MICROECONOMICS II.

MICROECONOMICS II.

Sponsored by a Grant TÁMOP-4.1.2-08/2/A/KMR-2009-0041 Course Material Developed by Department of Economics,

Faculty of Social Sciences, Eötvös Loránd University Budapest (ELTE) Department of Economics, Eötvös Loránd University Budapest

Institute of Economics, Hungarian Academy of Sciences Balassi Kiadó, Budapest

Author: Gergely K®hegyi Supervised by Gergely K®hegyi

February 2011

ELTE Faculty of Social Sciences, Department of Economics

MICROECONOMICS II.

week 3

General equilibrium theory, part 2

Gergely K®hegyi

Prepared by: Gergely K®hegyi, using Jack Hirshleifer, Amihai Glazer és David Hirshleifer (2009) Mikro- ökonómia. Budapest: Osiris Kiadó, ELTECON-könyvek (henceforth: HGH), and Kertesi Gábor (ed.) (2004) Mikroökonómia el®adásvázlatok. http://econ.core.hu/ kertesi/kertesimikro/ (henceforth: KG).

Equilibrium with production

One participant, two goods, one factor of production

First example

Robinson consumes sh and coconut, which he produces with labor.

• xK: Robinson's coconut consumption

• xH: Robinson sh consumption

• h: factor of production (working hours): ¯h= 10

• Prod. functions (constant return to scale):

xH= 10hH

xK= 20hK

• Resource constraint: hH+hK= 10

xH

10 +xK

20 = 10 x_K= 200−2x_H

Production-possibility set:

xK+ 2xH ≤200 x_K, x_H ≥0 Production-possibility frontier:

xK+ 2xH = 200 Marginal rate of transformation:

M RT = dxk

dxH

=−2

Second example

Robinson consumes sh and coconut, which he produces with labor.

• x_K: Robinson's coconut consumption

• xH: Robinson sh consumption

• h: factor of production (working hours): ¯h= 10

• Prod. functions (decreasing return to scale):

x_H=√ h_H xK=√

hK

• Resource constraint: h_H+h_K= 25

x²_H+x²_K = 25

Production-possibility set:

x²_K+x²_H ≤25 xK, xH ≥0 Production-possibility frontier (PPF):

x²_K+x²_H = 25 Transformation curve (implicit form of the PPF curve):

T(xH, xK) = 0 T(x_H, x_K) =x²_K+x²_H−25

Deriving the marginal rate of transformation from the transformation curve T(x_H, x_K) = 0

• Total dierential:

dT(x_H, x_K) = ∂T

∂xH

dx_H+ ∂T

∂xK

dx_K

• Along the curve dT(x_H, x_K) = 0

M RT = dxK

dx_H =−∂T /∂xH

∂T /∂x_K

• E.g.:

M RT =−2xH

2xK

=−xH

xK

Relation of the marginal rate of transformation and the marginal product xH =fH(hH) xK =fK(hK)

h_H =f_H⁻¹(x_H) h_K =f_K⁻¹(x_K) h_H+h_K = ¯h

T(x_H, x_K) =f_H⁻¹(x_H) +f_K⁻¹(x_K)−¯h

M RT =−∂T /∂xH

∂T /∂x_K =−df_H⁻¹/dxH

df_K⁻¹/dx_K

Since ^dh_dx = _dh/dx¹ (can be proven mathematically precisely), then ^df_dx⁻¹ = _mp¹ . Thus M RT =−1/mpH

1/mp_K =−mpK

mp_H

Role of the social planner

• Maximize: U(x₁, x₂)→max_x1,x2

• Subject to: T(x₁, x₂) = 0

• Lagrange-function: L=U(x₁, x₂)−λT(x₁, x₂) _∂x^∂L₁ =_∂x^∂U

1 −λ_∂x^∂T

1 = 0 _∂x^∂L₂ =_∂x^∂U

2 −λ_∂x^∂T

2 = 0

M RS=M RT

Decentralized decisions

• Robinson as producer (produces two goods with one factor):

Fix cost of production : F, price of production factor (labor) Maximize: Π =p1y1+p2y2−F →maxy₁,y₂

Subject to: T(y1, y2) = 0

Lagrange-function: L=p1y1+p2y2−F−λT(y1, y2) First order condition:

∗ _∂y^∂L

1 =p1−λ_∂y^∂T

1 = 0

∗ _∂y^∂L

2 =p₂−λ_∂y^∂T

2 = 0

M RT =−p₁ p2

• Robinson as consumer (income: income from labor (F)+ capital income (Π)):

Maximize: U(x₁, x₂)→max_x1,x2 Subject to: p₁x₁+p₂x₂=F+ Π

Lagrange-function: L=U(x1, x2)−λ(p1x1+p2x2−F−Π) First order condition:

∗ _∂x^∂L

1 =_∂x^∂U

1 −λp1= 0

∗ _∂x^∂L

2 =_∂x^∂U

2 −λp2= 0

M RS=−p1

p₂ M RS=M RT Equilibrium conditions:

y1(p1, p2) =x1(p1, p2);y2(p1, p2) =x2(p1, p2)

Two participants, two goods, one factor of production

Notation

• x^RK: Robinson's consumption of coconut

• x^R_H: Robinson's consumption of sh

• h^R: factor of production (¯h^R= 10)

• Production functions:

x^R_H= 10h^R_H x^RK= 20h^RK

• Resource constraint: h^R_H+h^R_K= 10

• Production-possibility set:

x^P_H+ 2x^P_K ≤200 x^P_H, x^P_K ≥0

• x^PK: Friday's coconut consumption

• x^P_H: Friday's sh consumption

• h^P: factor of production (h¯^P = 10)

• Production functions:

x^P_H= 20h^P_H x^P_K= 10h^P_K

• Resource constraint: h^P_H+h^P_K= 10

• Production-possibility set:

2x^R_H+x^R_K ≤200 x^R_H, x^R_K ≥0

Production-possibility frontier

x^R_H 10 +x^R_K

20 = 10 x^RK= 200−2x^RH

M RT^R=−2

x^P_H 20 +x^P_K

10 = 10 x^PK= 200−0,5x^PH

M RT^P =−0,5

Robinson has comparative advantage in coconut production, and Friday in sh production.

Utility of exchange

The utility of exchange comes from the division of labor.

Role of the social planner

Searching for Pareto-ecient allocations:

• Maximize: UA(x^A₁, x^A₂)→max_xA

1,x^A₂,x^B₁,x^B₂

• Subject to:

UB(x^B₁, x^B₂) = ¯UB

T(x1, x2) = 0 x1=x^A₁ +x^B₁ x₂=x^A₂ +x^B₂

• Lagrange-function:

L=U_A(x^A₁, x^A₂)−λ U_B(x^B₁, x^B₂)−U¯_B

−

−µT x^A₁ +x^B₁, x^A₂ +x^B₂

• First order conditions:

_∂x^∂LA 1

= ^∂U_∂xA^A 1

−µ_∂x^∂T

1 = 0

_∂x^∂LA 2

= ^∂U_∂xA^A 2

−µ_∂x^∂T

2 = 0 _∂x^∂LB

1

=−λ^∂U_∂xB^B 1

−µ_∂x^∂T

1 = 0 _∂x^∂LB

2

=−λ^∂U_∂xB^B

2

−µ_∂x^∂T

2 = 0

M RSA=M RT M RSB =M RT M RSA=M RSB =M RT

Two consumers, two producers, two goods, one factor of production

Decentralized decisions

The two companies (1 and 2) produces the two goods with one factor of production (labor). The two consumers (A and B) decide how much to consume besides their endowments, and how much labor to supply depending on their income, the prices of goods and the price of the factor of production. Consumers income stem from their labor (working for the companies) and as owners, from the prot of the companies.

• Product prices: p1, p2, price of factor of production: w.

• Produced quantities of products: y1, y2.

• Used quantities of factors of production: L1, L2

• Consumers' endowments: ω^A₁, ω₂^A, ω₁^B, ω₂^B

• Consumed quantities: x^A₁, x^A₂, x^B₁, x^B₂

• Oered labor: hA, hB

• The share of consumer Afrom the prot of rm 1 is: θ_A1

• Total prot of the companies are distributed among the consumers: θ_A1+θ_B1= 1, θ_A2+θ_B2= 1 Competitive mechanism I. (Optimal decision of companies):

• Maximize:

π1=p1y1−wL1→max

y1,L1

• Subject to: y1=f1(L1)

• First order condition: p1mpL₁ =w

• Solution:

Labor demand function:L1(p1, w) Supply function: y1(p1, w) Prot function: π1(p1, w)

• Maximize:

π2=p2y1−wL2→max

y2,L2

• Subject to: y2=f2(L2)

• First order condition: p2mpL₂ =w

• Solution:

Labor demand function:L2(p2, w) Supply function: y2(p2, w) Prot function: π2(p2, w)

Competitive mechanism II. (optimal decision of consumer): Aconsumer

• Maximize: UA(x^A1, x^A2, hA)→max_xA 1,x^A₂,h_A

• Subject to: p1x^A1 +p2x^A2 =p1ω1^A+p2ω2^A+whA+θA1π1+θA2π2

• First order condition: M RS₁₂^A =−^p_p¹

2, M RS_1h^A =−^p_w¹,(M RS_2h^A =−^p_w²)

• Solution:

Labor supply function.: hA(p1, p2, w) Demand function: x^A1(p1, p2, w), x^A2(p1, p2, w)

Competitive mechanism II. (optimal decision of consumer): B consumer

• Maximize: UB(x^B1, x^B2, hB)→max_xB 1,x^B₂,h_B

• Subject to: p1x^B₁ +p2x^B₂ =p1ω^B₁ +p2ω^B₂ +whB+θB1π1+θB2π2

• First order condition: M RS12^B =−^p_p¹

2, M RS_1h^B =−^p_w¹,(M RS_2h^B =−^p_w²)

• Solution:

Labor demand function:hB(p1, p2, w) Demand function: x^B1(p1, p2, w), x^B2(p1, p2, w)

Competitive mechanism III. (market equilibrium conditions):

• Product markets:

x^A₁(p1, p2, w) +x^B₁(p1, p2, w) =y1(p1, w) +ω₁^A+ω^B₁ x^A₂(p1, p2, w) +x^B₂(p1, p2, w) =y2(p2, w) +ω₂^A+ω^B₂

• Factor market (labor market):

L₁(p₁, w) +L₂(p₂, w) =h_A(p₁, p₂, w) +h_B(p₁, p₂, w)

• Parameters: p^∗₁, p^∗₂, w^∗

1. Consequence. Number of parameters and equations are equal.

1. Note. Since (product and factor) demand functions are zero order homogeneous (NO MONEY ILLU- SION), one of the products or factors can be used as numeraire. Let w=1˙ . Thus the system of equations seems to be over determined (more equations than parameters).

1. Statement. Walras-law (exchange economy with production): The total value of demanded and supplied goods equals, that is the aggregate market over demand is zero (with any price system):

p1z1(p1, p2, w) +p2z2(p1, p2, w) +wzh(p1, p2, w)≡0,

where ^z1(p1, p2, w) = x^A₁(p1, p2, w)−ω^A₁ +x^B₁(p1, p2, w)−ω^B₁ −y1(p1, p2, w), z2(p1, p2, w) = x^A₂(p1, p2, w)−ω₂^A+ x^B2(p1, p2, w)−ω2^B−y2(p1, p2, w)andzh(p1, p2, w) =L1(p1, p2, w) +L2(p1, p2, w)−hA(p1, p2, w)−hB(p1, p2, w). 1. Proof. Since equilibrium is based on optimal decisions, quantities and prices will satisfy consumer budget const- raints. Let's add the budget constraint of the two consumer and rearrange it:

p1x^A1 +p2x^A2 ≡p1ω^A1 +p2ω^A2 +whA+θA1π1+θA2π2

p1x^B₁ +p2x^B₂ ≡p1ω^B₁ +p2ω^B₂ +whB+θB1π1+θB2π2

p1(x^A1 +x^B1 −ω^A1 −ω^B1) +p2(x^A2 +x^B2 −ω2^A−ω2^B)≡

≡w(hA+hB) +π1(θA1+θB1) +π2(θA2+θB2) since rm prot is full allocated among consumers:

p1(x^A1 +x^B1 −ω^A1 −ω^B1) +p2(x^A2 +x^B2 −ω2^A−ω2^B)≡w(hA+hB) +π1+π2

Using the denition of the rm prot:

p1(x^A1 +x^B1 −ω^A1 −ω^B1) +p2(x^A2 +x^B2 −ω2^A−ω2^B)≡

≡w(hA+hB) +p1y1−wL1+p2y2−wL2

Rearranging it:

p1(x^A1 +x^B1 −ω^A1 −ω^B1 −y1) +p2(x^A2 +x^B2 −ω2^A−ω2^B−y2)+

+w(L1+L2−hA−hB)≡0, That is:

p1z1(p1, p2, w) +p2z2(p1, p2, w) +wzh(p1, p2, w)≡0.

2. Consequence. Due to the Walras-law the equilibrium conditions will not be independent (three equilib- rium equations, two price parameters). So one of the equations can be dropped, and the system will not be over determined.

Algorithm for search for equilibrium with production

1. Algorithm. • Writing up individual (producer and consumer) optimum equations

• Solving the producer optimum (supply, factor demand and prot functions)

• Solving the consumer optimum (demand and factor supply)

• Writing the market equilibrium equations (demand=supply on each market)

• Choosing the numeraire good (rewriting the demand and supply functions so that they depend on the price ratio)

• Dening equilibrium price (one of the equilibrium equations can be dropped)

• Finding the individual consumption and supply of goods.

2. Note. The above algorithm can be generalized to an N product, M consumer, R rm and K factor of production system.

Példa:

• Technology of two companies: y₁=√

L₁, y₂=√ L₂

• Utility functions of two consumers:

U_A=x^A₁x^A₂

h_A , U_B =x^B₁x^B₂ h_B

• Endowments of two consumers: ω^A₁ = 100, ω₂^A = 200, ω^B₁ = 300, ω^B₂ = 400, hA = 16, hB = 16 (both consumers work maximum 16 hours a day)

• θ_A1 = 0,2;θ_A2 = 0,8;θ_A1 = 0,6;θ_A2 = 0,4 (The A consumer receives 20% of consumers 1st rm prot,... etc.)

N goods, M consumers, R producers and K factors of production

General equilibrium of anN product,M consumer,Rrm andK production factor economy

• Parameters:

M ∗N (N pc. consumption good,M pc. consumers) R∗K (K pc. factor of production,R pc. rm) N pc. consumption price

K pc. factor of production price

Number of parameters: M ∗N+N+R∗K+K

• Equations:

M ∗N pc. individual optimum constraint (rst order constraints and budget constraints for the Lagrange-variables)

R∗Kpc. individual producer optimum constraint (rst order constraints+production functions+

rst order conditions for the Lagrange-variables)

N pc. equilibrium condition on the product markets: aggregate demand = aggregate supply K pc. equilibrium condition on the factor markets: aggregate demand = aggregate supply Number of equations: M∗N+N+R∗K+K

• So the number of equations and parameters are equal.

• BUT since only relative prices matter (demand and supply functions are zero order homogeneous) a numeraire can be choosen. (−1 parameter).

• So the system seems over determined. (More equations than parameters.)

• BUT due to the Walras-law, equilibrium equations are not independent!

• So the system is not over determined. Dropping one equilibrium equation, the equilibrium can be found according to the algorithm.

3. Note. Counting equations might lead to wrong conclusion however. Negative prices can turn out, since budget constraints and equilibrium conditions are inequalities and not equalities!→Problem of the existence of equilibrium (see below).

Fundamental theorems of the welfare state with production

2. Statement. The two fundamental theorems of the welfare state hold even with production included in the system.

2. Proof. Individual consumer optimums: M RS12^A = −^p_p¹

2, M RS_1h^A = −^p_w¹ and M RS12^B = −^p_p¹

2, M RS^B_1h = −^p_w¹. Thus

M RS₁₂^A =M RS₁₂^B M RS_1h^A =M RS_1h^B. From the production functions:

y1=f1(L1) y2=f2(L2) L1+L2=hA+hB

L1=f₁⁻¹(y1) L2=f₂⁻¹(y2)

F(y1, y2) ˙=f₁⁻¹(y1) +f₂⁻¹(y2)−hA−hB

Transformation curve:

T(y1, y2) ˙=F(y1+ω₁^A+ω₁^B, y2+ω^A₂ +ω^B₂)

Producer optimummp1= _p^w

1 ésmp2= _p^w

2. Thus M RT =−∂T /∂y1

∂T /∂y2

=mp2

mp1

=−w/p2

w/p1

=−p1

p2

,

hence M RS₁₂^A =M RS₁₂^B =M RT, which means the the rst theorem of the welfare state holds.

Let x^A₁, x^A₂, hA, x^B₁, x^B₂, hB, L1, L2, y1, y2 a random Pareto-ecient state. Let the numeraire bew=1˙ . Then let's choose prices p1, p2 so that: M RS^A_1h=M RS^B_1h=˙^p_w¹ andM RS₁₂^A =M RS₁₂^B =M RT=˙ −^p_p¹

2. Then let us reallocate endowments so that consumer budget constraints hold with the chosen prices. Thus the second theorem of the welfare state holds as well.

Main questions

Four main questions of the general equilibrium theory

• Existence: Is there an equilibrium?

• Eciency: Is the equilibrium ecient?

• Uniqueness: Is the equilibrium unique, or more price systems are possible?

• Stability: If the economy moves from the equilibrium (e.g. demand or technological shock), can it return?

Existence

The existence theorem of Arrow and Debreu (1954) says that if we include some limitations to the consumer and producer side, then competitive equilibrium exists. This does not come from the equation solving procedure. Mathematical foundations of the problem are rather complicated, but the conditions for existence are the following (only a list):

• Individual producers (rms) have a convex and closed production-possibility set, which included the origin. So returns to scale are not increasing and production can be shut down.

• The aggregate production set does not include the positive origin, so every production includes some factor use.

• Aggregate production is irreversible.

• The possible consumer sets are convex, closed and limited sets.

• The individual utility functions, representing preferences, are continuous and monotonic functions.

• Indierence surfaces are convex.

• Consumers have endowments.

• All rm prot is allocated among consumers in a xed ratio.

Eciency

What assumptions we have to make so that the welfare theorems hold?

E.g.: Without convexity the second theorem does not hold.

Uniqueness

What are the conditions for the equilibrium to be unique? (It is not indierent whether there is one or more price systems!)

Sidetrack:

Revealed preferences

p1x^∗₁+p2x^∗₂=p1ω1+p2ω2

p⁰₁x⁰₁+p⁰₂x⁰₂=p⁰₁ω1+p⁰₂ω2

p₁x⁰₁+p₂x⁰₂> p₁ω₁+p₂ω₂ p⁰₁x^∗₁+p⁰₂x^∗₂> p⁰₁ω₁+p⁰₂ω₂ p₁z⁰₁+p₂z₂⁰ >0 p⁰₁z₁+p⁰₂z₂>0

1. Denition. Az(p)over-demand function satises a weak axiom of revealed preferences (WARP), if by any p⁰6=κpprice system p⁰z(p)≥0 holds.

What are the conditions for the equilibrium to be unique? (It is not indierent whether there is one or more price systems!)

3. Statement. If in an economy the WARP holds for thez(p)over-demand function, then the competitive equilibrium is unique.

Stability

How does the economic system behave if it is not in an equilibrium? (Dynamic analysis) Price adaptation rule:

• IfD(p)−S(p)>0(over-demand), thenpincreases.

• IfD(p)−S(p)>0(over-supply), thenpdecreases.

2. Denition. The continuous and dynamic (Samuelson) price adaptation of a competitive economy:

˙

p(t) =dp(t)

dt =µ[D(p(t)−S(p(t)))] = (µZ(p(t))) With p0 equilibrium price: p(t) = 0˙ , thusD(p0) =S(p0).

Linear demand and supply, continuous price adaptation

D(p) ˙=A−Bp, S(p) =C+Dp (A, B, C, D >0)

˙

p(t) =A−C

| {z }

α

+ (−B−D)

| {z }

β

p

˙

p(t) =α+βp(t) (β <0)

Solution of the linear dierence equation: p(t) =e^βt+c₀. The equilibrium is stable ifβ <0. 3. Denition. Discrete and dynamic (Ezekiel) price adaptation rule of a competitive economy:

D(pt) Supply adapts to demand using the minus one period price:

S(p_t−1) In equilibrium: D(pt) =S(pt−1).

Linear demand and supply, discrete price adaptation (Cobb-web model) D(p_t) ˙=A−Bp_t, S(p_t−1) ˙=C+Dp_t−1 (A, B, C, D >0) In equilibrium the demanded and supplied quantities equal.

D(p_t) =A−Bp_t=C+Dp_t−1=S(p_t−1)

p_t=A−C B

| {z }

α

+

−D B

| {z }

β

p_t−1

Linear dierence equation:

pt=α+βp_t−1

Linear demand and supply, discrete price adaptation (Cobb-web model) Linear dierence equation:

pt=α+βp_t−1

Thept=pt−1 equilibrium is stable if|β|<1, so thatD < B (supply is less responsive to price changes, than demand), prices converge to the equilibrium.

The Samuelson-type continuous and dynamic price adaptation in a general equilibrium model:

˙

p_i(t) =dpi(t)

dt =µ_i[D_i(p₁(t), . . . , p_n(t))−S_i(p₁(t), . . . , p_n(t))]

(i= 1, . . . , n)

4. Statement. If the weak axiom of revealed preferences holds, then general equilibrium is stable with the Samuelson-type continuous price adaptation rule.

Market tests

• 'Tests' of the competitive market model.

• Sellers and buyers are randomly chosen

• Information: 'You have 100 tokens, which you can use to buy a maximum of 3 pieces from a good, with a price of 6. You can sell the rst piece to the test leader, for 16 tokens, the second for 11 tokens, and the third for 3 tokens. When you repay the 100 tokens to the test leader the remaining tokens is your prot, which you can exchange to dollars.'

Imperfect markets

Imperfect markets

• Transaction costs: contracting, that is the cost of exchange.

• Reasons:

Information asymmetry Property rights problems

Spatial distance bw. seller and buyer.

Etc.

Imperfect markets

Proportional transaction costs: We have to pay Gfee after each pieces of goods X.

Due to the proportionate transaction costs selling any buying price will depart. The larger the distance, the more likely individuals will opt for self-supply, and the smaller the total volume of the exchange.

Lump-sum transaction costs: Lump-sum transaction costs do not create disparities between selling and buying price. But they make consumers to exchange only in discrete periods of time. So both buyers and sellers are forced to keep endowments. Higher transaction costs and costs of keeping endowment will raise the probability of self-supply and thus decrease the volume of total exchange. In extreme cases the market exchange nill.

Role of money

• Money as the instrument of exchange

• Money as tool for keeping value (transitionally)

Money decreases the costs of exchange. (It does not decrease the costs of physically exchanging goods.

These exists in every economy with division of labor.) If only one good serves as the numeraire, less two

way transaction is necessary. Moreover such a numeraire enables a three-, or more-way exchange, which is nearly impossible with barter. For exchange endowment of goods might be needed. Costs of exchange is the lowest of there is a consensus about the numeraire, used as money, and which can also serve as tool for keeping value.

4. Note. To analyze money, we need to include time in our analysis.