80743422 General Equilibrium and Welfare Economics PDF

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ECON501: Lecture Notes Microeconomic Theory II by Jorge Rojas

Contents 1 General Equilibrium 1.1 Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Existence of Walrasian Equilibria . . . . . . . . . . . . . . . . . . . . . . 1.3 Existence of Walrasian Equilibria . . . . . . . . . . . . . . . . . . . . . .

2 3 3 4

2 Jones Model 2.1 Input endowment magnification effect . . . . . . . . . . . . . . . . . . . . 2.2 Output price magnification effect . . . . . . . . . . . . . . . . . . . . . . 2.3 Magnification effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6 6

3 Welfare Economics 3.1 FIRST THEOREM OF WELFARE ECONOMICS . . . . . . . . . . . . 3.2 SECOND THEOREM OF WELFARE ECONOMICS . . . . . . . . . . .

6 7 7

4 Public Goods and Externalities 4.1 Public Goods and Competitive Markets . 4.2 Externalities . . . . . . . . . . . . . . . . 4.2.1 Production Externality . . . . . . 4.2.2 Common Property Rights . . . . 4.2.3 Congestion Externality . . . . . .

. . . . .

8 9 10 10 11 11

5 Practical Themes 5.1 Intertemporal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Robinson Crusoe (Coop-structure) . . . . . . . . . . . . . . . . . . . . . 5.3 Fisher Separation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .

12 12 13 13

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

Abstract This is a summary containing the main ideas in the subject. This is not a summary of the lecture notes, this is a summary of ideas and basic concepts. The mathematical machinery is necessary, but the principles are much more important.1

1

General Equilibrium

The single-market story is a partial equilibrium model. While in the general equilibrium model all prices are variable, and equilibrium requires that all markets clear. There is a special case called “pure exchange” economy where all the economic agents are consumers. Each consumer i is completely described: 1. the preferences i or the corresponding utility function ui 2. the initial endowment ~ωi The consumption bundle is denoted by xi = (x1i , . . . , xki ) and an allocation is written as x = (xi , . . . , xn ). An allocation x is a collection of n consumption bundles describing what each of the n agents holds. Definition 1. Feasible Allocation. A feasible allocation is one that is physically possible, therefore: n X i=1

xi ≤

n X

~ωi

i=1

In a two goods, two agents economy, the Edgeworth box is a useful tool to represent the situation:

Figure 1: Edgeworth Box. 1

Without Equality in Opportunities, Freedom is the privilege of a few, and Oppression the reality of everyone else.

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

1.1

Walrasian Equilibrium

Let us define p = (p1 , . . . , pk ) which is exogenously given. Each consumer solves the problem: Maxxi ui (xi ) s.t. pxi = p~ωi Notice that for an arbitrary price vector p, it might not be possible to make the desired transactions for the simple reason that aggregate demand may not be equal to aggregate supply: n n X X ~ωi xi (p, p~ωi ) 6= i=1

i=1

Definition 2. Walrasian Equilibrium. We define a W.E. to be a pair (p∗ , x∗ ) such that: n X

xi (p, p~ωi ) ≤

i=1

n X

~ωi

i=1

p∗ is a Walrasian equilibrium of there is no good for which there is positive excess demand.

1.2

Existence of Walrasian Equilibria

We know that the demand functions are H.O.D. zero in prices. As the sum of homogeneous functions is homogeneous, then the aggregate excess demand function is also H.O.D. zero in prices. n X ∗ z(p ) = [xi (p∗ , p∗ ~ωi ) − ~ωi ] ≤ 0 i=1

So, z :

Rk+

∪ {0} 7→ R

k

Theorem 1. Walras’ Law: For any price vector p, we have p · z(p) = 0, i.e., the value of the excess demand is identically zero. The proof is direct. z = 0 since xi (·) must satisfy the budget constraint for each agent i. Corollary 1. Market Clearing: If demand equals supply in (k − 1) markets, and pk > 0 then demand must equal supply in the kth market. Definition 3. Free Goods: If p∗ is a Walrasian Equilibrium and zj (p∗ ) < 0, then p∗j = 0 In other words, if some good is in excess supply at a Walrasian equilibrium, it must be a free good. Definition 4. Desirability: If pi = 0, then zi (p) > 0 ∀i = 1, . . . , k. The following theorem is essential to solve applied problems since we will impose equality of the demand and the supply, almost all the time. Theorem 2. Equality of Demand and Supply. University of Washington

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

If all goods are desirable and p∗ is a Walrasian equilibrium, then z(p∗ ) = 0. The proof is done by contradiction. Since the aggregate excess demand z(p) is H.O.D. zero, we can express everything in terms of relative prices. Thus, we get: pˆi

pi = Pk

j=1

pˆj

,and therefore,

k X

pi = 1

i=1

So, p ∈ (k − 1)-dimensional unit simplex. S k−1 = {p ∈ Rk+ :

Pk

i=1

pi = 1}

Theorem 3. BROUWER FIXED-POINT THEOREM If f : S k−1 7→ S k−1 is a continuous function from the unit simplex to itself, there is some x ∈ S k−1 such that x = f (x). Another useful calculus tool is the Intermediate Value Theorem. If f is a real-valued continuous function on the interval [a, b], and I is a number between f (a) and f (b), then there is a c ∈ [a, b] such that f (c) = I . Theorem I below is one of the most important theorems in modern economics (in my opinion). This theorem can be used for the good or for the bad. So, be wise!

1.3

Existence of Walrasian Equilibria

If z : S k−1 7→ Rk is a continuous function that satisfies Walras’ law, i.e., p · z(p) ≡ 0, then there exists some p∗ ∈ S k−1 such that z(p∗ ) ≤ 0 I write the proof for this theorem given its importance in economics and the fact that the proof for this idea took around a hundred years to exist in its formal way. Proof: Define g : S k−1 7→ S k−1 by: gi (p) = Thus,

Pk

i=1

pi + max(0, zi (p)) P 1 + kj=1 max(0, zj (p))

∀i = 1, . . . , k

gi (p) = 1

The map g has a nice economic interpretation. Suppose that there is excess demand in some market, so that zi (p) ≥ 0, then the relative price of that good is increased. By the Brouwer’s fixed-point theorem there is a p∗ such that p∗ = g(p∗ ). So, pi∗ = =⇒ p∗i

k X

pi∗ + max(0, zi (p∗ )) P 1 + kj=1 max(0, zj (p∗ ))

∀i

max(0, zj (p∗ )) = max(0, zi (p∗ )) ∀i

j=1 University of Washington

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

we multiply by zi (p∗ ), we get: k  X max(0, zj (p∗ )) = zi (p∗ ) · max(0, zi (p∗ )) ∀i zi (p∗ ) · pi∗ j=1

now, we sum across all the agents, so we get: k X

k k X  X ∗ ∗ zi (p )pi = zi (p∗ ) · max(0, zi (p∗ )) max(0, zj (p )) · ∗

j=1

i=1

By Walras’ Law, we know that: =⇒

Pk

i=1

k X 

i=1

zi (p∗ )pi∗ = p∗ z(p∗ ) = 0

 zi (p∗ ) · max(0, zi (p∗ )) = 0

(1)

i=1

Each term of the sum in (1) is greater or equal to zero since each term is either 0 or z i2 (p∗ ) > 0 if zi (p∗ ) > 0. However, if any term were strictly greater than zero, the equality would not hold. Hence, every term of the summation must be equal to zero, so: zi (p∗ ) ≤ 0 ∀i = 1, . . . , k

2

Jones Model

The Jones Model assumes Constant Returns to Scale (CRS). This implies the zero-profit conditions. There are two inputs {L, T } that produce two outputs {M, F }. Output prices {pM , pF } are exogenously given, while the endogenous variables are {F, M, w, r}. To determine the technical coefficients, we solve the minimization cost problem for each firm (we have assumed that one firm produces only one output). Max wLi + rTi

(2)

s.t. G(Li , Ti ) = 1 where G(Li , Ti ) is the production function for good i. The solution to this problem corresponds to the technical coefficients, i.e., Li∗ = aLi and Ti∗ = aT i . Recall that apq is the amount of input p that is necessary to produce one unit of output q .

2.1

Input endowment magnification effect

Theorem 4. Rybczynski Theorem. An expansion in one factor (input) leads to an absolute decline in the output of the commodity that uses the other factor more intensively. Assuming that: aLM aLF ⇔ M is more labour intensive than F > aT F aT M If L increases, then F will decrease. The proof for theorem (4) is done using the full-employment conditions, and taking differentials. University of Washington

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

2.2

Output price magnification effect

Theorem 5. Stolper-Samuelson Theorem. Assume M is more labour intensive than F , and pF is constant. An increase in pM raises the return to the factor used intensively in M production by an even greater relative amount. M is labour intensive and pM increases =⇒

dw dpM > w pM

The proof for theorem (5) is done using the zero-profit conditions, and taking differentials. To avoid multiple equilibria, we impose that one output is more intensive in one input for all (w, r) ∈ R+2 . Thus, the lines intersect only ones.

2.3

Magnification effects

Input endowment magnification effect (3). General Rybczynski theorem.  aLF dF dL dT dM aLM   dL dT  =⇒ ∧ > > > > > aT M L F T L T aT F M

(3)

Output price magnification effect (4). General Stolper-Samuelson theorem.  aLF dpM  dpM dr aLM   dpF dw dpF > > > > =⇒ ∧ > aT F pF pF pM r w aT M pM

3

(4)

Welfare Economics

Definition 5. Pareto Efficiency A feasible allocation x is a weakly Pareto efficient allocation if there is no feasible ′ ′ allocation x such that all agents strictly prefer x to x. A feasible allocation x is a strongly Pareto efficient allocation if there is no feasible ′ ′ allocation x such that all agents weakly prefer x to x, and some agent strictly prefers ′ x to x. Suppose that preferences are continuous and monotonic. Then an allocation is weakly Pareto efficient if and only if it is strongly Pareto efficient. Definition 6. Walrasian Equilibrium. An allocation-price pair (x, p) is a Walrasian equilibrium if: P Pn xi ≤ ni=1 ~ωi (1) the allocation is feasible: i=1 ′

′

(2) If x i is preferred by agent i to xi , then p · xi > p · ~ωi , i.e., the agent is only better off with a bundle that cannot afford.

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

3.1

FIRST THEOREM OF WELFARE ECONOMICS

If (x, p) is a Walrasian equilibrium, then x is Pareto efficient. Proof: ′ Suppose is not, and let x be a feasible allocation that all agents prefer to x. Then by property 2 of the definition of Walrasian equilibrium, we have that: ′

p · xi > p · ~ωi

∀i = 1, . . . , n ′

summing over i = 1, . . . , n and using the fact that x is feasible, we have that: p

n X

~ωi = p

i=1

n X

′

xi >

i=1

n X

p · ~ωi

i=1

which is a contradiction. 

3.2

SECOND THEOREM OF WELFARE ECONOMICS

Suppose x∗ is a Pareto efficient allocation in which each agent holds a positive amount of each good. Suppose that preferences are convex, continuous, and monotonic. Then, x∗ is a Walrasian equilibrium for the initial endowments ~ωi = xi∗ ∀i = 1, . . . , n. Another version: The following lines are taken from MWG, pages 551-552. I ¯ ) an allocation , {Yj }Jj=1 , ω Definition 7. Given an economy specified by ({(Xi , i )}i=1 ∗ ∗ (x , y ) and a price vector p = (p1 , . . . , pL ) 6= 0 constitute a price quasi-equilibrium P with transfers if there is an assignment of wealth levels (w , . . . , w ) with 1 I i wi = P ∗ p·ω ¯ + j p · y j such that:

(i) ∀j, yj∗ maximises profits in Yj ; that is, p · yj ≤ p · y∗j

∀yj ∈ Yj

(ii) ∀i, if xi ≻i xi∗ then p · xi ≥ wi P P ∗ ¯ + j y∗j (iii) i xi = ω Theorem 6. Second fundamental theorem of Welfare Economics. I J Consider an economy specified by ({(Xi , i )}i=1 ,ω ¯ ), and suppose that every , {Yj }j=1 ′ ′ Yj is convex and every preference relation i is convex [i.e., the set {x i ∈ Xi : xi i xi } is convex for every xi ∈ Xi ] and locally non-satiated. Then, for every Pareto optimal allocation (x∗ , y∗ ), there is (exists) a price vector p = (p1 , . . . , pL ) 6= 0 such that (x∗ , y∗ , p) is a price quasi equilibrium with transfers.

Exercise. Calculating Pareto Efficient Allocations. ¯ T¯} (exogenously There are two goods {X, Y }, two individuals {A, B}, two inputs { L,

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

fixed), X = F (LX , TX ) and Y = G(LY , TY ). We solve the following optimization problem to find the locus of Pareto efficient allocations. Max uA (XA , YA ) s.t. uB (XB , YB ) = u¯B

(5)

F (LX , TX ) = XA + XB G(LY , TY ) = YA + YB LX + LY = ¯L TX + TY = T¯ We form the Lagrangean: L = uA (XA , YA ) + λ[¯ uB − uB (XB , YB )] + α[XA + XB − F (LX , TX )] +β[YA + YB − G(LY , TY )] + γ[L¯ − LX − LY ] + δ[ T¯ − TX − TY ] This leads to the efficient conditions. Efficiency in Consumption ∂uA ∂XA ∂uA ∂YA

∂F ∂LX ∂F ∂TX

=

∂uB ∂XB ∂uB ∂YB

=

α β

=

∂G ∂LY ∂G ∂TY

=

γ δ

Efficiency in Production

Efficient Product Mix ∂uA ∂XA ∂uA ∂YA

=

∂uB ∂XB ∂uB ∂YB

=

∂G ∂LY ∂F ∂LX

=

∂G ∂TY ∂F ∂TX

= MRT =

MCX MCY

where MRT is the usual “marginal rate of transformation” and MC is the “marginal cost”.

4

Public Goods and Externalities

Definition 8. A public good is a commodity for which use of a unit of the good by one agent does not preclude its use by other agents.2 Suppose that X is a public good (for example, education, healthcare, “national defense”) and Y is a private good (for example, wine, cigarettes, cars). There are two agents in this basic model i = A, B. They face the following maximization problem: Max uA (X, YA )

{X,YA }

s.t. uB (X, YB ) = u¯B F (TX ) = X G(TY ) = YA + YB TX + TY = T¯ 2

Mas-Colell, Whinston and Green (MWG), page 359.

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

We setup the following Lagrangean: L = uA (X, YA ) + λ[¯ uB −uB (X, YB )]+α[F (TX ) −X]+β[G(TY ) −YA −YB ]+γ[T¯−TX −TY ] After solving the FOC’s, we get a general result for Pareto efficiency in this basic and well-behaved model: ′ ∂uA ∂uB G (TY ) ∂X ∂X = + ∂uB (6) ∂uA F ′ (TX ) ∂YA ∂YB | {z } | {z } Marginal Benefit (agreggated)

Marginal Cost

Remark 1. Private goods and public goods have a different condition for optimality. Therefore, it is really important to make sure that public good is treated as such and not otherwise. As a society, we must define these goods. For instance, is education a public good or a private one? 1. Private goods =⇒ MRSA = MRSB = MRT 2. Public goods =⇒ MRSA + MRSB = MRT

4.1

Public Goods and Competitive Markets

This is the case in which we let the private sector, namely, firms to provide a public good. So, suppose the markets are competitive and we have two firms. One provides the public good and the other one the private good. Therefore, each firm solves their maximization problem, i.e., they try to maximise profits. In general terms, Max ΠX = px F (K, L) − wL − rK {K,L}

(7)

So, from (7) we get the price for the good X, and likewise for good Y . Now, assume that the consumers own the land T and the firms, so they can use the profits(profits are zero if F(·) has CRS). Then, the consumers have to solve the problem given by: Max ui (Xi + X−i , Yi ) ∀i = A, B

{Xi ,Y−i }

s.t. px Xi + py Yi = rTi + profits

(8)

After some manipulation, we get the condition MRSA = MRSB = MRT which is clearly different to the Pareto Optimal condition for public goods. Therefore, competitive markets will NOT be efficient in the provision of public goods. Notice that property rights do not play any role in this analysis. They will not change the main results. Remark 2. If the agents are identical in their utility functions, then there is no freeriders. However, if the agents have different utility functions, then there will be freeriders. In this situation, we apply the Kuhn-Tucker conditions to solve the problem. Moreover, the free-rider will be the agent that cares less about the public good (in a model in which there are only two goods, the public and the private ones, and only two agents. In more general setups, game theory and mechanism designed are needed). Remark 3. If individuals have identical homothetic utility functions, then we do not care about the distribution of wealth (result coming from Gorman form analysis). It seems that in Chile many leaders and economists believe that we all have the same preferences because inequality is monstrous (Chile is top 20!)3 . 3

Chile is the 17th most unequal country in the world, according to the CIA. Source https://www.cia.gov/library/publications/the-world-factbook/rankorder/2172rank.html and using other surveys we are top 10! University of Washington

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Microeconomic Theory II by Jorge Rojas

ECON501: Lecture Notes

4.2

Externalities

Figure (2) shows an example of an externality. There are two main types of externalities: Positive and Negative. A well-known example of negative externality in Chile is related to the mining companies based on foreign capital. The mining companies go to the source of mineral, extract the natural resources and generate a lot of pollution, and pay less than 5% in taxes. The farmers and peasants who live nearby see the death of their animals and the pollution of their water. An example of a positive externality is this note. I am writing it for me, but I share it with anyone who wants to download from Scrib.com. There are three main types of externalities: 1. Production externality 2. Common Property Rights 3. Congestion externality

Figure 2: Example of an externality.

4.2.1

Production Externality

Suppose that there are two firms i = A, B and they pr...