Relationships between Markets

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 22

Unit IV: Equilibrium

This Week:
General Equilibrium

Analyze a single market, taking everything determined outside that market as given

(prices of other goods,
consumer income, wages)

Last Week:
Partial Equilibrium

Today: examine linkages between markets

Analyze all markets simultaneously

Thursday: solve for equilibrium quantities in all markets simultaneously as a function only of production functions, resource constraints, and consumer preferences. 

(endogenize all prices, income, wages)

Today's Agenda

Part II: Supply Effects

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

Part I: 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

Section/Video: Resource Allocation

Demand Effects

Suppose two goods are complements.

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

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?

pollev.com/chrismakler

Why did the wage rate go up in this model?

“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.)

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.

Lecture Summary

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.

Homework for today's class is due Friday night.
Homework for Thursday's class will be due Sunday night.
Section this week is very important!

Econ 50 | Spring 2023 | Lecture 22

By Chris Makler

Econ 50 | Spring 2023 | Lecture 22

Relationships Between Markets

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