possibilities frontier
equivalent to y1 D max
N J
X j j;1 j;1 j j;2 j;2 f1 .I1 ; : : : ; IN / j D1 subject to
J
X f2 .I1 ; : : : ; IN / D y2 .w/ Lagrange mult. /
N D N
In; .w/ Lagrange mult. n/ j D1
J
X
j D1 j;2
j;1
In C In Production possibilities frontier
Lagrangean: LD J
X
j D1 J
X j j;2
j
j;1
j;1
j;2
f1 .I1 ; : : : ; IN / C .
f2 .I1 ; : : : ; IN / y2 /
N j D1 C N
X
nD1 FOC:
j;1
MPn C n D 0
j;1 MPn0 C n0 D 0
Hence,
j;1
TRSn;n0 Likewise, for consumption good 2. D j 0;1
TRSn;n0 : J
X
j;1
j;2
n .
In C In
j D1 N
In / T he boundary of t he production possibilities s et is called t he production
possibilities frontier. T his should be contrasted with t he production
function discussed earlier t hat depicts t he relationship between t he i nput
good and t he o utput good; t he production possibilities set depicts only t he
set of o utpr~tgoods t hat is feasible. ( In more advanced t reatments, b oth
i nputs and o utputs can b e considered p art of t he production possibilities
s et. b ut these t reatments cannot easily be handled with twodimensional
diagrams.) T he sha.pe of t he production possibilities set will deperid on t he n ature
of t he underlying technologies. If t he technologies for protlucing c ocon~its
and fish exhibit constant r eturns t o scale t he production possibilities set
will t ake a n especially simple form. Since by asslrmption t here is only one
i nput t o productionRobinsml's laborthe production functions for fish
and coconuts will b e simply linear functions of labor.
For example. suppose t hat Robinson car1 produce 1 0 ponnds of fish per Production possibilities frontier
Once we solve for the PPF, we get some implict equation: T .y1; y2/ D 0 iff .y1; y2/ 2 PPF: Marginal rate of transformation: MRT .y1; y2/ D @T
.y ; y /
@y1 1 2
@T
.y ; y /
@y2 1 2 Pareto efﬁciency
WIth production, Pareto efﬁciency requires not only that the MRS of the two
consumers is equalized, but also that they equal the MRT. PARETO EFFICIENCY 605 Production and the Edgeworth box. At each point on t he
production possibilities frontier, we can draw a n Edgeworth box
t o illustrate t he possible consumption allocations. T he Pareto set describes t he set of P areto efficient bundles given t he
amounts of goods 1 and 2 available, but in an economy with production
those amounts can themselves be chosen out of t he production possibilities
set. Which choices from t he production possibilities set will be P areto
efficient choices?
Let us think about t he logic underlying t he marginal r ate of substitution
condition. We argued t hat in a P areto efficient allocation, t he MRS of
consumer A had t o be equal t o t he MRS of consumer B: t he r ate a t which Competitive Equilibrium: Deﬁnition
Allocation:
.x A; x B ; I 1;1; : : : ; I J;1; I 1;2 ; : : : ; I J;2 ; y 1; : : : ; y J /;
where
i
i
x i D .x1; x2/; j j for i D A; B , j D 1; : : : ; J and ` D 1; 2. Prices for the consumption goods:
p D .p1; p2/
Prices for the inputs:
w D .w1; : : : ; wn/
such that... j;` j;`
y j D .y1 ; y2 / and I j;` D .I1 ; : : : ; In /: Competitive Equilibrium: Deﬁnition
.... such that: consumers take prices p as given and maximize their utility subject to
their budget constraint; ﬁrms take prices p and w as given and maximize their proﬁt subject to
their technological constraint; the markets for input goods and consumption goods clear. Competitive Equilibrium: Deﬁnition
But, wait, the model seems to be incomplete.
Where do the ﬁrms’ proﬁts go?
Who owns the ﬁrms?
Well, if this is a model of the whole economy, the ﬁrms must be owned by
either A or B , or both. Otherwise, we left the owner of our ﬁrm out of the
model, hence our model would be incomplete. Competitive Equilibrium: Deﬁnition
Thus, we need a new primitive in the model, namely the ownership shares: j;i D fraction of ﬁrm j that is owned by consumer i;
for i D A; B; j D 1; : : : ; J:
We must have: j;A C j;B D 1; j D 1; : : : ; J: Competitive Equilibrium: Deﬁnition
Also, we need to specify the endowments:
i
!` D endowment of consumption good ` of consumer i Ni
In D endowment of input good n of consumer i Competitive Equilibrium: Deﬁnition
Consumer maximization:
A
A
x A D .x1 ; x2 / solves: max uA.x1; x2/ subject to A
p1x1 C p2x2 D p1! A C p2!2 C N
X NA
w n In C nD1 and likewise for B .
Here, j designates the proﬁt of ﬁrm j in equilibrium. J
X
j D1 j;A j Competitive Equilibrium: Deﬁnition
Firm maximization:
j j j;1 j;1 j;2 j;2 y j D .y1 ; y2 /, I j;1 D .I1 ; : : : ; IN / and I j;2 D .I1 ; : : : ; IN / solves
j D max p 1 y1 C p 2 y2 N
X 2
1
wn.In C In / nD1 subject to
1
1
y1 D f1.I1 ; : : : ; IN /
2
2
y2 D f2.I1 ; : : : ; IN /: Competitive Equilibrium: Deﬁnition
Market Clearing: A
x1 C B
x1 D A
!1 C B
!1 C X j y1 j A
x2 X
j C B
x2 D A
!2 C B
!2 C X j y2 j
j;1
j;2
NA
NB
In C In D In C In ; for n D 1; : : : ; N: First Welfare Theorem with Production
Theorem. The allocation of every competitive equilibrium is Pareto efﬁcient. First Welfare Theorem with Production
Just like we did in the case of exchange economies, we will see two proofs
of this result.
The ﬁrst proof assumes that preferences and technology are smooth and
exploits ﬁrstorder conditions.
The second proof, which is more general, does not assume smoothness; it
uses a revealed preference argument. ...
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This note was uploaded on 03/01/2012 for the course ECON 121 taught by Professor Samuelson during the Spring '09 term at Yale.
 Spring '09
 SAMUELSON
 Microeconomics

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