University of Oslo · 3/37 A graphical GE Model with 2 goods A Model of Exchange A Model of Exchange and Production Returns to Scale and Production Possibility Frontier

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A graphical GE Model with 2 goodsA Model of Exchange

A Model of Exchange and ProductionReturns to Scale and Production Possibility Frontier

General Equilibrium and WelfareLectures 2 and 3, ECON 4240 Spring 2017

Summary of Snyder et al. (2015)

University of Oslo

24.01.2017 and 31.01.2017

Summary of Snyder et al. (2015) General Equilibrium

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Outline

General equilibrium: look at many markets at the same time. Hereall prices determined in the model. Goal:

Study effects that occur when changes in one market haverepercussions in other marketsStudy connections between markets for goods and markets forfactors of productionMake general welfare statements about how well a marketeconomy performs


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Outline

We proceed by steps:A graphical model of General Equilibrium with 2 consumptiongoods and 2 factors of productionA mathematical model of exchange with n goodsA mathematical model of production and exchange with ngoods


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Perfectly Competitive Price System

In all the general equilibrium models we borrow many assumptionsfrom the partial equilibrium analysis:

All individuals are price-takers, and utility-maximizersAll firms are price-takers and profit-maximizersZero transaction costsPerfect information


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A graphical GE Model with 2 goods

Many identical individuals, many identical firms2 goods (x and y) and 2 inputs (capital and labor)The endowments of capital and labor are fixedFor simplicity, all individuals have identical endowments ofcapital and labor, and they all own equal shares of each firm


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We first focus on the supply side. As firms are profit maximizers, inequilibrium inputs will be used in an efficient ways to produce thetwo goods (efficient = it is not possible to produce more of onegood without producing less of the other)

We draw an "Edgeworth box", where the horizontal sidemeasures total labor endowment, the vertical side total capitalendowmentAny point in the box is a full-employment allocation of inputsEach point in the Edgeworth box measures how much of eachfactor is devoted to the production of x and how much isdevoted to the production of yNothing in the Edgeworth about how much of each factor isused in each single firm (for now it does not matter)


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In the Edgeworth box we can draw isoquants for good x andisoquants for good y

DefinitionAn isoquant (for good x) shows those combinations of capital andlabor that can produce a given level of good x.

Note that these are aggregate isoquants, not at single firmlevelThe efficient allocations are the ones where the isoquants forthe production of the two goods are tangent


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The information about all possible efficient allocations can be usedto construct a production possibility frontier in the graph ofcombinations of good x and good y

DefinitionThe production possibility frontier is the curve in the x-y spacethat delimits all combinations of x and y that can be produced,given the initial endowment of capital and labor


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Efficient Allocations of Inputs

DefinitionThe rate of product transformation (RPT) between two outputsis the negative of the slope of the production possibility frontier forthose outputs:

RPT =−dydx (along the production possibility frontier)

Pick any 2 prices for the inputs

For any point (xa,ya) on the production possibility frontier thetotal cost for producing xa units of good x and ya units of y isthe same.Note that RPT =−MCx

MCy


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Efficient Allocations of Inputs

Concave frontier ↔ MCxMCy

increases as x increases and y decreases

Possible reasons behind concavity of frontier:

Diminishing returnsSpecialized inputsDiffering factor intensities


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Determination of Equilibrium Prices

At any given (good) price ratio pxpy

firms altogether produce

some amount (x f ,y f ) such that(a) (x f ,y f ) lies on the production possibility frontier,(b) at (x f ,y f ) the production possibility frontier has slope−px

py.

In order to find the equilibrium price ratio (denote it p∗xp∗y) we

need to consider the demand sideNote that the competitive markets only determine equilibriumrelative prices, not absolute prices (more on this in a bit)


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In the x− y space, we can draw individuals’ indifference curves(aggregating individuals’ indifference curve is not a trivialissue, here for simplicity say that if (xb,yb) is consumedaltogether and there are n individuals, then each individualgets xb/n units of good x and yb/n units of good y)For any pair of prices px and py , individuals altogether demandsome amount (xc ,y c) such that the indifference curve passingthrough the point (xc ,y c) has slope −px

py


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A pair of prices are equilibrium prices if the overall quantitieschosen by firms coincide with the overall quantities chosen byindividuals: (x f ,y f ) = (xc ,y c) = (x∗,y∗)What about the budgets of individuals? How do individualspay for x and y?In equilibrium total revenues (=sum of revenues of all firms)are x∗px + y∗py .These revenues cover the costs of inputs and if anything is leftthey become profits. Whether the revenues are used to covercosts or they are profits they all go to individuals anywayFor individuals, the revenue from selling labor and capital,together with the profits, add up to x∗px + y∗py


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General Equilibrium Models in International Trade

So far we learned that the equilibrium price ratio p∗x/p∗y

persists as long as technologies or preferences do not change.We can apply this model to questions of International TradeNote that the model can be made more realistic (and moreuseful) by assuming that there are more inputs (e.g. high andlow skilled labor) and individuals differ in the endowments ofthe inputs


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General Equilibrium Models in International Trade

The Corn Law Debate: focus on tariffs on grain imports.Grain = x , manufactured goods = yWith tariffs high enough to completely prevent trade:equilibrium "E", domestic price ratio: pE

xpEy

Removing tariffs: pAx

pAy, grain imports of xB − xA, financed with

export of manufacturing of yA− yB

Result of tariff removal: lower (relative) price of grain, lessgrain production, less farmers, less rent for land owners(different from textbook story)


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General Equilibrium Models In International Trade

US elections: trade was an important issueIdea: trade affects the relative incomes of various factors ofproductionIn the US exports use intensively skilled labor, while importsare products that require low skilled laborMore free trade: increasing relative wage of high skilledworkers (more inequality (in the US - worldwide not clear))


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A Mathematical Model of Exchange: Setting

Model of exchange = no production, goods already existn goods, m individualsx i : vector of consumption of individual i = 1..m (vector of sizen)x i : vector of endowment of individual i (vector of size n)Here the same individuals can be on both sides of differentmarkets; for simplicity imagine an individual sells all herendowment and uses the revenues to buy whatever sheconsumesMarket value of individual’s endowment: px i

Budget constraint of individual i : px i ≤ px i


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Equilibrium and Walras’ Law

Vector of demand of individual i : x i (p,px i )

These demands are homogeneous of degree 0:

x i (p,px i ) = x i (tp, tpx i ) ∀t > 0

DefinitionWalrasian equilibrium is an allocation of resources and anassociated price vector p∗ such that quantity demanded andquantity supplied of each good coincide:

Σmi=1x

i (p∗,p∗x i ) = Σmi=1x

i (1)


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Equilibrium Existence

Does an equilibrium exists? Not obvious as:(1) from last slide corresponds to n equationsThe equations in (1) need not be linear (x i (·) is a vector of nequations, not necessarily linear)These equations are not independent: they are related byWalras’ Law


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Walras’ Law

DefinitionWalras’ law :

Σmi=1px

i = Σmi=1px

i

Walras Law is a direct consequence of the individual’s budgetconstraints: for every individual i = 1, ..m we have px i = px i :adding over the m individuals we have Walras’ law


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Existence of Equilibrium

Intuitive evidence seems to suggest equilibrium existsProof of existence:

As demands are homogeneous of degree 0, and budgetconstraints are not affected if all prices are multiplied by aconstant t > 0, we can "normalize" pricesOne way to normalize prices is, for example, to multiply everyprice by 1

Σnk=1pk

, thus getting p′i = piΣn

k=1pk

This is called in jargon a normalization as Σni=1p

′i = 1 (note

that relative prices are unaffected by the normalization)


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Proving existence ↔ showing there exists a vector p∗ such that

z(p∗) := Σmi=1x

i (p∗,p∗x i )−Σmi=1x

i = 0

Starting from an arbitrary vector of prices p0, define p1 as(textbook expression is wrong)

p1 = f (p0) :=

[p01 + αz1(p0)

Σnk=1(p0

k + αzk(p0)), ..,

p0n + αzn(p0)

Σnk=1(p0

k + αzk(p0))

]where α is some positive constantFunction f is continuous and maps normalized prices ontoother normalized prices (= if p0 is a vector of normalizedprices, so is p1)


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Brouwer Fixed Point Theorem

Function f meets the conditions of Brouwer fixed pointtheorem for existence of fixed point p∗ (see next slide)A fixed point of function f is a vector p∗ such that p∗ = f (p∗)

TheoremBrouwer Fixed Point Theorem: Any continuous function from aclosed compact set onto itself has a fixed point such that f (x) = x.

(No proof)


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DefinitionS is a compact subset of Rn if S is closed (= contains itsboundaries) and bounded (= pick any point in Rn: along any "ray"starting from that point and going in any direction there are pointsoutside S)

In our case, we consider a subset of Rn: [0,1]n


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Welfare Properties of Competitive Equilibria

After establishing existence of competitive equilibria, we shouldask ourselves whether these equilibria lead to socially desirabledivisions of resources (divisions of resources= allocations)Welfare economics studies criteria for choosing amongalternative allocationsAside: when considering 1 consumer, any allocation thatmaximizes his utility is clearly the best one: definition ofefficiency is UNIQUE. When considering n consumers, thereare many different definitions of efficiency


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DefinitionAn allocation of the available goods in an exchange economy isPareto efficient if it is not possible to devise an alternativeallocation in which at least one person is better off and no one isworse off.

Other measures of efficiency require to formally define a socialwelfare function:

DefinitionA social welfare function is a scheme from ranking potentialallocations of resources based on the utility they provide toindividuals.Social Welfare = SW [U1(x1),U2(x2), ..,Um(xm)]


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Note: for any "reasonable" SW (.) any allocation thatmaximizes SW is Pareto EfficientPareto Efficiency is a VERY limited way to rank allocations:e.g. crazily unequal allocations can be Pareto Efficient(My) view: Pareto efficiency is a necessary but not sufficientrequirement for an allocation to be acceptableUtilitarian SW function

SW (U1,U2, ..,Um) = U1 +U2 + ...+Um

Maximin SW function

SW (U1,U2, ..,Um) = min [U1,U2, ...,Um]


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First Theorem of Welfare Economics

TheoremFirst Welfare Theorem. Every Walrasian equilibrium is ParetoEfficient.

Proof by contradiction. (LOTS of typos in the textbook proof)Consider a Walrasian equilibrium with prices p∗ and allocationxk (k = 1, ..,m).Suppose there exists an alternative feasible allocation, x̃k ,such that a consumer i is better off and everyone is at least aswell off as with xk . Then:p∗x̃ i > p∗x i (Suppose instead p∗x̃ i ≤ p∗x i : then individual i couldafford x̃ i , and as we assume individual i is better off with x̃ i then with x i

we should conclude that x i is not the optimal choice for consumer i . Wereached a contradiction as x i IS optimal. So it must be the case thatp∗x̃ i > p∗x i )


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First Theorem of Welfare Economics

p∗x̃k ≥ p∗xk for k = 1..m (Again, suppose instead that p∗x̃k ≤ p∗xk

for some k: then consumer k could ensure the same utility obtained fromconsuming xk at a lower cost by purchasing x̃k , and with what she savesshe could buy more of some good thus increasing overall utility, so xk isnot her optimal choice. This is a contradiction as xk is the optimalchoice for k = 1..m.)Σm

i=1x̃i = Σm

i=1xi (This is the definition of feasibility.)

Multiply both sides of the last equality by p∗:Σm

i=1p∗x̃ i = Σm

i=1p∗x i

But the previous 2 inequalities, together with Walras Law,imply: Σm

i=1p∗x̃ i > Σm

i=1p∗x i , hence we reach a contradiction


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Second Welfare Theorem

TheoremSecond Welfare Theorem. For any Pareto Optimal allocation ofresources, there exists a set of initial endowments and a relatedprice vector such that this allocation is also a Walrasian Equilibrium.

We provide a graphical proof of the theorem using anEdgeworth box.A model of exchanges is sufficient to talk about welfareproperties of different allocation, but if we think of policyinstruments, such as taxes and subsidies, used to obtain thedesired allocation, then we need to account for the role ofthese policies on the incentive to produce


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A Mathematical Model of Production and Exchange

n goods (including factors of production).Goods consumed and used as factor of production can comefrom the household endowments or can be produced by somefirmr profit-maximizing firms. Production function of firm j : n×1column vector y j .Positive entries in y j : outputs, negative entries: inputsProfit of firm j : πj(p) = py j

Firms can choose y j = 0: exit (long run perspective) (TYPOS inbook)Firms are owned by individuals. Simplification: each individuali owns a share si of every firm (simplification: same share ofeach firm)


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Budget Constraints and Walras’ Law

Budget constraint of individual i :px i (p) = siΣr

j=1pyj(p) +px i (p)

Note slight inconsistency: we allow for positive profits (like inshort run) but we allow for firms exit (like in the long run)Walras law:

px(p) = py(p) +px ,

where:x(p) = Σm

i=1xi (p),

y(p) = Σrj=1y

j(p) andx = Σm

i=1xi .


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Walrasian Equilibrium

Equilibrium price vector p∗ at which quantity demanded equalsquantity supplied in all markets simultaneously:

x(p∗) = y(p∗) + x .

(One) use of these models: look at effect on wages of chancesin exogenous factors


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Walrasian Equilibrium

Going back to the first model of GE considered (goods x andy , inputs K and L) let’s look at the relation between returnsto scale and the slope of the production possibility frontierx = kα

x lαx and y = kα

y lαy

2α = 1: constant returns to scale (RTS), 2α < 1: decreasingRTS, 2α > 1: increasing RTS,Full employment of both inputs: lx + ly = l and kx +ky = k


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Contract Curve

First we characterize the set of efficient allocations of inputs(that is, the contract curve)Slope isoquant of x . Rate of Technical Substituttion:

−dxdlxdxdkx

=−αkαx lα−1

xαkα−1

x lαx

=−kxlx

Slope isoquant of y : −kyly

On the contract curve the 2 slopes coincide: kxlx

=kyly

Substituting the "full employment" constraints:kxlx

= k−kxl−lx→ kx

lx= k

l (and kyly

= kl )


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Production Possibility Frontier

On the conctract curve, both production functions can beexpressed as a function of lx only:x(lx) = (k

l )α l2αx

y(lx) = (kl )α l2α

y = (kl )α (l − lx)2α

Slope of the production possibility frontier:dydx |contract curve = y ′(l)

x ′(l) =− y1− 12α

x1− 12α

< 0

d2yd2x |contract curve =−(1− 1

2α) y−

12α

x1− 12α

dydx + (1− 1

2α) y1− 1

2α

x2− 12α


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Production Possibility Frontier

2α < 1 (decreasing RTS) → d2yd2x |contract curve < 0 (concave

production possibility frontier (PPF))

2α = 1 (constant RTS) → d2yd2x |contract curve = 0 ("straight"

PPF)

2α > 1 (increasing RTS) → d2yd2x |contract curve > 0 (convex PPF)


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University of Oslo · 3/37 A graphical GE Model with 2 goods A Model of Exchange A Model of Exchange and Production Returns to Scale and Production Possibility Frontier