i.2 general equilibrium and welfare - universitetet i oslo · piacquadio & traeger:...

Piacquadio & Traeger: Equilibrium, Welfare, & Information. UiO Spring 2018. 1/38

I.2 General Equilibrium and Welfare

ECON 4240 Spring 2017

Snyder et al. (2015), chapter 12

University of Oslo

1.2.2018


Outline

General equilibrium: look at many markets at the same time.All prices determined simultaneously in the model.Useful to:

◮ Study effects that occur when changes in one market haverepercussions in other markets

◮ Study connections between markets for goods and markets forfactors of production

◮ Make general welfare statements about how well a marketeconomy performs


Outline

We proceed in 3 steps:

1. A graphical model of General Equilibrium with 2 consumptiongoods and 2 factors of production

2. A mathematical model of exchange with n goods

3. A mathematical model of production and exchange with n

goods


Perfectly Competitive Price System

We continue assuming that

◮ All individuals are price-takers, and utility-maximizers

◮ All firms are price-takers and profit-maximizers

◮ Zero transaction costs

◮ Perfect information

as we did in the partial equilibrium analysis.


A graphical GE Model with 2 goods

Assumptions (graphical model):

◮ Many identical individuals, many identical firms (two types)

◮ 2 goods (x and y) and 2 inputs (capital and labor)

◮ The endowments of capital and labor are fixed

◮ For simplicity, all individuals have identical endowments ofcapital and labor, and they all own equal shares of each firm



For now, focus on the supply side:

As firms maximize profits, and we assume increasing returns in allinputs, endowments will be fully employed in equilibrium.

◮ We draw an “Edgeworth box”:

◮ the horizontal side measures total labor endowment,◮ the vertical side total capital endowment

◮ Any point in the box is a full-employment allocation of inputs

◮ Each 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 y

(Edgeworth box does not tell how much of each factor is used by an individual

firm, only what is used by a sector producing a particular good.)



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

Definition 1

An 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 firmlevel

◮ The efficient allocations are the ones where the isoquants forthe production of the two goods are tangent

(efficient:not possible to produce more of one good without producing less of the other)



Moving to the output plane:

We use the information about the efficient allocations in theEdgeworth box to construct the production possibility frontier inthe graph of combinations of good x and good y .

Definition 2

The production possibility frontier shows all combinations ofgoods x and y that can be produced efficiently given a fixedamount of inputs (capital and labor).


Efficient Allocations of Inputs

Definition 3

The 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


Efficient Allocations of Inputs

Concave frontier ↔ MCx

MCyincreases as x increases and y decreases

Possible reasons behind concavity of frontier:

◮ Diminishing returns

◮ Specialized inputs

◮ Differing factor intensities


Determination of Equilibrium Prices

◮ For any given (good) price ratio pxpy

, the firms produce

amounts (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 side

◮ Note that the competitive markets only determine equilibriumrelative prices, not absolute prices (more on this in a bit)



◮ In the x-y space, we can draw the consumers’ indifferencecurves(It’s aggregate demand. For simplicity we assumed identical individuals

(and identical endowments). Then, if (xc ,yc) is total consumption, each

individual gets xc/n units of good x and yc/n units of good y)

◮ For any pair of prices px and py , individuals aggregate demand(xc ,y c) is such that the indifference curve passing through thepoint (xc ,y c) has slope −px

py



◮ 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 in the end.

◮ Thus, for individuals, the revenues from selling labor andcapital, together with the profits, add up to x∗px + y∗py . Sodoes the cost of buying the goods.


Summary

So far we learned

◮ how fixed input endowments can be used to produce either oftwo goods (Edgeworth box)

◮ how the input distribution across the sectors translates intothe production possibility frontier

◮ how the implied rate of product transformation has to equalthe price ratio of the two goods,which also has to equal the consumers’ willingness to

substitute among the two goods

◮ the resulting equilibrium price ratio p∗x/p∗y is determined by the

underlying technology and preferences.

Now

◮ We will apply the model to questions of International Trade


General Equilibrium Models in International Trade

◮ The Corn Law Debate: focus on tariffs on grain imports.(Britain, 200 years ago. Similar: US presidential talk...now)

◮ Grain = x , manufactured goods = y

◮ With tariffs high enough to completely prevent trade:

equilibrium “E ”, domestic price ratio: pExpEy

◮ Removing tariffs: pAxpAy

, grain imports of xB − xA,

financed with export of manufacturing of yA− yB



Result of tariff removal:

◮ average consumer is better off (higher indifference curve)

◮ land owners are worse off:

1. lower (relative) price of grain, less grain production, less rentfor land owners.

2. More suitable (& historic) story:Take labor and land (instead of capital and land).

3. General story:Return of factor used more intensively in the sector whoseprice falls (grain) will drop. (Stolper-Samuelson theorem)



◮ US talk & recent elections: trade was and is an important issue

◮ Idea: trade affects the relative incomes of various factors ofproduction

◮ In the US exports use intensively skilled labor,while imports are products that require low skilled labor

◮ Tariffs could be used to increase the relative value of low skill

labor (and potentially bring back corresponding production)

◮ More free trade: increasing relative wage of high skilled workers→ more inequality in the US - worldwide not so clear


A Mathematical Model of Exchange: Setting

Beyond the graphical model...

◮ Model of exchange = no production, goods already exist

◮ n goods, m individuals

◮ x i : vector of consumption of individual i = 1..m (vector of size n)

◮ 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 her endowmentsand uses the revenues to buy whatever she consumes

◮ Market value of individual’s endowment: px i

◮ Budget constraint of individual i : px i ≤ px i


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

Definition 4

A Walrasian 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)


Equilibrium Existence

Does an equilibrium exist?

◮ Equation (1) from last slide corresponds to n equations for nunkown prices

→ Might be tempting to think it’s a simple exercise...

BUT NOTE:

◮ The equations in (1) need not be linear(x i (·) is a vector of n equations which are generally non-linear)

◮ These equations are not independent:they are related by Walras’ Law


Walras’ Law

Definition 5

Walras’ law: m

∑i=1

px i =m

∑i=1

px i

Analogous formulation:

The value of excess demands must sum to zero:

m

∑i=1

px i −m

∑i=1

px i = 0

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.


Existence of Equilibrium

Preparation:

Definition 6

S is a compact subset of Rn if S is closed (= contains itsboundaries) and bounded (= pick any point in R

n: along any "ray"starting from that point and going in any direction there are pointsoutside S)

We will consider a subset of Rn: [0,1]n.

Theorem 1

Brouwer Fixed Point Theorem: Any continuous function from a

closed compact set onto itself has a fixed point such that f (x) = x .

(No proof)

i.2 general equilibrium and welfare - universitetet i oslo · piacquadio & traeger:...

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