Relationships between Markets and General Equilibrium

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 15

Where We Are

Today:
General Equilibrium

Monday:
Autarky

Given resource constraints, production functions, and utility functions, solve for the bundle the market would "choose" to produce in competitive equilibrium.

(endogenize all prices, income, wages)

Given resource constraints, production functions, and utility functions, solve for the bundle a single agent would choose to produce and consume. 

Where We Are

Today:
General Equilibrium

Given resource constraints, production functions, and utility functions, solve for the bundle the market would "choose" to produce in competitive equilibrium.

(endogenize all prices, income, wages)

Part I: Investigate relationships between markets

Part II: The circular flow and general equilibrium

Part I: Investigate Relationships between Markets

Supply Effects and the PPF

How consumer’s utility functions
(treating goods as complements or substitutes) determine how a shift
in the supply of one good
affects other markets

Demand Effects

How profit-maximizing firms choose
the point along the PPF that maximizes GDP 

How firms’ demands for resources
determine how a shift
in the demand for one good
affects other markets

Demand Effects

Suppose peanut butter and grape jelly
are complements.

What happens in both markets 
if there is a supply shift
in the market for one of the goods?

For example: suppose much of the grape crop is destroyed in a fire.

Grape Jelly

Peanut Butter

Before we get to markets: WWCD?

Equilibrium in One Market

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1)
D_1(p_1,p_2)
=
S_2(p_2)
D_2(p_1,p_2)
=
S(p)
D(p)
=

Market for Good 2

Market for Good 1

Equilibrium in Two Markets with Related Demand

S_1(p_1) = {p_1 \over a}
D_1(p_1,p_2) = {m \over p_1 + p_2}
S_2(p_2) = {p_2 \over b}
D_2(p_1,p_2) = {m \over p_1 + p_2}

Supply Effects

Consider two goods (“guns” and “butter”) which are unrelated
but which both use the same resource (e.g. labor) in production.

What happens in both markets 
if there is a demand shift
in the market for one of the goods?

Why did the wage rate go up?

“Chain of Causality"

  • Increase in demand for good 1:
    • Movement up along the supply curve for good 1 → produce more good 1, increase \(p_1\)
    • Increase \(p_1\) → outward shift of the demand for labor from firms producing good 1
  • Increase in labor demand → equilibrium wage increases for all firms
  • Increase in wage affects supply of good 2:
    • Movement up along the labor demand curve for firms producing good 2
    • Inward shift of the supply curve for good 2 → produce less good 2
  • Secondary effect: increase in wage also reduces supply of good 1
  • Cumulative effect:
    • Produce more good 1 and less 2.
    • Use same amount of labor in total → stay along the PPF

Notation

\(Y_1\) = total amount of good 1 produced by all firms in an economy

\(Y_2\) = total amount of good 2 produced by all firms in an economy

\(GDP(Y_1,Y_2) = p_1Y_1 + p_2Y_2\)
= market value of all final goods and services produced in an economy

Resource Allocation

Narrow question: 
How many productive resources should we devote to a single good?

Broader question:
How should we allocate productive resources across goods?

Firms will choose the quantity at which \(p = MC\)

Firms will choose the point along the PPF at which \({p_1 \over p_2} = MRT\)

GDP maximizing point!!

Firms will choose the point along the PPF at which \({p_1 \over p_2} = MRT\)

How will they do this?

In this lecture, we'll show:

MRT = {MC_1 \over MC_2}
p_1 = MC_1
p_2 = MC_2
=MRT

PROFIT MAX FOR GOOD 1

PROFIT MAX FOR GOOD 2

Input prices signal resource constraints, keep production on PPF.

MRT = {MC_1 \over MC_2}

Case 1: Labor is the only input

c(q) = w \times L(q)
MC = {dc \over dq} = w \times {dL \over dq}
MC = w \times {1 \over MP_L}
MP_L = w \times {1 \over MC}
MP_{L2} = w \times {1 \over MC_2}
MP_{L1} = w \times {1 \over MC_1}
\displaystyle{{MP_{L2} \over MP_{L1}} = {MC_1 \over MC_2}}
MRT = {MC_1 \over MC_2}

Case 2: More than one input

c_1(Y_1) + c_2(Y_2) = \overline C

Let's write the market value of all resources in the economy as \(\overline C\).

Can therefore write the PPF as the set of all possible combinations of output, \((Y_1,Y_2)\), such that

By the implicit function theorem,

|\text{slope of PPF}| = {c_1^\prime(Y_1) \over c_2^\prime(Y_2)}

How do firms
maximize GDP?

For a given set of prices \((p_1,p_2)\), what combination of outputs \((Y_1,Y_2)\) on our PDF would maximize GDP?

\(GDP(Y_1,Y_2) = p_1Y_1 + p_2Y_2\)

(Assume labor is the only input.)

What is the slope of an iso-GDP line?

Conditions for
GDP Maximization

{p_1 \over p_2} = {MC_1 \over MC_2}

TANGENCY CONDITION

CONSTRAINT CONDITION

L_1(Y_1) + L_2(Y_2) = \overline L

Firms in industry 1 set \(p_1 = MC_1\)

Firms in industry 2 set \(p_2 = MC_2\)

How does competition achieve this?

Wages adjust until the
labor market clears

Main Takeaways

Another expression for the MRT
is the ratio of marginal costs:

MRT = {MC_1 \over MC_2}

Given prices \(p_1\) and \(p_2\), GDP is maximized at the point on the PPF where

MRT = {p_1 \over p_2}

Profit-maximizing firms,
acting in their own self-interest,
respond to prices by producing the
GDP-maximizing combination of outputs.

Summary of Part I

Markets are interrelated,
both because consumers buy multiple goods
and multiple firms compete for the same resources (e.g. labor).

Profit-maximizing firms,
acting in their own self-interest (not coordinating!),
respond to prices by "choosing" the point along the PPF where MRT = price ratio.

Changes in the price ratio cause firms to shift along the PPF,
toward the good whose relative price has increased
and away from the good whose relative price has decreased.

Part II: The Circular Flow and General Equilibrium

  • The Circular Flow Model
  • Conditions for General Equilibrium
  • Equilibrium and Disequilibrium

The Circular Flow

In our consumer theory, we've treated income as exogenous.

In our producer theory, we've treated wages as exogenous.

We've also assumed firms are maximizing profits, but haven't said where those profits go.

Crazy thought: what if the money firms pay for labor becomes the income of workers?

...and their profits become the income of the owners/shareholders of the firm?

Consumers

Good 1 Firms

Market for Good 1

Market for Good 2

Market for Labor

Good 2 Firms

Money flows clockwise

Goods, labor flow counter-clockwise

General Equilibrium: Everyone optimizes, all markets clear simultaneously.

Competitive Equilibrium

Review: Autarky (Chuck on a desert island)

We sometimes call the autarky model the "centralized" model: if there were a single agent making a decision, what would they do?

Similarly, we call competitive equilibrium a "decentralized" model, because lots and lots of individuals are making small decisions that add up to what "society chooses"

1. Given prices \(p_1,p_2\), firms will choose the point \((Y_1^*,Y_2^*)\) along the PPF where \(MRT = \frac{p_1}{p_2}\)

2. All money received by firms \((p_1Y_1^* + p_2Y_2^*)\) will become income \(M\) for consumers.

3. Given prices \(p_1,p_2\) and income \(M\), the consumer will choose the point \((X_1^*,X_2^*)\) along the budget line where \(MRS = \frac{p_1}{p_2}\)

4. At equilibrium prices, markets clear (\(X_1^* = Y_1^*\) and \(X_2^* = Y_2^*\)) so \(MRS = MRT\).

5. In disequilibrium, there is a shortage in one market and a surplus in the other, pulling the system toward equilibrium.

Overview of General Equilibrium

1. Given prices \(p_1,p_2\), firms will choose the point \((Y_1^*,Y_2^*)\) along the PPF where \(MRT = \frac{p_1}{p_2}\)

2. All money received by firms \((p_1Y_1^* + p_2Y_2^*)\) will become income \(M\) for consumers.

3. Given prices \(p_1,p_2\) and income \(M\), consumers will choose the point \((X_1^*,X_2^*)\) along the budget line where \(MRS = \frac{p_1}{p_2}\)

Equilibrium in and Disequilibrium in the Short Run

MRS =
p_1 = MC_1
p_2 = MC_2
= MRT

If consumers and firms all face the same price, and if they choose the same quantity in response to that price, then MRS = MRT.

u(X_1,X_2) = \alpha \ln X_1 + (1-\alpha) \ln X_2

Key Takeaways

In general equilibrium, everything having to do with money has been endogenized.

 

We are left with the same things Chuck had on his desert island:
resources, production technologies, and preferences.

As an individual in autarky, Chuck solved his maximization problem by setting
the marginal benefit of any activity he undertook equal to its opportunity cost.

Markets solve the problem of how to resolve scarcity in the same way:
by having everyone equate their own MB or MC to a common price,
which represents the opportunity cost of using resources in some other way.

Econ 50 | Fall 2022 | 15 | General Equilibrium

By Chris Makler

Econ 50 | Fall 2022 | 15 | General Equilibrium

Bringing supply and demand together

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