Introductory Microeconomics - Elsevier...Introductory Microeconomics Prof. Wolfram Elsner Faculty of...

Introductory Microeconomics

Prof. Wolfram ElsnerFaculty of Business Studies and Economics

iino – Institute of Institutional and Innovation Economics

The Ideal Neoclassical Market and General Equilibrium

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies

Chapter 05: Ideal Neoclassical Market and General Equilibrium22.07.2014

2

Readings for this lecture

Mandatory reading this time:

The Ideal Neoclassical Market and General Equilibrium, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 97-128.

The lecture and the slides are complements, not substitutes

An additional reading list can be found at the companion website

http://www.amazon.com/The-Microeconomics-Complex-Economies-Institutional/dp/0124115853/ref=sr_1_1?ie=UTF8&qid=1396453322&sr=8-1&keywords=Microeconomics+of+complex+economies

http://booksite.elsevier.com/9780124115859/


Chapter 05: Ideal Neoclassical Market and General Equilibrium

What is a neoclassical method?

To develop a “pure theory” Axiomatic approach Taking as little as possible as given To construct an optimal equilibrium model In a price-quantity world

Neoclassical economics typically ends up in price-quantity spaces employing the mathematics of maximization under restrictions (Lagrange algorithm). Economism in that sense is the idea of a “pure” economy, isolated from the rest of society and from the natural environment.

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Introduction – neoclassical method



What is a model?

Imaginary Analogy Metaphor Relating variables to one another Diverging in the number of variables, agents and relations

Perfect competition and perfect information are assumed at least as a benchmark to which more specific models with imperfect competition or imperfect information are compared

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Introduction - model



… as the study of all interdependent ideal partial markets

Comprehensive formulation of neoclassical economics Simultaneous equilibrium in all partial markets Neoclassical paradigm cannot be tested and rejected

straightforwardly “data” are always selected Stylized facts

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General Equilibrium Theory (GET)…



Fictitious auctioneer (computes market-clearing prices) No direct interdependence (only indirect via the ‘market’)

Walras was the first to emphasize the interdependence of individual partial markets in his Elements of pure economics (1874).

Neoclassical or marginalist revolution Marginal utility (and marginal productivity in production) Scarcity (focus on the allocation of scarce resources)

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GET – Walrasian Economics



Optimization problems Consumer theory Production theory

Utility/ profit maximization problems

Cost minimization problems

Partial and general equilibrium

Limitiations Theory of the second best, Markets of lemons

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Overview



All agents (firms and households) have all relevant information Past, present and future Reflected in prices At no costs

Preferences and production technologies are exogenous given

Technology can be accessed and employed by everyone without problems

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Perfect Information and Perfect Competition



No endogenous mechanism that causes a change in prices

Exogenous change in preferences or technology will result in a shift of the equilibrium

Such an equilibrium situation, as the “first welfare theorem” shows, is Pareto-optimal

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Competitive Equilibrium



Descartes division of the world into an inner subjective domain and an outer objective domain

The mind as having values and being subjective The individual with reference to reason and consciousness

Cogito ergo sum – I think, therefore I am

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Preferences



The Cartesian dualism is the root of the neoclassical conception of the individual

Focus shifted from individual toward her individualistic choice No explicit consideration of interaction among individuals No explicit consideration of the institutional environment

Individual decision making

Completely detached from her social structure

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Cartesian dualism



We assume that the individual has a rational preference relation over the set of all possible choices. Let 𝑥, 𝑦, 𝑧 be mutually exclusive alternatives in the set of possible choices 𝑋A rational preference relation is characterized by

Completeness: for 𝑥, 𝑦 ∈ 𝑋, 𝑥 ≽ 𝑦 ∨ (read: or) 𝑦 ≽ 𝑥 or both

Transitivity: for 𝑥, 𝑦, 𝑧 є 𝑋, 𝑥 ≽ 𝑦 ∧ (read: and) 𝑦 ≽ 𝑧 ⇒ 𝑥 ≽ 𝑧

(Reflexivity: 𝑥 ~ 𝑥 ∀ (read: for all) 𝑥 є 𝑋)

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A rational preference relation



An agent with lexicographic preferences will choose the bundle that offers the largest amount of the first good x1 no matter how much of the other good is in the bundle.

(𝑥1, 𝑥2) ≻ (𝑥´1, 𝑥´2) if 𝑥1 ≻ 𝑥´1

Rational, but cannot be represented by a continuous function

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Lexicographic preferences



14

Notation I

Symbol Meaning

𝑝 = (𝑝1, 𝑝2, … , 𝑝𝑁) Price vector

ℝ𝑁, ℝ+𝑁 Real numbers, positive real numbers (size of

commodity space and consumption set)

𝑥𝑖´ = (𝑥1𝑖 , 𝑥2𝑖 , … , 𝑥𝑁

𝑖 ) є ℝ+𝑁 Commodity bundle or demand individual 𝐼

𝑥´ = (𝑥1, 𝑥2, … , 𝑥𝐼) є ℝ+𝐼𝑁 Allocation

𝜔𝑖´ = (𝜔1𝑖 , 𝜔2𝑖 , … , 𝜔𝑁

𝑖 ) Endowment individual 𝐼

𝜔 =

𝑖=1

𝐼

𝜔𝑖 , 𝜔 = 𝑝 𝜔Aggregate endowment, wealth



15

Notation II

Symbol Meaning

𝑥, 𝑥𝑛𝑖 , 𝑝 Equilibrium allocation, equilibrium demand agent

𝑖 good 𝑛, equlibrium price vector

𝑢 … , 𝑣(… ) Direct and indirect utility functions

𝑥 … , ℎ(… ) Marshallian and Hicksian demand

e(… ) Expenditure function

𝜆, Λ(… ) Lagrange multiplier, Lagrangian function

𝐾, 𝐿, 𝑟, 𝑤 Production factors (capital, labor), Factors productivities(unit interest and unit wage)

𝑦(𝐾, 𝐿) Production function



Unrestricted utility/profit-functions of this type have no maxima but individuals usually are restricted at least by a limited budget

Hence, neoclassical microeconomics needs a method for optimization under restrictions

The Lagrangian method is a method to find local maxima and minima (originally from classical mechanics, developed by Joseph-Louis Lagrange in correspondence with Leonhard Euler, 1755)

Restrictions are exogenously imposed

Note that the method requires the solution to be on every one of the restrictions (i.e. all restrictions are binding)

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Lagrangian method



Continuous

Strictly increasing in every argument 𝜕𝑢/𝜕𝑥𝑛 > 0

At least differentiable twice on the interior of ℝ+𝑁

Strictly concave 𝜕2𝑢/𝜕2𝑥𝑛 < 0

Satisfies lim𝒙𝒏→𝟎𝜕𝑢/𝜕𝑥𝑛 = +∞

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Utility function



Marginal utility is positive, but decreasing. Quantities and prices will be strictly positive in equilibrium.

This leads to a budget set

𝐵𝑖 = 𝑥𝑖 ∈ ℝ+𝑁: 𝑝𝑥𝑖 ≤ 𝑝𝜔𝑖

… that typically takes the form of a linear decreasing budget restriction

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Marginal utility



The Lagrangian function is constructed as follows:

max𝑥1,𝑥2,𝜆Λ = 𝑢 𝑥1, 𝑥2 − 𝜆(𝑝1𝑥1 + 𝑝2𝑥2 − 𝜔)

Assuming the utility function is concave, first order conditions are necessary and sufficient conditions – a system of three equations:

(1) 𝜕 Λ

𝜕𝑥1=𝜕𝑢

𝜕𝑥1− 𝜆𝑝1 = 0; (2)

𝜕 Λ

𝜕𝑥2=𝜕𝑢

𝜕𝑥2− 𝜆𝑝2 = 0;

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The Lagrange Multiplier Approach



(3) 𝜕 Λ

𝜕𝜆= − 𝑝1𝑥1 − 𝑝2𝑥2 + 𝜔 = 0

Dividing the first and the second equation – At the optimum, the ratio of prices equals the ratio of marginal costs:

𝜕𝑢/𝜕𝑥1𝜕𝑢/𝜕𝑥2=𝜆𝑝1𝜆𝑝2=𝑝1𝑝2

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The Lagrange Multiplier Approach



The solution to the UMP (utility maximization problem) is a bundle of commodities as a function of prices and wealth:

𝑥𝑖 = 𝑥𝑖(𝑝, 𝜔)

Using the Lagrange method, we maximize:

Λ = 𝑢 𝑥 − 𝜆(𝑝𝑥 − 𝜔)

First order conditions (FOC): 𝜕𝑢

𝜕𝑥𝑛= 𝜆𝑝𝑛 ∀𝑛 ∈ 𝑁; 𝑝𝑥 = 𝜔

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Walrasian or Marshallian demand function



Marginal utilities are equalized formally:

𝜕𝑢/𝜕𝑥𝑛1𝑝𝑛1=𝜕𝑢/𝜕𝑥𝑛2𝑝𝑛2

−𝜕𝑢/𝜕𝑥𝑛1𝜕𝑢/𝜕𝑥𝑛2

= −𝑝𝑛1𝑝𝑛2= 𝑀𝑅𝑆

The FOCs of the UMP give us 𝑁 + 1 variables (𝑥1, 𝑥2, … , 𝑥𝑁 and 𝜆), a system that is solvable in principle.The relations of marginal utilities (and good prices) is called the marginal rate of substitution (MRS)

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Equilibrium conditions for consumers



23

The utility function is given by:

𝑢 𝑥1, 𝑥2 = 𝛼 ln 𝑥1 + (1 − 𝛼) ln 𝑥2

subject to 𝑝1𝑥1 + 𝑝2𝑥2 ≤ 𝜔.We set up the Lagrange function

max𝑥1,𝑥2,𝜆𝛼 ln 𝑥1 + 1 − 𝛼 ln 𝑥2 + 𝜆(𝜔 − 𝑝1𝑥1 − 𝑝2𝑥2)

and derive the FOCs:

(1) 𝛼

𝑥1= 𝜆𝑝1 (2)

1−𝛼

𝑥2= 𝜆𝑝2

(3) 𝑝1𝑥1 + 𝑝2𝑥2 = 𝜔

An

Ex

amp

le



24

We combine the first two to get:

𝑝1𝑥1 =𝛼

1 − 𝛼𝑝2𝑥2

And substitute into the third, 𝑝1𝑥1 = 𝜔 − 𝑝2𝑥2We obtain:

𝛼

1 − 𝛼𝑝2𝑥2 = 𝜔 − 𝑝2𝑥2

This equations can be solved for 𝑥2 as a function of prices and wealth:

𝑥2 𝑝1, 𝑝2, 𝜔 = 1 − 𝛼𝜔

𝑝2

An

Ex

amp

le



25

Inserting and solving for demand of 𝑥1, we get:

𝑥1 𝑝1, 𝑝2, 𝜔 = 𝛼𝜔

𝑝1

The last two equations are the individual´s Walrasianor Marshallian demand functions, telling us, how much of each good the individual will demand as a function of prices and wealth.

An

Ex

amp

le



26

The consumer´s utility at the optimum is given by the indirect utility function:

𝑣 𝑝, 𝜔 = 𝑢 𝑥1, 𝑥2 = 𝛼 ln 𝛼𝜔

𝑝1+ (1 − 𝛼) ln (1 − 𝛼)

𝜔

𝑝2

An indirect utility function gives utility as a function of prices and income.

The direct utility function gives utility as a function of goods consumed

An

Ex

amp

le



27

Marsh

alliand

eman

d

x1

x2

budget line

Tangential point –

optimum consumption

bundle x1, x2

Indifference curves



The dual problem to utility maximization is a minimization of expenditure for reaching a specific utility level u.The general form:

min 𝑒(𝑝𝑖 , 𝑥𝑖) subject to 𝑢 𝑥 ≥ 𝑢

Given an expenditure function 𝑒(𝑝, 𝑢) that shows the minimum expenditure required for reaching a certain utility level, the relation between Hicksian and Marshallian demand is given by:

ℎ 𝑝, 𝑢 = 𝑥 𝑝, 𝑒 𝑝, 𝑢

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Marshallian Demand, Hicksian Demand



Given the indirect utility function just derived before, we can also state:

𝑥 𝑝, 𝜔 = ℎ(𝑝, 𝑣(𝑝, 𝜔))

The two kinds of demand can be related by what is termed the Slutsky equation. Split changes in uncompensated demand

Substitution of the (after the price changes) relatively cheaper goods for relatively more expensive ones, the substitution effect

Change in the consumer´s purchasing power following the price changes, the income effect

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Slutsky Equation



𝜕𝑥𝑖 𝑝, 𝜔

𝜕𝑝𝑗=𝜕ℎ 𝑝, 𝑢

𝜕𝑝𝑗𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑜𝑛 𝐸𝑓𝑓𝑒𝑐𝑡

−𝜕𝑥𝑖 𝑝, 𝜔

𝜕𝜔𝑥𝑗(𝑝, 𝜔)

𝐼𝑛𝑐𝑜𝑚𝑒 𝐸𝑓𝑓𝑒𝑐𝑡

Using Shephard’s Lemma 𝜕𝑒 𝑝,𝑢

𝜕𝑝𝑗= ℎ𝑗(𝑝, 𝑢) the derivation follows with

the definition of the indirect utility above (𝑥𝑗 𝑝, 𝜔 = ℎ𝑗(𝑝, 𝑣(𝑝, 𝜔))) from

the total differential of the Hicksian demand function (with 𝑑𝑢 = 0):

𝑑ℎ𝑖 𝑝, 𝑢 =𝜕𝑥𝑖(𝑝,𝑒 𝑝,𝑢 )

𝜕𝑝𝑗𝑑𝑝𝑗 +

𝜕𝑥𝑖(𝑝,𝑒 𝑝,𝑢 )

𝜕𝑒(𝑝,𝑢)

𝜕𝑒 𝑝,𝑢

𝜕𝑝𝑗𝑑𝑝𝑗 +

𝜕𝑥𝑖(… )

𝜕𝑒(… )

𝜕𝑒 …

𝜕𝑢𝑑𝑢

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Slutsky Equation



31

IE an

d S

E in

dem

and

ch

ang

es

x1

x2

IE SE

Hicksian demand

(compensated)

New optimum



As an example for firm´s production possibilities is often used the Cobb-Douglas technology, which is defined by the following production function for 𝛼 ∈ (0,1) and two inputs (capital 𝐾, labor 𝐿):

𝑦 = 𝐹 𝐾, 𝐿 = 𝐴𝐾𝛼𝐿1−𝛼

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The Production Function



33

Co

bb

-Do

ug

las p

rod

uctio

n fu

nctio

n



If we assume, that the production function is differentiable, the MRTS tells us at which rate one input can be exchanged for another one without altering the quantity of output.

Setting the equal to zero and sticking to the Cobb-Douglas production function for the derivation, with 𝐴 = 1:

𝑑𝑦 = 𝛼𝐿

𝐾

1−𝛼

𝑑𝐾 + 1 − 𝛼𝐾

𝐿

𝛼

𝑑𝐿 = 0

Resolving for 𝑑𝐿

𝑑𝐾gives

𝑑𝐿

𝑑𝐾= −𝛼𝐿

𝐾

1−𝛼

1−𝛼𝐾

𝐿

𝛼 = −𝛼

1−𝛼

𝐿

𝐾

1−𝛼+𝛼

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Marginal rate of technical substitution (MRTS)



The absolute value of the scope of the slope of the isoquant:

𝑀𝑅𝑇𝑆𝐾𝐿 =𝑑𝐿

𝑑𝐾= −𝛼

1 − 𝛼

𝐿

𝐾

… or sometimes also given in terms of the absolute value,

𝑀𝑅𝑇𝑆𝐾𝐿 = −𝑑𝐿

𝑑𝐾= −

𝛼

1−𝛼

𝐿

𝐾

Formally, isoquant 𝑆(𝑦) and input requirement set 𝐼(𝑦) are defined as:

𝑆 𝑦 = { 𝐿, 𝐾 : 𝐹 𝐾, 𝐿 = 𝑦}𝐼 𝑦 = { 𝐿, 𝐾 : 𝐹 𝐾, 𝐿 ≥ 𝑦}

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MRTS



36

Isoq

uan

tsan

d in

pu

t sets

α=0,5



The law of diminishing returns states, that if all but one input is fixed, the increase in output from an increase in the variable input decline. For the production function, this implies:

Strictly increasing in both dimensions 𝜕𝐹

𝜕𝐾> 0 and

𝜕𝐹

𝜕𝐿> 0

Concave in both dimensions𝜕2𝐹

𝜕2𝐾< 0 and

𝜕2𝐹

𝜕2𝐿< 0

(Note that as the function is also twice statically differentiable and its slope becomes infinite near zero (𝐾 = 0 or 𝐿 = 0) it mathematically resembles the neoclassical utility function)

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Law of diminishing returns



Starting with the firm´s cost-minimization problem:

min𝐾,𝐿𝑟𝐾 + 𝑤𝐿 subject to 𝐹 𝐾, 𝐿 ≥ 𝑦

we construct the Lagrangian

Λ 𝐾, 𝐿, 𝜆 = 𝑟𝐾 + 𝑤𝐿 + 𝜆(𝐹 𝐾, 𝐿 − 𝑦)

To figure out which point on the isoquant is cost minimizing, we take the FOCs and set them to zero:

(1) 𝜕Λ

𝜕𝐾= 𝑟 + 𝜆

𝜕𝐹

𝜕𝐾= 0 (2)

𝜕Λ

𝜕𝐿= 𝑤 + 𝜆

𝜕𝐹

𝜕𝐿= 0

(3) 𝐹 𝐾, 𝐿 − 𝑦 = 0

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Cost Minimization and Cost Functions



The first two FOCs can conveniently be rearranged to

𝑟 = 𝜆𝜕𝐹

𝜕𝐾and w=𝜆

𝜕𝐹

𝜕𝐿

Combining the two equations we get the firm´s optimality condition for the relative amounts of inputs used:

𝑟

𝑤=

𝜕𝐹𝜕𝐾𝜕𝐹𝜕𝐿

= − 𝑀𝑅𝑇𝑆𝐾𝐿

At the optimum, the relative price of inputs has to equal the MRTS.

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40

Inp

ut req

uirem

ent

set, iso

qu

ant, relativ

e prices



Deriving the firm´s cost function with using the Cobb-Douglas function (𝛼 = 0.5):

𝐶 𝑟, 𝑤, 𝑦 = min𝐾,𝐿𝑟𝐾 + 𝑤𝐿 subject to 𝑦 ≥ 𝐾0.5𝐿0.5 = 𝐾𝐿

The constraint must be binding, 𝑦 ≥ 𝐾0.5𝐿0.5. Solving for 𝐾 yields

𝐾 = 𝑦2

𝐿which can be substituted into the cost function:

𝐶 𝑟,𝑤, 𝑦 = min𝐾,𝐿𝑟 𝑦2

𝐿+ 𝑤𝐿

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41




Taking the FOC for labor we have

𝐿 𝑟, 𝑤, 𝑦 =𝑟

𝑤

0.5

𝑦.

The conditional demand for capital is derived in the same way; we obtain

𝐾 𝑟,𝑤, 𝑦 =𝑤

𝑟

0.5

𝑦.

and can rewrite the cost function as

𝐶 𝑟, 𝑤, 𝑦 =w𝐿 𝑟, 𝑤, 𝑦 + 𝐾 𝑟,𝑤, 𝑦 = 2𝑟0.5𝑤0.5 𝑦.

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Using the cost function, we now turn to the firm´s profit maximization problem. A firm´s profits are defined as revenue minus cost:

Π 𝑦, 𝐿, 𝐾, 𝑤, 𝑟, 𝑝0 = 𝑝0𝑦 − (𝑤𝐿 + 𝑟𝐾)

The firm has now the task to choose the level of output.Substituting the cost function for (𝑤𝐿 + 𝑟𝐾) and taking input prices (𝑤 and 𝑟) as given:

max𝑦≥0Π = 𝑝0𝑦 − 𝐶(𝑟, 𝑤, 𝑦)

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Profit Maximization



The Cobb-Douglas production function exhibits constant returns to scale since:

𝐹 𝑐𝐾, 𝑐𝐿 = 𝐴 𝑐𝐾 0.5 𝑐𝐿 0.5 = 𝑐𝐹(𝐾, 𝐿)

This means that the cost function is linear and average costs as well as marginal costs are constant.

𝐴𝐶 =𝐶 𝑟,𝑤, 𝑦

𝑦

𝑀𝐶 =𝜕𝐶 𝑟, 𝑤, 𝑦

𝜕𝑦

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Profit Maximization



45

Existence of Equilibrium: Constant Returns to Scale



46

Existence of Equilibrium: DecreasingReturns to Scale



47

Existence of Equilibrium: IncreasingReturns to Scale



48

Partial Equilibrium: Stylized Demand and Supply Functions



49

Partial Equilibrium



Instead of just looking at one market in isolation, as we did before, we look at the complete set of markets now.

In equilibrium, marginal rates of substitution (MRS) and technical substitution (MRTS) are equal:

𝑀𝑅𝑆12 = − 𝜕𝑢 𝜕𝑥1

𝜕𝑢 𝜕𝑥2

= −𝑝1𝑝2= − 𝜕𝑦𝜕𝑥1

𝜕𝑦𝜕𝑥2

= 𝑀𝑅𝑇𝑆12

22.07.2014

General Equilibrium



The combination of equilibrium prices and allocations ( 𝑝, 𝑥)as Arrow-Debreu equilibrium.

We assume that all markets are in equilibrium, i.e. excess demand 𝑧 (demand minus supply) is equal to zero:

𝑧𝑛 𝑝 =

𝑖=1

𝐼

𝑥𝑛𝑖 (𝑝) −

𝑖=1

𝐼

𝜔𝑛𝑖 = 0 ∀𝑛 ∈ {1,… ,𝑁}

A corollary of this is Walras´ Law, stating that if 𝑁 − 1markets are in equilibrium, the 𝑁th market must also be in equilibrium

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Walras´ Law



If 𝑁 – 1markets are in equilibrium, the Nth market must also be in equilibrium. This follows from simple accounting. If we sum up all individual budgets constraints, we see that total expenditure has to equal total receipts, i.e.,

𝑝

𝑖=1

𝐼

𝑥𝑖 𝑝 = 𝑝

𝑖=1

𝐼

𝜔𝑖

For excess demand functions, Walras´ Law implies that they sum up to zero:

𝑛=1

𝑁

𝑧𝑛 𝑝 = 0

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Walras´ Law



For the analysis of general equilibrium, this result proves useful since it implies that if all market but one are in equilibrium, the last market also has to be in equilibrium:

𝑧𝑛 𝑝 = 0 ∀𝑛 ∈ 1,… ,𝑁 − 1 ⇒ zN 𝑝 = 0

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Walras´ Law



For now, we just assume that there exists an equilibrium with positive prices and state some properties of this equilibrium:

Theorem 1 (first welfare theorem):A competitive equilibrium allocation is an efficient allocation.

Theorem 2 (second welfare theorem):Every efficient allocation can be transformed into a competitive equilibrium allocation by appropriate transfers

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Welfare Theorems



Existence Convexity of preferences (concave utility function) Convex production set

Uniqueness Either the initial allocation is already Pareto-efficient Or all goods are substitutes (positive cross price elasticities) Then the excess demand curve crosses the x-axis only once

Stability Conditions for uniqueness are sufficient for global stability If excess demand becomes zero only once, this equilibrium will be stable

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Conditions for Walras´ common equilibrium of Sonnensch./Mantel/Deb.



The conditions that have to be fulfilled for an existing, unique, and stable equilibrium are called SMD conditions.

The GET is a special exception, not the rule for the condition of a perfect rational agent. For all other cases the system could react differently, i.e., multiple equilibria, no equilibrium, or instable equilibria etc.

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Sonnenschein-Mantel-Debreu



To which degree the general equilibrium model can serve to inform relevant policy decisions in the real world?

Some distortion is present in the setup

Optimal conditions in all other relevant areas will not guarantee a second best result (i.e. there will not be an efficient allocation even between the undistorted elements) as the theory of the second best shows

Some first-best conditions are violated, there may be numerous second best results that can be reached

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The General Theory of the Second Best



We have a function 𝐹(𝑥1, … , 𝑥𝑛 ) of 𝑛 variables 𝑥𝑖 that is to be maximized (minimized) and a constraint 𝐿 𝑥1, … 𝑥𝑛 = 0that has to be taken into account in that operation.

The optimization gives the FOCs (with 𝐹𝑖 as 𝜕𝐹𝑖

𝜕𝑥𝑖etc.)

𝐹𝑖 − 𝜆𝐿𝑖 = 0 𝑖 = 1,… , 𝑛

and, from these, the relative conditions for optimality,

𝐹𝑖𝐹𝑛=𝐿𝑖𝐿𝑛

i = 1,… , n.

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In Lipsey and Lancaster’s theory of the second best, this first best solution is excluded by an additional constraint of the form:

𝐹1𝐹𝑛= 𝑘𝐿1𝐿𝑛with k ≠ 1.

This additional constraint changes the optimization problem to:

min 𝐹 − 𝜆1𝐿 − 𝜆2𝐹1𝐹𝑛− 𝑘𝐿1𝐿𝑛

(or equivalently for maximization problems).𝜆1 and 𝜆1 will generally be different from λ.

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The FOCs here are:

𝐹𝑖 − 𝜆1𝐿𝑖 − 𝜆2𝐹𝑛𝐹1𝑖 − 𝐹1𝐹𝑛𝑖

𝐹𝑛2 − 𝑘

𝐿𝑛𝐿1𝑖 − 𝐿1𝐿𝑛𝑖

𝐿𝑛2 = 0

Denoting 𝐹𝑛𝐹1𝑖−𝐹1𝐹𝑛𝑖

𝐹𝑛2 = Qi and

𝐿𝑛𝐿1𝑖−𝐿1𝐿𝑛𝑖

𝐿𝑛2 = 𝑅𝑖 we can write:

𝐹𝑖𝐹𝑛=𝐿𝑖 +

𝜆2𝜆1 𝑄𝑖 − 𝑘𝑅𝑖

𝐿𝑛 +𝜆2

𝜆1 𝑄𝑛 − 𝑘𝑅𝑛

The Pareto optimum conditions (achieved in the first best result above) can therefore only be attained if 𝜆2 = 0 (i.e. if the additional condition is irrelevant).

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Differences in the information (on quality) available to sellers and potential buyers of a good.

Sellers will exploit their information advantage and sell low-quality goods at market price

Buyers use some statistics from observed frequencies of good and bad products (lemons) to form an idea of expected quality

The example will show that these information sets directly lead to the collapse of the market (and without a market, no efficient or Pareto optimal allocation of goods can be established).

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Asymmetric Information – The Markets for Lemons



Quality 𝑞 of goods is distributed uniformly; sellers will not sell under value, i.e. the seller price is 𝑝𝑠 ≥ 𝑞 for each seller.

Buyers are willing to pay a higher price 𝑘𝑞 (with 1 < 𝑘 < 2)over their expected quality; since the average value of the goods

is 1

2 𝑖 𝑞𝑖, the buyer’s reservation prices will be 𝑝𝑑 ≤ 𝑖

𝑘

2𝑞𝑖

Since 𝑘 < 2 there are sellers with quality 𝑞 > 𝑖𝑘

2𝑞𝑖 who will

exit the market which leads to a further fall in average quality and sends the market into a downward spiral

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Assume the following situation: Two groups of agents, 1 and 2. Distinguished by the utility they gain from consumption of a specific good 𝑞𝑖 is the indicator of the quality of a particular unit of the good 𝑀 being a bundle of the rest of goods

𝑈1 = 𝑀 + 𝑖=1𝑛 𝑞𝑖 and 𝑈2 = 𝑀 + 𝑖=1

𝑛 3

2𝑞𝑖

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Spending one more unit on bundle 𝑀 increases utility by one unit𝑑𝑈𝑖𝑑𝑀= 1

Spending on the second good, the utility effect depends on the quality

𝑞 > 𝑝 : The purchase is worthwhile of type 1 agents

3𝑞

2> 𝑝 : worthwhile for type 2 agents

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Let 𝑌1 and 𝑌2 denote the income of all types 1 and 2 agents, respectively. Then, demand 𝐷 for the good from type 1 agents is of the form:

𝐷1 =𝑌1

𝑝𝑖𝑓 𝑞 ≥ 𝑝

𝐷1 = 0 otherwise

For type 2 agents, we have the following analogous expressions

𝐷2 =𝑌2

𝑝𝑖𝑓3

2𝑞 ≥ 𝑝

𝐷2 = 0 otherwise

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Accordingly, total demand 𝐷 is:

𝐷 =𝑌1+𝑌2

𝑝𝑖𝑓 𝑞 ≥ 𝑝

𝐷 =𝑌2

𝑝𝑖𝑓3

2𝑞 ≥ 𝑝 ≥ 𝑞

𝐷 = 0 otherwise (i. e. if 𝑝 >3

2𝑞)

With uniform quality distribution, the average quality at any price 𝑝

would be 𝑞 =𝑝

2; therefore the demand would be 𝐷 = 0.

No trade will never take place, even though at any price there would be someone willing to pay the asked price if quality could be assured

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This chapter provided a short introduction to neoclassical microeconomics, optimization methods, and GET

Some critique (SMD) and two specific limitations (second best and markets for lemons) were included

Further extensions (intertemporal optimization, growth models) and limitations (imperfect information) are introduced in the chapter (but are not included in this presentation)

Chapters 6 and 7 of the textbook extend this and go more into detail on the critique of GET and on oligopoly models.

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Conclusion & Limitations of GET



68

Readings for the next lecture

Compulsory reading:

Critiques of the Neoclassical ‘Perfect Market’ Economy and Alternative Price Theories, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 129-155.

For further readings visit the companion website

http://www.amazon.com/The-Microeconomics-Complex-Economies-Institutional/dp/0124115853/ref=sr_1_1?ie=UTF8&qid=1396453322&sr=8-1&keywords=Microeconomics+of+complex+economies

http://booksite.elsevier.com/9780124115859/

Introductory Microeconomics - Elsevier...Introductory Microeconomics Prof. Wolfram Elsner Faculty of...

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