General Equilibrium

Dr. Andrew McGeeSimon Fraser University

Interconnectedness of markets

• Suppose new research is released showing that wine consumption is good for one’s overall health.

• Demand for wine increases. Price of wine increases. Marginal revenue product of wine workers increases. More workers and firms enter this market.

• Cloth, used in wine production, also sees its demand increase, and similarly its price will go up and the marginal revenue product of workers in the cloth industry will go up.

• The new firms and workers entering these industries, however, come from somewhere—namely other markets. Hence the developments in one market will affect many other markets.

General equilibrium model

• Models all markets at once• Markets include not just markets for

consumption goods but factor markets (e.g., capital and labor)

• In the equilibrium of a general equilibrium model, Qs=Qd for all markets simultaneously– There is some vector of equilibrium prices such

that all markets clear

Assumptions of general equilibrium model

• All markets are perfectly competitive (i.e., selling homogenous products, many buyers, many sellers, zero transaction costs, firms are price-takers)– Law of one price must hold

• Consumers maximize their utility• Firms maximize their profits

General equilibrium model with 2 goods

• Assume that all individual have identical preferences concerning goods X and Y.

• Assume that there are fixed amount of inputs (capital and labor) available for the production of X and Y.

• Edgeworth box: represents all possible allocations of inputs to the production of X and Y with dimensions given by the total amounts of capital and labor available

Edgeworth box

Totalcapital

Total labor0x

0y

Where point A is in the Edgeworth box tells you how much labor and capital are being devoted to the production of X and how much to Y

A

Labor in X production

Capital in X production

Labor in Y production

Capital in Y production

At any point in the Edgeworth box, all of the capital and labor in the economy is fully employed

All possible allocations of K & L are depicted in the Edgeworth box

Efficient allocations

• Many allocations of K & L to the production of X and Y are inefficient insofar as more of both X and Y could be produced by allocating K & L to the production of X and Y differently

• Assume that perfectly competitive markets result in efficient allocations of inputs

• Overlay isoquant maps on top of the Edgeworth box

Efficient allocations

0x

0y

Red: isoquants for YBlue: isoquants for X

A

Points such as A are clearly inefficient because more X and more Y can be produced given society’s endowments of K & L.

The green line represents the locus of all efficient allocations of K & L—allocations at which there exists no allocation at which more X and more Y can be produced relative to the current outputs. Notice that the isoquants are tangent to each other at these efficient allocations, meaning that MRTSx=MRTSy.

Contract curve

Production possibility frontier (PPF)

Y

X

0x

0y

Note that at each efficient allocation in an Edgeworth box, there are corresponding output levels of X & Y. The graph below depicts those output levels. Note that the bottom left-hand corner of an Edgeworth box corresponds to the case in which no X is produced and only Y is produced.The production possibility frontier shown here depicts the alternation combinations of two outputs that can be produced with fixed quantities of inputs if those inputs are employed efficiently. The output bundle A corresponds to the inefficient input combination A in the Edgeworth box.A

The slope of the PPF reflects the rate at which X can be substituted for Y when total resources are held constant. For instance, the slope might be -1/4 near 0x, meaning 1 unit of Y can be sacrificed in order to produce 4 more units of X. Near Oy, this slope might be -5, meaning that 5 units of Y must be sacrificed to produce 1 more unit of X. The negative of the slope is known as the rate of product transformation (RPT).

Shape of the PPF

• The PPF on the last slide exhibits an increasing RPT. The sacrifice required in terms of Y given up increases as you try to produce more X (and vice versa).

• The RPT can be expressed as the ratio of marginal costs of X and Y:

• Suppose labor is the only input and the marginal unit of X requires 1 unit of labor and the marginal unit of Y requires 2 units of labor. Then one must give up 2 units of X to get 1 more unit of Y.

• Why should we expect the ratio of MCx to MCy to increase as X output expands and Y output contracts?

Diminishing Returns

• If the production processes of both X and Y exhibit diminishing returns to scale, then increasing X output will increase MCx, while reducing Y output will decrease MCy, which would cause the production function to exhibit increasing RPT.

• But economists don’t find DRS theoretically compelling….

Specialized inputs

• Suppose some inputs are better suited for X and others better suited for Y (e.g., marshlands for the production of rice and prairies for the production of wheat). Producing more X requires using more inputs ill-suited for the production of X and better suited for the production of Y and vice versa.

• But we have assumed inputs are homogenous (i.e., land is land is land).

Differing factor intensities

• If the two output goods use K & L in different proportions, then the PPF will be bowed outward

• Suppose you produce x1 and y4 using one allocation of inputs K & L and x3 and y2 using another allocation of inputs.

• An output combination between these two output combinations is given by

Why is the PPF bowed outward?Y

X

0x

0y

y4

y2

x1 x3

𝑥1+𝑥32

,𝑦4+𝑦22

=𝐵

If we assume CRS, then this output combination is possible (because ½ the inputs used to produce x1 will produce ½ of x1 and likewise for the other output levels). We also know that all K& L will be used at this new allocation (though we can’t see that in this graph).

B

We know that bundle B uses ½ of the K used in x1 and ½ the K used in x3 to produce the new amount of X and similarly ½ the L used in x1 and ½ the L used in x3 to produce the new amount of X. Similarly for the amounts of K & L used to produce Y.

Why is the PPF bowed outward?

0x

0y

x1y4

x3

y2

This input bundle, C, involves using ½ the K used to produce output combination P1 and ½ the K used to produce the output combination P2.

P1

P2

Notice that C is not on the contract curve; it is not efficient. It is possible to produce more of both X and Y. Thus there must exist output bundles to the northeast of B on the PPF in the preceding slide. If the K/L mix (factor intensity) does not vary between production of X and Y, the contract curve and PPFs will be linear

CTotalcapital

Total labor

PPF & Opportunity cost

• For a society with finite resources, producing more X requires producing less Y if all resources are fully employed.

• The slope of the PPF gives the opportunity cost of producing an additional unit of X in terms of units of Y foregone and vice versa

Equilibrium prices

• Given that society faces tradeoffs in the production and consumption of X and Y, how does it decide how much X & Y to produce (and consume)?

• The prices of X and Y must adjust so that the quantity demanded of each equals the amount produced by the society.

Determination of equilibrium prices

Y

X

0x

0y

U4U3U2

U1

PPF

At price ratio px/py, firms produce (x1, y1), but consumers demand (x1’, y1’)—supply does not equal demand and markets do not clear.

x1

y1

x1’

y1’

px/py

x*

y*

px*/py*

Why is the tangency between the PPF and the price ratio the profit maximizing choice for firms? Because at this bundle the price ratio equals the ratio of marginal costs (all firms are earning zero profit by assumption).

Note that the red lines here are budget constraints. X and Y sell for pxX+pyY, but in this economy the producers are also the consumers, so whatever they earn in revenues is returned to consumers through wages and rents from capital.

Given that these red lines are budget constraints, the point of tangency with the indifference curves is the utility maximizing consumption bundle (from 1st ½ of course).

Determination of equilibrium prices

Y

X

0x

0y

U4U3U2

U1

PPFx1

y1

x1’

y1’

px/py

x*

y*

px*/py*

At px/py, there will be excess demand for X and excess supply of Y. Markets do not clear. In order for the markets to clear, the price of X must go up, and the price of Y must go down (or some combination) such that the price ratio gets steeper. A higher price for X both reduces demand for X and increases its supply.

At px*/py*, the supply of X equals the demand for X and similarly for Y. This is a general equilibrium: all markets clear simultaneously.

Persistence of general equilibria

• Consistent with the definition of an equilibrium, a GE will tend to persist unchanged without a change in the underlying economic fundamentals

• The economic fundamentals that could lead to a change in the equilibrium prices and quantities are preferences and technologies

Technological Change

Y

X

0x

0y

U2

PPF1PPF2

U3 Here, producers get better at producing X as evidenced by the fact that they can produce more X along with any given amount of Y still using the same fixed quantities of K & L. This is what technological progress looks like!

Notice that the market clearing price ratio changes following this technological change. Society gets better at producing X; its relative price falls in equilibrium.

Change in preferences

Y

X

0x

0y

U2

Original preferences Suppose people decide that they like X

less. This would lead to “flatter” indifference curves. Notice that the price of X declines relative to that of Y in equilibrium (the price ratio gets flatter) and less X is consumed.

New equilibrium price ratio

Old equilibrium price ratio

xoldxnew

yold

ynew

New preferences, U2’

A simple general equilibrium model:Production

• Suppose that the production functions for goods x and y depend only on labor:

, • Suppose that there are 100 units of labor available

in the economy:

• Implies the production possibility frontier:

• Producers maximize profits

A simple general equilibrium model:Consumption

• Consumers’ utility is given by: • Implies demand functions:

• Consumers maximize their utility subject to their budget constraint

• How much do consumers have to spend? Labor income plus profits

A simple general equilibrium model:Equilibrium

• Utility maximization requires:

• Profit maximization requires

• Given the PPF ,

• In equilibrium, , so y=x in equilibrium• Implies that and .• Note that an equilibrium is defined by an

allocation (x*,y*) and a price vector (px,py)

A simple general equilibrium model: Consumers’ budget constraint

• Total income=labor income + profits• Labor income=100w• What are profits? Consumers are the producers in

this economy, so we just need to calculate profits:

• Because , , so profits=100w• Total labor income=200w• Can verify that this income will allow consumers to

spend 100w on each good to get units of each at a price of

Corn Laws• 19th century British laws placing high tariffs on

grain imports (1815-1846)• Served the interests of landowners, who

benefits from higher grain prices• Disliked by manufacturers• Assuming Corn Laws prevented trade (which

they didn’t), repealing the Corn Laws would have an unambiguously positive effect on consumers, but what about the effects on factor owners (landowners and capitalists)?

An argument for trade

Y

X

0x

0y

PPF1

U1

U2

AE B

yAyE

yB

xA xE xB

In the absence of trade, a society is limited to consuming only those bundles which it can produce. In this economy under autarky (i.e., a no trade regime), the GE would occur at point E. If trade with other countries/economies is allowed, then this country could consume bundle B while producing bundle A. It would be a net exporter of good Y while being a net importer of good X.

Winners & losers from trade?

0x

0y

x1y4

x3

y2

P1

P2

The movement from E to A corresponds to a move from P2 to P1;it causes the K to L ratio to increase in both industries. This means that the relative price of capital must fall (or the relative price of labor increases).

Total labor Thus in this simple, stylized model, trade benefits laborers while hurting capital owners.

Totalcapital

Total labor

Walrasian Equilibrium

• Suppose there is no production in the economy. The economy is just endowed with a certain amount of goods X and Y.

• Individuals hold endowments of X & Y and then trade among themselves.

• Once trade among individuals occurs, a Walrasian equilibrium is an allocation of resources and an associated price vector (i.e., set of prices for all goods in the economy) such that the demand for each good equals the economy’s total endowment for that good (i.e., markets clear).

Trading: how do we get from an initial allocation to an equilibrium allocation

0B

0A

𝑥=10

𝑦=6

(4,4)

(8,2)

Price vector px/py has slope -1/2: you can trade on unit of x for 2 units of y

Initial allocation

Equilibrium allocation

Here, A gives B 4 units of X in exchange for 2 units of Y. In this sense the price vector defines the trades that can be made and the set of possible final allocations

Mathematical model of exchange (no production)Vector notation

• Bold letters denote vectors, arrays of values:

• Vector addition is only defined for vectors with the same dimension (n x 1)

• Dot product of two vectors (like multiplication):

Mathematical model of exchange (no production)

• In this model of exchange: n goods, m individuals• Individual i’s utility from consumption:• where is the vector indicating how much of every

one of the m goods consumer i consumes• Individuals are born with some endowment of

each of the n goods: • Individuals are free to exchange their

endowments among each other

Exchange

• Individuals are price-takers: they face a price vector indicating the prices of each good which dictate the exchanges that individuals can make (as we saw a few slides ago)

• Consumer’s budget constraint: Total value of purchases cannot exceed value of endowment

• Earlier in the course we saw that demand functions are homogeneous of degree 0:

Walrasian Equilibrium

• A Walrasian equilibrium in an economy with m consumers is an allocation of resources and an associated price vector such that

Normalizing prices• The fact that demand is homogeneous of degree

0 in prices means that the scale of price is arbitrary, so we will assume prices are normalized so that they sum to one.

• That is, suppose prices are given by . We can always deflate these prices in the following manner to get a new set of n prices that sums to one:

• Notice that relative prices are unaffected by such re-scaling:

Existence of a general equilibrium

• We have not yet shown that an equilibrium price vector exists. That is, we must show that there exist prices such that

• That is, given prices there are no excess demands for any of the n goods

Existence “proof”

• Let denote the set of excess demands for the n goods. We want • Suppose we start out an arbitrary set of normalized prices, .

Define a new set of normalized prices, , as

• Note that all prices in and are between 0 and 1 because of the normalization assumption.

• Because the demand functions are continuous, Brouwer’s fixed point theorem implies that there exists a price vector such that

meaning that at some price vector the excess demands will be zero (because it must be true in the above from the fixed point theorem that )

Fixed point theorem

1

1x

f(x)

0

45-degree linex=f(x)f(x*)

x*

Fixed pointAny continuous function that maps from a closed compact set (here the interval [0,1]) onto that same set ([0,1]) will have a fixed point where x=f(x).

Existence of GE proof

• Existence proof relied on a number of non-trivial assumptions:

1. Price-taking by all individuals2. Homogeneity of degree 0 demand functions3. Continuity of demand functions4. Budget constraints5. Walras’ law: value of all quantities demanded

equals value of all endowments

First Fundamental Theorem of Welfare Economics

• FFWT: Every Walrasian equilibrium is Pareto efficient

• Pareto efficient allocation: An allocation of the available goods in an exchange economy is Pareto efficient if it is not possible to devise an alternation allocation in which at least one person is better off and no one is worse off.

• FFWT does not imply that every Walrasian equilibrium is socially desirable.

FFWT• Given initial endowments, individuals trade. They move

from their initial endowment to some final allocation of goods. The equilibrium that emerges (at the equilibrium prices) will be Pareto efficient insofar as none of the traders can be made better off without making another trader worse off.

• Pareto efficient equilibria can be highly inequitable (points A or C, for instance)

• The location of the WE depended substantially on the initial endowment, E

• The initial endowment constrains the set of possible WE (between A & C). There may exist other far more socially desirable but not attainable Pareto efficient allocations.

FFWT in a 2 person, 2 good exchange economy

𝑥𝐵𝑥𝐴

𝑥

𝑦𝐵

𝑦 𝐴

𝑦

0B

0A

Initial endowment

Px/Py

Walrasian equilibrium (WE): after trade, A and B end up here given prices Px and Py

Notice that the WE is Pareto efficient (FFWT)

A

C

Region between A & C is the Pareto-improving set.

Proof of FFWT• Suppose generates a Walrasian equilibrium in which allocation of the n goods to the

kth consumer is given by (k=1,…,m). • Now assume that there exists another allocation (k=1,…,m) such that for at least one

person i, is preferred to . For this person, it must be true that (1)

Meaning that this bundle is more expensive than the equilibrium bundle, because otherwise this person would have purchased in the first place• If all other individuals are equally well off under , it must be true for them that

(2)If not true, these other individuals could not have been minimizing their expenditures at • To be feasible, the new allocation must obey the quantity constraints:

• Multiplying the above by , we get:

• (1) and (2) along with Walras’ law applied to the original equilibrium imply that

• This is a contradiction; therefore no such alternative allocation exists. We have shown that any Walrasian equilibrium is also Pareto efficient.

Second Fundamental Theorem of Welfare Economics

• SFWT: For any Pareto optimal allocation of resources, there exists a set of initial endowments and a related price vector (Px/Py in the 2-good case) such that this allocation is also a Walrasian equilibrium

• Implication: if you can adjust initial allocations (i.e., through taxes and other forms of coercion), then any Pareto efficient allocation can be sustained as a Walrasian equilibrium

SFWT in a 2-good, 2-person exchange economy

𝑥

𝑦𝐵

𝑦 𝐴

𝑦

0B

0A

Initial endowment

Px/Py

A

C

Q

Q is thought to be more socially desirable than any WE between A and C

If one can re-allocate goods from the initial endowment to any point such as Z on the price line passing through the tangency between UA and UB, then Q can be sustained through trade as a WE

Z

UA

UB

Example of SFWT in a 2-person (A & B) exchange economy

• Suppose and and • For any given amount of x and y consumed by B, A

maximizes his/her utility by solving the Lagrangian: or

• Note that we only need to solve for one person’s maximization problem as knowing A’s allocation determines B’s allocation

Example of SFWT in a 2-person (A & B) exchange economy

• The first order conditions for a maximum are:

• Dividing one FOC by the other yields:

• This equation determines all Pareto optimal allocations, because the FOC maximize A’s utility subject to the constraint that B cannot be made worse off than some reference point.

Example of SFWT in a 2-person (A & B) exchange economy

• Suppose—for whatever reason—we want to split x evenly between A and B: Then the equation at the bottom of the previous slide implies that in order for the allocation to be Pareto optimal. At this allocation, .

• To find the equilibrium price ratio at this allocation, we solve for the MRS for A & B:

• As expected, the MRS are equal at this allocation and in equilibrium

• To show the SFWT, I need only show that there exists some initial allocation that would lead to this equilibrium allocation and price vector after trade

Example of SFWT in a 2-person (A & B) exchange economy

• This equilibrium price ratio allows them to trade 8 units of y for every 10 units of x. You have a point on the line (the equilibrium allocation) and the slope (the equilibrium price) so any allocation on the line defined by this information will work as an initial allocation that would lead to this equilibrium.

• Suppose the initial endowments are • To get to (500,200), A trades B 120 units of Y in

exchange for 150 units of X. That is, for every 8 units of Y A gives up, he/she receives 10 units of X as expected.

Social welfare functions• There are many Pareto-efficient allocations available in an economy. By

the SFWT, we can achieve any of these allocations with appropriate adjustments to initial allocations.

• How do we choose among these Pareto-efficient allocations?• Imagine there were a “social planner” whose goal is to choose the “best”

allocation. This planner maximizes “social welfare”:

• The social welfare function ranks possible allocations according to the planner’s “preferences.”

• What are the planner’s preferences?– Surely any welfare maximum must also be Pareto efficient.– Does the planner wish to minimize inequality by ranking low allocations with

extreme allocations (i.e., some people get a lot, others get little)? Should the planner only care about the poorest individual given an allocation? Or should the planner prefer allocations that result in old people getting a lot, young people getting a little? Or should the planner only care about the sum of the individual utilities? And so on…

Insights from GE models

• In economic models, we take preferences and production technologies as given. With these in hand, all prices are necessarily determined.

• Firms and productive inputs are owned by households. All income (profits and returns to inputs) accrues to households.

• Bottom line in policy evaluations is the utility of households. Firms and governments are just intermediaries.

• All taxes distort economic decisions along some dimensions. The welfare costs of such distortions must be weighed against the benefits of such taxes (whether from public good provision or equity-enhancing transfers)