Microeconomics 3

Economics Programme, University of Copenhagen

Spring semester 2006

Week 10

Lars Peter Østerdal

Today�s programme

� Production economies.

� Theme 3: Market imperfections. Introduction

� Public goods.

General equilibrium in production economies

We now consider a more general (and perhaps more relevant) the model.

We go from an exchange economy to an economy with production.

Modelling issues:

� Production technology

� Firm behavior

� Ownership

Production technology

m �rms

k goods (a good can be consumed by consumers, it can be an input factor toproduction, or both)

A net output vector for �rm j is a vector yj = (y1j ; :::; ykj ).

Negative entries: net input (ex., steel, crude rubber, labor, ...)

Positive entries: net output.(ex., cars)

Yj is the the production possibilities set for �rm j.

Example: One input, one output

Y

Input

Output

Production function (one output only):

f(x) = fmax y j (y;�x) 2 Y g

Transformation function (multiple outputs):

We say that y in Y is e¢ cient if there is no y0 2 Y such that y0 � y, y0 6= y.

A transformation function is a function T such that T (y) � 0 for all y 2 Yand T (y) = 0 if and only if y is e¢ cient:

Our notation is very convenient:

If p is a price vector, then pyj is pro�t associated with production plan yj.

Firm behavior:

Assume that each �rm solves

maxpyj

subject to yj 2 Yj

Remark 1: This problem has generally no solution under increasing returnsto scale (why? ).

Remark 2: This problem has generally not a unique solution under constantreturns to scale (why? )

Y

Input

Output

Isoprofit curves

Profitmax

Solving this pro�t-maximization problem gives us net supply yj(p) for �rm j.

For m �rm

y(p) =mXj=1

yj(p)

is aggregate net supply.

Y =Pmj=1 Yj is the aggregate production possibilities set.

i.e. y 2 Y if and only if y 2 Pmj=1 yj, where yj 2 Yj.

NB: No production externalities in this model! (why? )

Theorem: The following is equivalent:

i) Aggregate production y maximizes aggregate pro�t,

ii) Each �rm�s production yj maximizes its individual pro�t.

Intuitively, this theorem is rather clear given our assumption that there are noproduction externalities. (make sure you understand this intuition).

We now look at labor supply and pro�t distribution.

Labor supply (input to production) is modelled as follows:

A consumer has L units of time available (ex. L = 24):

Divide between labor l and leisure L = L� l.

Example:

The consumer cares about leisure L and a consumption good c.

w: wage rate.

p: price of consumption good.

c: endowment of consumption good.

Consumer�s problem:

maxu(c; L)

subject to pc = pc+ w(L� L); 0 � L � L; 0 � c.

(we can rewrite budget constraint: pc+ wL = pc+ wL).

Distribution of pro�ts

Consumers own the �rms.

We assume that ownerships are historically given and �xed (thus consumerscannot buy and sell stocks in this model).

Tij is consumer i�s share of the pro�ts of �rm j.

(remember that we have n consumers and m �rms)

Pni=1 Tij = 1, j = 1; :::;m.

Consumer i�s total pro�t income:

Pni=1 Tijpyj(p)

and his/her budget constraint is:

pxi = p!i +Pni=1 Tijpyj(p)

Consumer i�s demand: xi(p):

Aggregate demand: x(p) =Pni=1 xi(p)

Aggregate excess demand: z(p) = x(p)� y(p)| {z }remember:input negative!

� !.

Remark about notation: Varian writesX(p) instead of x(p), and he sometimeswrites Y(p) instead of y(p) (you can write it the way you prefer as long asyou make sure the meaning is clear).

Walras�law holds again: pz(p) = 0 for all p.

The reason is the same: If each consumer satis�es his/her budget constraintwith equality, then the economy as a whole also satis�es the aggregate budgetconstraint with equality.

Walrasian Equilibrium (production economy): (x�;y�;p�) is a W.E. if z(p�) �0.

i.e., if supply � demand on all markets, when consumers maximize utility givenp� and �rms maximize pro�ts given p�.

Theorem: Existence of Walras Equilibrium (Arrow-Debreu 1954):

An equilibrium exists if the following is satis�ed (see next two slides):

For each consumer i:

1. The consumption set is closed, convex, and bounded from below (ex. Rk+).

2. There is no satiation consumption bundle.

3. Preferences are continuous. That is, the sets fxijxi � x0ig and fxijx0i �xig are closed for each x0i.

4. The initial endowment is in the interior of the consumption set.

5. xi �i x0i implies txi + (1 � t)x0i � x0i for any 0 < t < 1. (convexitycondition).

Firms:

6. For each �rm j; 0 2 Yj (always possible to produce nothing).

7. Y =Pmj=1 Yj closed, convex (guarantees continuity of �rms aggregate

net supply correspondence)

8. Y \ (�Y ) � f0g:That is, the only vector y 2 Y for which �y 2 Y isy =f0g (irreversible production).

9. Y � �R+. (free disposal)

The welfare theorems can be generalized:

First theorem of welfare economics: If (x;y;p) is a W.E. then (x;y) isweakly Pareto e¢ cient:

Proof: Somewhat similar to that for an exchange economy. Read yourself.

Second theorem of welfare economics: Suppose that (x�;y�) is a feasi-ble Pareto e¢ cient allocation in which each consumer holds strictly positiveamounts of each good, and where preferences are convex, continuous, andstrongly monotonic. Suppose further that Yj is convex for j = 1; :::;m. Thenthere exists some vector p � 0; and a suitable reallocation of initial wealth(i.e. a reallocation of pro�t shares and endowments), such that (x�;y�;p) isa W.E.

(skip the proof)

Robinson Crusoe economy.

Suppose that unit price of labor w and unit price of consumption good is 1.

Initial endowment: (0; L)

Illustrate in a "Koopmans diagram":

Labor (input)

Leisure

Consumption

L

Real equlibrium profit

Equilibriumconsumption

Income

Slope=-w/1=-w

Theme 3: Market imperfections.

Market imperfections: Deviations form the assumptions of perfect competition(as in Varian ch. 17 & 18).

3.a. Public goods. Varian ch. 23

3.b. Externalities. Varian ch. 24

3.c. Externalities in production economies. PNS

Public goods

Until now, we have assumed that goods are ordinary private consumption goods:

In particular, they are excludable (people can be excluded from consuming it)and rival (one person�s consumption reduces the amount available to others).

Public goods are non-excludable and non-rival.

Classical examples:

� Streetlights (non-excludable, non-rival)

� Lighthouse (non-excludable, non-rival)

� Clean air (non-excludable, non-rival)

� Military defence (non-excludable, non-rival)

� Basic research (non-excludable, non-rival)

Some good are non-rival but excludable ("club good"):

� TV broadcast (non-rival, excludable if coded or requires payment of licencefee)

� Music and other digital goods that can be downloaded on the web.

Some good are rival but non-excludable:

� Highways (non-excludable unless charging tolls to road users, rival)

� Libraries (non-excludable unless charging fees, rival)

� Beaches (usually non-excludable, rival)

There are many in-between cases. Whether a good is rival/non-rival andexcludable/non-excludable is often a matter of interpretation/judgement. Strictlyspeaking, we need an explicit model (as at next slide) for an exact investigation.

Note that if a good is provided by the public, it doesn�t have to be a publicgood:

Some good of often provided by the public, but still ordinary private in nature(ex. many kinds of health care goods)

As we will see, competitive markets may not be good for allocating publicgoods.

First, we will establish conditions that can tell us under what circumstances itis e¢ cient to provide a public good.

E¢ cient provision of a discrete public good

Model with two agents and two goods.

xi consumption of private good (think of it as money spent on private con-sumption)

wi initial endowment of private good.

G public good (non-excludable, non-rival)

gi agent i�s contribution to public good.

xi = wi � gi

ui(G; xi) (assume strictly increasing in xi)

G =

(1; if g1 + g2 � c0; g1 + g2 < c.

If there is (g1; g2); g1 + g2 � c,

such that

u1(1; w1 � g1) > u1(0; w1)

and

u2(1; w2 � g2) > u2(0; w2):

Then providing the public good (with payments (g1; g2)) is Pareto e¢ cient.

Let ri be i�s reservation price; that is, i�s maximum willingsness-to-pay forthe public good.

ui(1; wi � ri) = ui(0; wi)

If ui(1; wi � gi) > ui(0; wi) for both i, we have:

ui(1; wi�gi) > ui(0; wi) = ui(1; wi�ri). And since ui is strictly increasingin its second argument:

wi � gi > wi � ri for both i.

This implies:

ri > gi:

Therefore:

r1 + r2 > g1 + g2 � c.

That is: the sum of reservation prices is greater than c:

Conversely, suppose that we have r1 + r2 > c.

For "i > 0, de�ne gi = ri � "i:

Then we can select "1; "2 su¢ ciently small such that r1 + r2 > g1 + g2 > c:

Thus:

ui(1; wi � gi) > ui(0; wi)

and providing public goods with payments g1; g2 is a Pareto improvement (rel-ative to not providing it):

Conclusion:

1. and 2. is equivalent:

1. The sum of reservation prices is greater than the cost of providing thepublic good.

2. There exists payments g1, g2, g1+ g2 � c, such that providing the publicgood is a Pareto improvement (relative to the situation with no paymentand no provision of public goods).

Private provision of a discrete public good

How e¤ective is the private market at providing a public good?

It can be very ine¤ective:

Example:

Two agents

r1 = r2 = 100

c = 150:

(NB: r1 + r2 > 150)

Suppose that each agent decides independently whether or not to buy the publicgood.

If consumer 1 buys the good, he/she get 100 worth of bene�t, but pays 150for getting it.

If consumers 1 buys the good, and consumer 2 does not buy, then consumer 2gets 100 worth of bene�t and pays nothing (i.e. consumer 2 free rides)

agent 1=agent 2 buy don�t buybuy �50,� 50 �50; 100

don�t buy 100;�50 0; 0

In this game, for each player the strategy "don�t buy" is a strictly dominantstrategy.

So (don�t buy; don�t buy) is the only outcome that survives elimination ofstrictly dominated strategies. In particular it is the only Nash Equilibrium ofthe game.

NB: This game is not a Prisoner�s Dilemma game, although it has some simi-larities (why?).

Point:

Usual market behavior (purely independent decisions) will generally not resultin e¢ cient provision of public goods.

We may therefore look for another (better?) mechanism than the competitivemarket.

We shall now investigate some alternatives.

Voting for discrete public goods

Will voting result in e¢ cient provision?

An interesting question because this is indeed as very popular mechanism.

Example:

Society with three individuals.

Cost of public good: c = 99.

Reservation prices: r1 = 90, r2 = 30, r3 = 30.

Procedure 1: If the majority votes in favor if provision, then split the costequally, i.e. each individual pays 33.

Note that r1 + r2 + r3 > c, i.e. provision is e¢ cient.

But only individual 1 votes in favor of provision) public good is not provided.

Ine¢ cient outcome, because providing public good and paying, say:

g1 = 50,g2 = 25; g3 = 25;

is a Pareto improvement.

Procedure 2:

� Individuals state their private willingness-to-pay for public good.

� Public good provided if sum of WTP exceeds the cost of the public good.

� The cost shares are �xed (same as under Procedure 1):

Then individual 1�s has an incentive to overstate his/her WTP (the higher thebetter for individual 1).

Individual 2 and 3 have an incentives to understate their WTP (the lower thebetter for individual 2 and 3).

This is a silly game, and this procedure clearly doesn�t work well.

Procedure 3:

Same as Procedure 2, but now each agent pays the stated WTP.

Suppose that stated WTP � true WTP for each individual, and the sum ofstated WTP = c.

Then we have e¢ cient provision, and stated WTP constitute a Nash Equilibriumin this game.

Problems:

� Many Nash-equilibria (e¢ cient outcome requires coordination).

� Also ine¢ cient Nash-Equilibria (typically there is a Nash equilibrium whereall agents tell that their WTP=zero)

� True-telling is not an equilibrium.

Conclusion:

There are serious problems with all these procedures.

We need a more sophisticated mechanism.

Nb: we now jump to Varian section 23.8 (come back to 23.4-23.7 later).