General Equilibrium and Welfare Economics

ECON501: Lecture NotesMicroeconomic Theory II

by Jorge Rojas

Contents

1 General Equilibrium 21.1 Walrasian Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Existence of Walrasian Equilibria . . . . . . . . . . . . . . . . . . . . . . 31.3 Existence of Walrasian Equilibria . . . . . . . . . . . . . . . . . . . . . . 4

2 Jones Model 52.1 Input endowment magnification effect . . . . . . . . . . . . . . . . . . . . 52.2 Output price magnification effect . . . . . . . . . . . . . . . . . . . . . . 62.3 Magnification effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Welfare Economics 63.1 FIRST THEOREM OF WELFARE ECONOMICS . . . . . . . . . . . . 73.2 SECOND THEOREM OF WELFARE ECONOMICS . . . . . . . . . . . 7

4 Public Goods and Externalities 84.1 Public Goods and Competitive Markets . . . . . . . . . . . . . . . . . . . 94.2 Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Production Externality . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Common Property Rights . . . . . . . . . . . . . . . . . . . . . . 114.2.3 Congestion Externality . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Practical Themes 125.1 Intertemporal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Robinson Crusoe (Coop-structure) . . . . . . . . . . . . . . . . . . . . . 135.3 Fisher Separation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1


by Jorge Rojas

Abstract

This is a summary containing the main ideas in the subject. This is not asummary of the lecture notes, this is a summary of ideas and basic concepts. Themathematical 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 equilib-rium model all prices are variable, and equilibrium requires that all markets clear.

There is a special case called “pure exchange” economy where all the economicagents 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 writtenas x = (xi, . . . ,xn). An allocation x is a collection of n consumption bundles describingwhat each of the n agents holds.

Definition 1. Feasible Allocation.

A feasible allocation is one that is physically possible, therefore:

n∑i=1

xi ≤n∑

i=1

~ωi

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

Figure 1: Edgeworth Box.

1Without Equality in Opportunities, Freedom is the privilege of a few, and Oppression the reality ofeveryone else.

University of Washington Page 2


by Jorge Rojas

1.1 Walrasian Equilibrium

Let us define p = (p1, . . . , pk) which is exogenously given. Each consumer solves theproblem:

Maxxiui(xi)

s.t. pxi = p~ωi

Notice that for an arbitrary price vector p, it might not be possible to make the desiredtransactions for the simple reason that aggregate demand may not be equal to aggregatesupply:

n∑i=1

xi(p,p~ωi) 6=n∑

i=1

~ωi

Definition 2. Walrasian Equilibrium.

We define a W.E. to be a pair (p∗,x∗) such that:

n∑i=1

xi(p,p~ωi) ≤n∑

i=1

~ωi

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 homo-geneous functions is homogeneous, then the aggregate excess demand function is alsoH.O.D. zero in prices.

z(p∗) =n∑

i=1

[xi(p∗,p∗~ωi)− ~ωi] ≤ 0

So, z : Rk+ ∪ {0} 7→ Rk

Theorem 1. Walras’ Law: For any price vector p, we have p ·z(p) = 0, i.e., the valueof the excess demand is identically zero.

The proof is direct. z = 0 since xi(·) must satisfy the budget constraint for each agenti.

Corollary 1. Market Clearing: If demand equals supply in (k − 1) markets, andpk > 0 then demand must equal supply in the kth market.

Definition 3. Free Goods: If p∗ is a Walrasian Equilibrium andzj(p

∗) < 0, then p∗j = 0

In other words, if some good is in excess supply at a Walrasian equilibrium, it mustbe 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 imposeequality of the demand and the supply, almost all the time.

Theorem 2. Equality of Demand and Supply.



by Jorge Rojas

If all goods are desirable and p∗ is a Walrasian equilibrium, then z(p∗) = 0. Theproof is done by contradiction.

Since the aggregate excess demand z(p) is H.O.D. zero, we can express everything interms of relative prices. Thus, we get:

pi =pi∑kj=1 pj

,and therefore,k∑

i=1

pi = 1

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

i=1 pi = 1}

Theorem 3. BROUWER FIXED-POINT THEOREM

If f : Sk−1 7→ Sk−1 is a continuous function from the unit simplex to itself, there issome x ∈ Sk−1 such that x = f(x).

Another useful calculus tool is the Intermediate Value Theorem. If f is a real-valuedcontinuous function on the interval [a, b], and I is a number between f(a) and f(b), thenthere is a c ∈ [a, b] such that f(c) = I.

Theorem I below is one of the most important theorems in modern economics (in myopinion). This theorem can be used for the good or for the bad. So, be wise!

1.3 Existence of Walrasian Equilibria

If z : Sk−1 7→ Rk is a continuous function that satisfies Walras’ law, i.e., p · z(p) ≡ 0,then there exists some p∗ ∈ Sk−1 such that z(p∗) ≤ 0

I write the proof for this theorem given its importance in economics and the fact thatthe proof for this idea took around a hundred years to exist in its formal way.

Proof:Define g : Sk−1 7→ Sk−1 by:

gi(p) =pi + max(0, zi(p))

1 +∑k

j=1 max(0, zj(p))∀i = 1, . . . , k

Thus,∑k

i=1 gi(p) = 1

The map g has a nice economic interpretation. Suppose that there is excess demand insome 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,

p∗i =p∗i + max(0, zi(p

∗))

1 +∑k

j=1 max(0, zj(p∗))∀i

=⇒

p∗i

k∑j=1

max(0, zj(p∗)) = max(0, zi(p

∗)) ∀i



by Jorge Rojas

we multiply by zi(p∗), we get:

zi(p∗) · p∗i

[ k∑j=1

max(0, zj(p∗))]

= zi(p∗) ·max(0, zi(p

∗)) ∀i

now, we sum across all the agents, so we get:[ k∑j=1

max(0, zj(p∗))]·

k∑i=1

zi(p∗)p∗i =

k∑i=1

zi(p∗) ·max(0, zi(p

∗))

By Walras’ Law, we know that:∑k

i=1 zi(p∗)p∗i = p∗z(p∗) = 0

=⇒k∑

i=1

[zi(p

∗) ·max(0, zi(p∗))]

= 0 (1)

Each term of the sum in (1) is greater or equal to zero since each term is either 0 orz2i (p∗) > 0 if zi(p

∗) > 0. However, if any term were strictly greater than zero, theequality 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-profitconditions. 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 foreach 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 problemcorresponds to the technical coefficients, i.e., L∗i = aLi and T ∗i = aT i. Recall that apq isthe 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 thecommodity that uses the other factor more intensively.

Assuming that:aLMaTM

>aLFaTF

⇔M is more labour intensive than F

If L increases, then F will decrease.

The proof for theorem (4) is done using the full-employment conditions, and takingdifferentials.



by Jorge Rojas

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 inpM raises the return to the factor used intensively in M production by an even greaterrelative amount.

M is labour intensive and pM increases =⇒ dw

w>

dpMpM

The proof for theorem (5) is done using the zero-profit conditions, and taking differ-entials.

To avoid multiple equilibria, we impose that one output is more intensive in one inputfor all (w, r) ∈ R2

+. Thus, the lines intersect only ones.

2.3 Magnification effects

Input endowment magnification effect (3). General Rybczynski theorem.

(aLFaTF

>aLMaTM

)∧(dL

L>

dT

T

)=⇒ dF

F>

dL

L>

dT

T>

dM

M(3)

Output price magnification effect (4). General Stolper-Samuelson theorem.

(aLFaTF

>aLMaTM

)∧(dpFpF

>dpMpM

)=⇒ dw

w>

dpFpF

>dpMpM

>dr

r(4)

3 Welfare Economics

Definition 5. Pareto Efficiency

A feasible allocation x is a weakly Pareto efficient allocation if there is no feasibleallocation x

′such that all agents strictly prefer x

′to x.

A feasible allocation x is a strongly Pareto efficient allocation if there is no feasibleallocation 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 weaklyPareto 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:

(1) the allocation is feasible:∑n

i=1 xi ≤∑n

i=1 ~ωi

(2) If x′i is preferred by agent i to xi, then p ·x′i > p · ~ωi, i.e., the agent is only better off

with a bundle that cannot afford.



by Jorge Rojas

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 · x′i > p · ~ωi ∀i = 1, . . . , n

summing over i = 1, . . . , n and using the fact that x′

is feasible, we have that:

pn∑

i=1

~ωi = pn∑

i=1

x′

i >

n∑i=1

p · ~ωi

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 amountof each good. Suppose that preferences are convex, continuous, and monotonic.Then, x∗ is a Walrasian equilibrium for the initial endowments ~ωi = x∗i ∀i = 1, . . . , n.

Another version: The following lines are taken from MWG, pages 551-552.

Definition 7. Given an economy specified by ({(Xi,�i)}Ii=1, {Yj}Jj=1, ω) an allocation(x∗,y∗) and a price vector p = (p1, . . . , pL) 6= 0 constitute a price quasi-equilibriumwith transfers if there is an assignment of wealth levels (w1, . . . , wI) with

∑iwi =

p · ω +∑

j p · y∗j such that:

(i) ∀j, y∗j maximises profits in Yj; that is,

p · yj ≤ p · y∗j ∀yj ∈ Yj

(ii) ∀i, if xi �i x∗i then p · xi ≥ wi

(iii)∑

i x∗i = ω +

∑j y∗j

Theorem 6. Second fundamental theorem of Welfare Economics.

Consider an economy specified by ({(Xi,�i)}Ii=1, {Yj}Jj=1, ω), and suppose that every

Yj is convex and every preference relation �i is convex [i.e., the set {x′i ∈ Xi : x′i �i xi}

is convex for every xi ∈ Xi] and locally non-satiated. Then, for every Pareto optimalallocation (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.There are two goods {X, Y }, two individuals {A,B}, two inputs {L, T} (exogenously



by Jorge Rojas

fixed), X = F (LX , TX) and Y = G(LY , TY ). We solve the following optimization problemto find the locus of Pareto efficient allocations.

Max uA(XA, YA) (5)

s.t. uB(XB, YB) = uB

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

=∂uB

∂XB

∂uB

∂YB

=α

β

Efficiency in Production

∂F∂LX

∂F∂TX

=∂G∂LY

∂G∂TY

=γ

δ

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 “marginalcost”.

4 Public Goods and Externalities

Definition 8. A public good is a commodity for which use of a unit of the good by oneagent does not preclude its use by other agents.2

Suppose that X is a public good (for example, education, healthcare, “nationaldefense”) and Y is a private good (for example, wine, cigarettes, cars). There are twoagents in this basic model i = A,B. They face the following maximization problem:

Max{X,YA}

uA(X, YA)

s.t. uB(X, YB) = uB

F (TX) = X

G(TY ) = YA + YB

TX + TY = T

2Mas-Colell, Whinston and Green (MWG), page 359.



by Jorge Rojas

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 andwell-behaved model:

∂uA

∂X∂uA

∂YA

+∂uB

∂X∂uB

∂YB︸︷︷︸Marginal Benefit (agreggated)

=G′(TY )

F ′(TX)︸︷︷︸Marginal Cost

(6)

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 nototherwise. As a society, we must define these goods. For instance, is education a publicgood 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 publicgood and the other one the private good. Therefore, each firm solves their maximizationproblem, i.e., they try to maximise profits. In general terms,

Max{K,L}

ΠX = pxF (K,L)− wL− rK (7)

So, from (7) we get the price for the good X, and likewise for good Y . Now, assume thatthe consumers own the land T and the firms, so they can use the profits(profits are zeroif F(·) has CRS). Then, the consumers have to solve the problem given by:

Max{Xi,Y−i}

ui(Xi +X−i, Yi) ∀i = A,B

s.t. pxXi + pyYi = rTi + profits (8)

After some manipulation, we get the condition MRSA = MRSB = MRT which isclearly different to the Pareto Optimal condition for public goods. Therefore, competitivemarkets will NOT be efficient in the provision of public goods. Notice that propertyrights 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 free-riders. However, if the agents have different utility functions, then there will be free-riders. 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 modelin 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 notcare 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 samepreferences because inequality is monstrous (Chile is top 20!)3.

3Chile is the 17th most unequal country in the world, according to the CIA. Sourcehttps://www.cia.gov/library/publications/the-world-factbook/rankorder/2172rank.html and using othersurveys we are top 10!


https://www.cia.gov/library/publications/the-world-factbook/rankorder/2172rank.html


by Jorge Rojas

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 relatedto the mining companies based on foreign capital. The mining companies go to thesource of mineral, extract the natural resources and generate a lot of pollution, and payless than 5% in taxes. The farmers and peasants who live nearby see the death of theiranimals 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 produce goods X and Y , respectively.Firm A pollutes in its production process and this pollution affects the profits of firm B.

Production and PollutionFirm A Firm Bprice good X: p price good Y: qcost function: C(X) cost function: D(Y, s)Pollution: s = s(X)

So, in this situation, the methodology of analysis is standard. First, we analyse each firmseparately obtaining the “individual optimal outcome”. Second, we put the two firmstogether and we maximise joint profits. Thus, we obtain the “social optimal outcome”.In general, the profits coming from both firms working together will be greater thansumming the profits from each firm working on its own (Cooperation is positive ifnot needed).

Private marginal cost: C′(X)

Social marginal cost:C′(X) + ∂D

∂ss′(X)← also called true MC


http://www.scribd.com


by Jorge Rojas

To solve the problem of externalities there are three main economic tools that can bedeployed by the government or by the private counterparts:

1. Taxes (You subtract τX from the polluter’s profit function)

2. Licenses

3. Agreements between the private agents

Theorem 7. COASE THEOREM: Under zero transaction costs, if trade of theexternality can occur, then bargaining will lead to an efficient outcome no matter howproperty rights are allocated.4

4.2.2 Common Property Rights

An example of this situation is fishing on international waters. Suppose that each agenthas only one boat, and the cost of sending the boat is constant. Then:Equilibrium: R(X) = CPareto Efficiency: Max XR(X)−XCIn general, when there is a negative externality, the Pareto efficient outcome of fishing(production) will be less than the competitive one.

Figure 3: Social optimal versus private optimal. Graph from http://tutor2u.net

4.2.3 Congestion Externality

Suppose that there is a bridge with some demand for trips. The demand function is givenby:

X = α− βt α, β > 0⇒ t =α−Xβ

where X is the number of crossings and t the time per trip. In addition, there is acongestion function

t = γ + δX γ, δ > 0

4MWG, page 357, second paragraph.


http://tutor2u.net


by Jorge Rojas

The conditions for this problem will be given by:For Equilibrium:

α−Xβ︸︷︷︸

Marginal benefit

= γ + δX︸︷︷︸Marginal cost

For Pareto Efficiency: First, we need to calculate the “true marginal cost”. This isdone as X × private cost = X · (γ + δX), then we just differentiate to get the MC. Onthe benefit side, there is no change, since the externality is fully negative.

γ + 2δX︸︷︷︸Social marginal cost

=α−Xβ︸︷︷︸

Marginal benefit (unchanged)

To achieve the Pareto efficient outcome the government could install a toll. The ideais to make the users pay the “true value” of their trips. Thus, the toll price should beτ = PSoc − POp (assuming we already translated time to money).

Figure 4: Congestion in a bridge.

5 Practical Themes

5.1 Intertemporal Approach

There is one agent that consumes at two points in time and the interest rate, r, isexogenously given. The agent, therefore, solves the problem:

Max{C0i,C1i} u(C0i, C1i) (9)

s.t. p0C0i + p1C1i = p0ω0i + p1ω1i

or equivalently, C0i + C1i

1+r= ω0i + ω1i

1+rsince p0

p1= 1 + r.



by Jorge Rojas

5.2 Robinson Crusoe (Coop-structure)

There is one consumer and one firm that produces only one good with only one input(labour). C: consumptionL: leisure per dayH: hours of work per dayQ = F (H): output producedThe firm wants to maximise profits (it wants “more and more”):

Max{H}

Π = pF (H)− wH (10)

while the consumer (who owns the firm) solves:

Max{C,L} u(C,L) (11)

s.t. pC = pC0 + wH + Π∗(p, w)

H + L = 24hrs

Remark 4. A brief note on Returns to Scale.

1. DRS⇒ profits are positive and the setup is compatible with “perfect competition”.

2. CRS ⇒ profits are zero (zero-profit conditions) and is compatible with P.C.

3. IRS ⇒ firm may run into infinite size, but this setup is NOT compatible with P.C.since the SOC’s are never satisfied. (IRS implies a monopoly, duopoly or anothersetup).

5.3 Fisher Separation Theorem

In a Robinson Crusoe two-periods type model, but with n agents. Each agent i solvesthe problem:

Max{C0i,C1i} u(C0i, C1i) (12)

s.t. C0i + C1i

1+r= ω0i + ω1i

1+r+ Fi(I0i)

1+r− I0i

where Q1i = Fi(I0i) and I0i is the amount that is invested at t = 0 to produce Q1i. Agenti has an initial endowment ωi

Irving Fisher showed that this problem can be solved in two stages:

1. maximize the value produces by the firm, i.e., Fi(I0i)1+r

− I0i

2. using I∗0i solve the whole consumer problem

We define the net borrowing for i as NBi = C∗0i + I∗0i(r)−ω0i. In equilibrium we get that∑Ii=1 NBi = 0.


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General Equilibrium and Welfare Economics