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MT1 General Equilibrium Notes

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These notes were prepared based on the lectures and supplemented by information from textbooks and tutorials where parts of the lecture were unclear. Graphs, equations, and bullet-point explanations included. Prepared by a first class Economics and Management student for the FHS Microeconomics pape...

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  • June 27, 2024
  • 32
  • 2022/2023
  • Class notes
  • Simon cowan
  • Mt1 general equilibrium
  • Unknown
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MT1 Microecons (General Equilibrium)
What would we like to know?
 What is an equilibrium when there are multiple markets, and what does it look like?
 How do goods markets interact with factor (input) markets?
 Can we ensure that an equilibrium exists, that it's unique, and that there's a reasonable process
for getting there?
 What are the welfare properties of an equilibrium?

We will look at
 2-person exchange economies with two goods
 Robinson Crusoe economies: one person, who both produces and consumes
 Two issues in international trade: how changes in goods prices affect factor prices, and how
changes in resource endowments affect production patterns
 How opening up an economy to international trade expands consumption possibilities and leads
to some specialization in production
 How to find a general equilibrium, and some thoughts on existence and uniqueness

Lecture 1: General Equilibrium Introduction
Outline
 The Edgeworth box
 General equilibrium framework
 Existence of competitive equilibrium in exchange economy
 Extension to production economy with one input factor and one good

From partial to general equilibrium
 How price change of one product affects demand and supply for all other products
 An economy is in general equilibrium if every market in the economy is in partial equilibrium
(including goods and factor markets)

General Equilibrium Model components
 Households: demand goods, supply inputs (including capital via investment like share
ownership) and own firms.
 Firms: hire inputs, produce goods, and earn profits which are distributed to households.
 Markets where goods and inputs are traded.
 General Equilibrium is also called Competitive Equilibrium and Walrasian Equilibrium.

Assumptions
 No frictions: no externalities or distortionary taxes
 Price-taking: no market power

Exchange Economy
 Exchange Economy: households/consumers endowed with consumption goods, which they
trade among themselves. There is no production.
 Setting it up:

, o Two goods: x1 and x2
o Two agents: Alice and Bob
o Alice and Bob have endowments of goods
o Alice and Bob maximize utility subject to their budget constraints
o They can trade goods between themselves without any distortions, such as taxes or
external costs, taking prices as given
 Notation




o
o Demand depends on prices and income (given by p*w)
o Bold represents vectors
 Alice's problem




o
o Maximise utility subject to consumption value ≤ endowment value
o Dotted lines show initial endowment point
o The constraint is a line through the initial endowment point, with a slope equal to the
price ratio (sell endowment of one good and buy the other good)
o Note how vector notation is used above
 Bob's problem




o

Plotting and solving an Edgeworth box

, Edgeworth box




o
o Oa (origin for Alice) and Ob (origin for Bob)
o The height of the box is the total endowment of w 2 (𝑤2𝑎 + 𝑤2𝑏), while the length of the
box is the total endowment of w 1.
o Hence, the Edgeworth box is constructed such that the endowment points for both Alice
and Bob are at the same point in the box
o The black line is the budget constraint for both Alice and Bob. It cuts through their
endowment point and has the slope of the price ratio
o Blue points show Alice and Bob's optimum consumption (tangency of utility function
and constraint)
 Not an equilibrium: markets do not clear due to a mismatch between demand and endowment
o Good 1: demand exceeds endowment (sum of blue lines exceeds the length of the box)





o Good 2: endowment exceeds demand (sum of blue lines is below the height of the box)





 To ensure markets clear and get to equilibrium, change the goods' relative prices to rotate the
budget line such that total demand = total supply

, o Adjust towards the equilibrium by increasing p1 relative to p2





 This reduces demand for good 1 which currently has excess demand, and
increases demand for good 2 which currently has excess endowment
 The budget line rotates around the endowment point
 Notice the 2 optimal tangency points go closer together
o Markets clear where both Alice and Bob's demand point overlap (purple point)





 Competitive equilibrium is (𝒙*,𝒑*) such that:
 𝒙𝒂 (𝒑*, 𝒑* ⋅ 𝒘𝒂) + 𝒙𝒃 (𝒑*, 𝒑* ⋅ 𝒘𝒃) = 𝒘𝒂 + 𝒘𝒃
 In words, at equilibrium, Demand = endowment for both goods
 The competitive equilibrium vector x* has 4 values (2 for Alice, 2 for
Bob). Competitive equilibrium includes information on prices and
demands.
 Slopes of both utility functions = slope of the budget line
 General main result: Suppose that in an exchange economy, consumers’ preferences are
continuous, monotonic, and convex, and 𝒘> 𝟎 then competitive equilibrium exists!

Assumptions for nicely behaved preferences
 Continuous preferences (“No jumps”)
o No break in utility function
o Differentiable (can compute MRS)
 Monotonic preferences (“More is better”):
o Utility is strictly increasing in both goods
o Binding budget constraint
 Convex preferences (“Combinations are better”)

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