Eco 300 Intermediate Micro - University at Albany, SUNYaj4575/eco300_2010/LectureNotes/...Eco 300...

Eco 300 Intermediate Micro

Instructor: Amalia Jerison

Office Hours: T 12:00-1:00, Th 12:00-1:00,and by appointment

BA 127A, [email protected]

A. Jerison (BA 127A) Eco 300 Spring 2010 1 / 43

Pindyck and Rubinfeld, Chapter 16

General Equilibrium

Markets for different goods often are interdependent.

For instance, a change in the price of one good can affect demandfor another if they are complements or substitutes.

An increase in demand for an input can cause both the inputprice and the output price to rise.

General equilibrium analysis takes the relationships betweendifferent markets into account.


In general equilibrium analysis, prices and quantities in allmarkets are determined simultaneously.

Feedback effects are considered.

Feedback effects are the effects on price or quantity in one marketresulting form price or quantity changes in another market, whichmay have been due to a price or quantity change in the originalmarket.


Example. Suppose the US government taxes oil imports.

Supply curve for oil shifts to the left.

Demand for natural gas (a substitute for oil) increases, then priceof natural gas increases.

Due to higher natural gas price, demand for oil rises, increasingthe oil price more.

This process of affecting each other continues until eventually themarkets reach an equilibrium.

In equilibrium, quantity demanded and quantity supplied areequal in both markets.


It is not feasible to study the impact of one market’s price changeon all other markets.

To study the effects of a change in price of one market, choose afew closely related markets and see the effect on them.


Consider two interdependent markets: the competitive marketsfor DVD rentals and for movie theater tickets.

The two goods are substitutes (though not perfect).

So a change in price in one of the markets will probably affect theother market, and have feedback effects on the first market.


To get the general equilibrium effects of a tax on movie tickets, wehave to find the equilibrium prices and quantities of movies andDVDs simultaneously.

We have to find the demand and supply curves for movies ticketsassuming that equilibrium price of DVDs is PD, and PD must bethe equilibrium price of DVDs given the intersection of thosedemand and supply curves for movie tickets.

This is a solution to four equations (supply equal to demand inboth markets, and each quantity equals a function of price in bothmarkets) in four unknowns.


Exercise:

Consider the markets for textbooks and college education. Thesetwo goods can be thought of as complements.

Suppose that textbooks become available online for a muchreduced price.

What are the effects on the college education market and on thetextbook market?


Efficiency in exchange

With no externalities and complete information, the outcome of acompetitive market is efficient, since it maximizes the sum ofproducer and consumer surplus.

To analyze efficiency in more detail, consider a pure exchangeeconomy: There are two consumers (or countries) who can tradebetween themselves.

The definition of efficiency we use in general equilibrium (Paretoefficiency) will be slightly different from the efficiency defined bysurplus maximization. But surplus maximization efficiencyimplies Pareto efficiency.


The consumers start out with an endowment (initial allocation) oftwo goods.

Suppose that the endowment is such that both consumers canmake themselves better off by trading with each other.

This means that the endowment is Pareto inefficient. An Paretoefficient allocation is one where no agent can be made better offwithout another agent being made worse off.

Eventually a sequence of mutually beneficial trades will result in aPareto efficient allocation.


Suppose there are two goods, food and clothes. There are twopeople, James and Karen.

Let James have initially 7 units of food and 1 unit of clothes.Karen initially has 3 units of food and 5 units of clothes.

To find out whether trading can improve their well-being, we needto know their preferences over food and clothes.

Suppose that at her allocation, Karen’s MRS of food for clothing(= MUF/MUC) is 3. That is, she is willing to give up 3 units ofclothes for one unit of food.

James’ MRS of food for clothing is 1/2, so he is willing to give uponly 1/2 unit of clothing for an additional unit of food.


Because their MRSs are different, mutually advantageous tradecan be made. James values clothing more than Karen, and Karenvalues food more than James.

If Karen offers James 1 unit of clothing for 1 unit of food, Jameswill accept, and both will be better off: Karen only gave up 1 unitof clothing when she was willing to give up 3, and James got 1unit of clothing for 1 unit of food when he was willing to give up 2units of food.

If MRSs are different and both consumers have positive amountsof both goods (not a corner solution), then mutually beneficialtrade can be made.

If an interior allocation is Pareto efficient, the two consumers’MRSs are equal.


The Edgeworth box diagram

This diagram shows which trades can occur, and which willallocate the goods efficiently among the consumers.

The horizontal axis shows the amount of food (total is 10 units)and the vertical axis shows the amount of clothing (total is 6units).

The amount of food held by James is represented by thehorizontal distance from point OJ (the left, lower corner of thebox). The amount of clothing held by James is shown by thevertical distance from OJ .

The amount of food held by Karen is shown by the horizontaldistance from point OK (the right upper corner of the box). Theamount of clothes held by Karen is shown by the vertical distancefrom OK .


We can draw indifference curves for the two consumers in theEdgeworth box (to draw Karen’s indifference curves, we have tomeasure amounts of food and clothing from OK).

The negative of the slope of James’ indifference curve at theendowment point is his MRS at that point. Same for Karen.

The two indifference curves passing through the endowment pointintersect, and are not tangent.

By moving into the area between the two indifference curves, bothconsumers can be made better off.


OJ

OK

James’ food

James’clothes

Karen’sclothes

Karen’s food


Suppose that Karen and James make a mutually beneficial trade.Even if a new allocation makes both better off than theendowment point, the new allocation is not necessarily efficienteither.

When no more mutually beneficial trade can be made, theallocation is efficient.

There can be many possible efficient outcomes of mutuallybeneficial trades between James and Karen. Some are preferableto Karen over others, and some are preferable to James overothers.


The contract curve shows all possible efficient allocations. Theseare the allocations from which no mutually beneficial trade can bemade.

Ignoring corner solutions, the contract curve can be found byfinding all points of tangency between their indifference curves(the points where the indifference curves have the same slope).

Once the allocation is a point on the contract curve, no one canbe made better off without making the other worse off.


In real-life situations, it can be possible to improve someone’s wellbeing without making anyone worse off when the losers from aproposed change are compensated for their loss.

For example, if the quota on steel imports into the United Statesis eliminated, the total surplus in the steel market will increase.

Consumers will get a bigger selection of cars at lower prices, butsome workers will lose their jobs as a result.

However if some of the increase in total surplus is used tocompensate the American steel workers in some way (like jobrelocation subsidies), consumers would be better off and steelworkers no worse off (note that this assumes that markets clear,so there is no unemployment and the steel workers immediatelyfind new jobs).


Consumer equilibrium in a competitive market

In a competitive market each buyer and seller takes the price asgiven – they act as if they have no effect on the price.

But we can represent a competitive equilibrium by a two-personEdgeworth box if the two consumers behave competitively (actingas if they have no effect on the price). The goal is to show that a

market equilibrium outcome is Pareto efficient.


Equilibrium in a pure exchange system is defined by the followingconditions:

1. Each consumer is maximizing utility subject to the budgetconstraint.

2. The amount of each good that is supplied equals the amountdemanded (markets clear).


Each consumer will choose their optimal point on the price line (itis just like a budget line, passing through the endowment point).Prices will adjust so that markets clear (there is no simple model

to describe how this happens):

When there is excess demand for food (the quantity demandedexceeds the quantity supplied), price of food relative to clotheswill rise.

When there is excess supply of food (the quantity supplied exceedsthe quantity demanded), price of food relative to clothes will fall.


Equilibrium is where two indifference curves are tangent to a linethrough the endowment point (there is not necessarily anequilibrium, and there may be more than one).

With convex indifference curves and positive endowments of bothgoods there is at least one equilibrium.

Below is an example of three equilibria.


OJ

OK

James’ food

James’clothes

Karen’sclothes

Karen’s foodThree equilibria


Exercise: If the endowment point is Pareto efficient, how manyequilibria are there?


If the slope of the price line is different from the slope of theindifference curves at a point of tangency between the indifferencecurves, then the price is not consistent with equilibrium (if thatpoint is interior).

For example, suppose the price of food is 1/2 and the price ofclothing is 3 (so that 1 unit of clothing can be exchanged for 6units of food).

Then James (2 units of food for 1 unit of clothes), is willing to sellclothes at this price – he would get 6 units of food for a unit ofclothes, whereas he only needs to get 2). However, Karen (3 units

of clothes for 1 unit of food) is not willing to exchange food forclothes at this price. She would need to get 3 units of clothes for aunit of food, but only gets 1/6 units of clothes per unit of food.


A price like this is not consistent with equilibrium, because theamount of food supplied is greater than the amount of fooddemanded.

The relative price of food will fall, until amount demanded equalsamount supplied.


OJ

OK

James’ food

James’clothes

Karen’sclothes

Karen’s food


The allocation in a competitive equilibrium with no externalitiesand complete information is Pareto efficient – First theorem ofwelfare economics.

Sketch of proof:

1. At a competitive equilibrium, each consumer is maximizingutility over his/her budget set.

2. So to make one consumer better off, the allocation must bemoved outside that consumer’s budget set.

3. But then we will be in the interior of the other consumer’sbudget set.

4. By nonsatiation, this is worse for this consumer than the bestallocation on the budget line. So no one can be made better offwithout the other being made worse off.


Summary of results:

At an interior solution,

1. Because the indifference curves are tangent, all consumers’marginal rates of substitution at the equilibrium allocation areequal.

2. Because each indifference curve is tangent to the price line,each person’s MRS of clothing for food (how much food willing togive up for an additional unit of clothing) equals the ratio of thetwo prices (PC/PF ).

The MRS of food for clothing equals PF/PC .


Efficiency in production

Now consider the efficient use of inputs in production.

Assumptions:

- two inputs, labor and capital, in fixed quantity

- two outputs, food and clothing

- Consumers own the inputs and earn income by selling them.


To find how to combine inputs efficiently, we must know thedifferent combinations of inputs that can produce a givencombination of both outputs.

An allocation of inputs to the production process is technicallyinefficient if an output can be increased or an input decreasedwithout changing the other levels of inputs and outputs.

An allocation of inputs to the production process is technicallyefficient if it is not technically inefficient.


Example of an inefficient allocation of inputs:

Suppose that a firm produces paper and cardboard using labor,wood and machines.

The firm is producing 1000 units of paper and 500 units ofcardboard per day, using 20 units of labor, 20 machine hours and200 units of wood.


But suppose that labor and machines are not being combined inthe most effective way, because people are not being matched tothe machine that they best know how to operate.

If machines and workers were combined optimally, the sameoutput could be produced using 18 units of labor, 15 machinehours and 200 units of wood.

Then the firm’s allocation is technically inefficient. It is producingat a higher cost than it needs to, using more resources thannecessary.


The production possibilities frontier shows all combinations offood and clothes that can be produced with fixed inputs of laborand capital (with fixed technology).

Usually the production possibilities frontier is bowed out. Thismeans that society must give up more and more clothes for anadditional unit of food as the amount of food increases – as morefood is produced, resources that are less suited to producing foodare used.

The marginal rate of transformation (MRT) is the magnitude ofthe slope of the production possibilities frontier.

MRTS, MRS and MRT can all be referred to as generalmarginal rate of transformation (GMRT).


A characterization of Pareto efficient allocations:

Consider all pairs of goods and all pairs of agents. If agent i has ahigher GMRT of food for clothes than another agent, then agenti’s trade in clothes is zero (if the agent is a firm, they do notproduce it; if the agent is a consumer, they do not consume it).

If agent i has a higher GMRT of labor for capital than anotheragent, then agent i’s trade in capital is zero (if the agent is a firm,they do not use it; if the agent is a consumer, they do not supplyit).

In other words, all GMRTs over any two goods are equal unlessthe outcome is a corner solution.


Now we want to show that under certain conditions, a competitiveequilibrium in an economy with production is Pareto efficient.

Competitive equilibrium in an economy with production satisfiesthe following conditions:

1. Firms choose outputs and inputs to maximize profits.

2. Consumers maximize utility subject to their budget constraint,which is determined from prices and their wealth obtained fromselling labor and capital and getting profits (consumers own thefirms).

3. The input and output markets clear.


Assume that each consumer has at least one good that is desirable– more of that good is always better. Consider the net trades ofeach consumer.

When the consumer sells a good, it is counted as a negativequantity. When the consumer buys a good, it is counted as apositive quantity.

The value of the net trade of a consumer equals the price of eachgood times the consumer’s net trade in the good, added up overall the goods (this is either zero or positive, because we assumefor now they can’t borrow).

For the value of the net trade to be positive, the consumer mustreceive positive profits from the firms – this is their only othersource of income.


The value of the net trade of a consumer equals the profit thatthe consumer receives from firms.

The sum all the values of net trade over all consumers equals thesum of the profits over all the firms.

Start with an equilibrium allocation A. Consider a Paretosuperior allocation A’. This new allocation has to be moreexpensive than A, using the assumption of at least one desirablegood for each consumer.

We want to show it is impossible with given technology andendowments for the economy to produce A’.


Because A’ is more expensive than A, the sum of the values of thenet trades for A’ is greater than that for A.

Thus the profits if A’ is produced must be greater than the profitsif A is produced.

But that is impossible as the firms were maximizing profit tobegin with (since they are in competitive equilibrium).

So a competitive equilibrium allocation in a production economyis Pareto efficient.


But a Pareto efficient allocation need not be an equilibrium.Consider the following one-consumer, one-producer example.Assume that both the producer and the consumer are

price-taking, that is, they behave competitively.

There is one consumption good and one input, labor (or leisure).The consumer sells its labor (leisure) to the firm and buys theconsumption good from the firm.

The consumer own the firm and receives the firm’s profits (thismay seem inconsistent with the assumption that they areprice-taking, but we can imagine that there are many replicationsof the consumer and of the firm).


Leisure

Indifference curve

Production possibilities frontier

Consumptiongood


The above allocation A is Pareto efficient, because the consumeris at the optimal point on the production possibilities frontier andthus cannot be made better off.

But A is not an equilibrium for any set of prices. Given anybudget line not tangent to the indifference curve passing throughA, there is a better point for the consumer on the budget line.

The budget line tangent to the indifference curve passing throughthat allocation is not consistent with profit maximization.


Explanation:

A budget line shows all points that give the firm a particularprofit level. Any line with slope −w/p, where w is the wage rateand p the price of the output, is an isoprofit line.

As the firm moves up and to the left on a budget line, it buysmore labor and sells more output.

The value of the additional output sold equals the additional wagepayment, because if labor is increased by 1 unit, the output isincreased by w/p units. Selling that additional output at price pgives the firm additional revenue equal to w.

So the change in profit is zero.

Thus, at the given prices there are points of production close to Athat give the firm higher profits. There are no prices consistentwith equilibrium at A.


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