COMPETITIVE MARKETS.

(Partial Equilibrium Analysis)

Consider an economy with:

I consumers, i = 1, ...I.

J firms, j = 1, ...J.

L goods, l = 1, ...L.

Initial endowment of good l in the economy: ωl ≥ 0.

Consumer i’s:

Consumption set Xi ⊂ RL.

Utility function: ui : Xi→ R.

Production technology of firm j : Yj ⊂ RL.

yj ∈ Yj is a production vector yj = (y1j, ...yLj) ∈RL, j = 1, ...J.

Total (net) availability of good l in the economy: ωl +PJj=1 ylj, l = 1, ....L.

Pareto Optimality.

Definition. An economic allocation (x1, ..., xI, y1, ...yJ)is a specification of a consumption vector xi ∈ Xi foreach consumer i = 1, ...I, and a production vector yj ∈Yj for each firm j = 1....J.

The allocation is feasible if

IXi=1

xli ≤ ωl +JX

j=1

ylj for l = 1, ...L.

Definition. A feasible allocation (x1, ..., xI, y1, ...yJ) isPareto optimal (or, Pareto efficient) if there is no otherfeasible allocation (x01, ..., x0I, y01, ...y0J) such that

ui(x0i) ≥ ui(xi) for all i = 1, ...I,

and

ui(x0i) > ui(xi) for some i = 1, ...I.

Competitive Equilibria.Competitive market economy: initial endowments andtechnological possibilities (firms) are owned by consumers.

Consumer i initially owns ωli ≥ 0 of good l, , l =1, ...L,where

IXi=1

ωli = ωl.

Initial enowment vector of consumer i:

ωi = (ω1i, ...., ωLi).

In addition each consumer i owns a share θij of firm j

giving her a share θij of the profits of firm j, j = 1, ...J .

IXi=1

θij = 1.

A market exists for all goods.

All consumers and producers act as price takers i.e., as-sume that market prices are unaffected by their actions.

Denote vector of prices: p = (p1, ...pL).

Definition. The allocation (x∗1, ..., x∗I, y∗1, ...y∗J) and aprice vector p∗ ∈ RL+ constitute a competitive (or, Wal-rasian) equilibrium if the following conditions are satis-fied:

(i) Profit maximization: For each firm j, y∗j solves

maxyj∈Yj

(p∗yj).

(ii) Utility maximization: For each consumer i, x∗i solves

maxui(xi)

s.t.

p∗xi ≤ p∗ωi +JX

j=1

θij(p∗y∗j ).

(iii) Market clearing: For each good, l = 1, ...L,

IXi=1

x∗li = ωl +JX

j=1

y∗lj.

Sometimes we permit excess supply in equilibrium withprice of the good being zero; assuming free disposal.

If goods are "desirable", for example if marginal utility isalways strictly positive, then this possibility is ruled out.

Note: if p∗ >> 0 and (x∗1, ..., x∗I, y∗1, ...y∗J) is a competi-tive equilibrium then so does the allocation (x∗1, ..., x∗I, y∗1, ...y∗J)and price vector αp∗ for any α > 0.

So, we can always normalize prices without loss of gen-erality.

Lemma. If the allocation (x1, ..., xI, y1, ...yJ) and pricevector p >> 0 satisfy the market clearing condition (iii)for all goods l 6= k, and if every consumer’s budget con-straint is satisfied with equality so that

pxi = pωi +JX

j=1

θij(pyj), for all i = 1...I, (1)

then the market for good k also clears (i.e., (iii) holdsfor l = k).

Proof. Adding (1) over i = 1...L we get

p[IX

i=1

(xi − ωi − (JX

j=1

θijyj))] = 0

so that

LXl=1

pl(IX

i=1

xli − ωl −IX

i=1

JXj=1

θijylj) = 0

i.e.,LXl=1

pl(IX

i=1

xli − ωl −JX

j=1

ylj) = 0

or,

Xl 6=k

pl(IX

i=1

xli−ωl−JX

j=1

ylj) = −pk(IX

i=1

xki−ωk−JX

j=1

ykj)

and as the left hand side is zero, the RHS is zero andsince pk > 0, we have

(IX

i=1

xki − ωk −JX

j=1

ykj) = 0.

Partial Equilibrium Analysis.

Analysis of market for one (or several) goods that form asmall part of the economy.

Marshall (1920): consider one good that accounts forsmall fraction of consumer’s total expenditure.

The wealth (or income) effect on the demand for thegood can be negligible.

Substitution effect of change in the price of the good isdispersed among all goods and so prices of other goodsare approximately unaffected.

So, for the analysis of this market, we can take prices ofall other goods as fixed.

Expenditure on all other goods taken to be a compositecommodity - the numeraire.

The Basic Quasi-linear Model:

Consumers i = 1, ...I.

Two commodities: good l and the numeraire.

xi : consumer i0s consumption of good l.

mi : consumer i0s consumption of the numeraire (i.e.,expenditure on all other goods).

Consumption set of consumer i : R× R+.

(Allow negative consumption of the numeraire good -"borrowing" - assumption avoids dealing with corner so-lution).

Utility function:

ui(mi, xi) = mi + φi(xi), i = 1, ...., I

Assume:

φi(.) is bounded above, twice continuously differentiable,

φi(0) = 0,

φ0i(xi) > 0, φ”i (xi) < 0,∀xi ≥ 0.

Quasi-linear formulation: no wealth effect.

Normalize price of the numeraire good to equal 1.

Let p be the price (relative price) of good l,

Then, one can think of φi(xi) as measuring utility interms of the numeraire good

.

Firm j = 1, ...J, produces qj units of good l using (atleast) amount cj(qj) of the numeraire good.

cj(qj) : "cost function" of firm j.

Technology of firm j:

Yj = {(−zj, qj) : qj ≥ 0, zj ≥ cj(qj)}.

Assume:

cj : R+→ R+ is twice differentiable.

c0j(qj) > 0 and c”j(qj) ≥ 0 at all qj ≥ 0.

[Think of cj(qj) as derived from a cost minimizationproblem with fixed input prices.]

Non-decreasing marginal cost curve (allows for constantand decreasing returns to scale).

Also continuity of cj at 0 rules out any fixed cost that isnot sunk

(cost can be avoided by producing zero).

Initial endowment: No initial endowment of good l.

Consumer i’s initial endowment of the numeraire good :ωmi > 0.

Let

ωm =IX

i=1

ωmi

be the total endowment of the numeraire good in theeconomy.

Competitive Equilibrium:

Profit max.

Given equilibrium price p∗ for good l, firm j’s equilibriumoutput q∗j solves

maxqj≥0

[p∗qj − cj(qj)]

Necessary and sufficient first order condition:

p∗ ≤ c0j(q∗j ), if q∗j = 0 (2)

= c0j(q∗j ), if q∗j > 0. (3)

Utility max.

Given p∗ and the solution to the firms’ profit maximiza-tion problems, consumer i’s equilibrium consumption (m∗i , x∗i )solves:

maxmi∈R,xi∈R+

[mi + φi(xi)]

s.t.

mi + p∗xi ≤ ωmi +JX

j=1

θij(p∗q∗j − cj(q

∗j ))

Budget constraint holds with equality in any solution tothe above problem.

Rewrite the problem without of loss of generality as oneof choosing only the consumption of good l:

maxxi∈R+

[ωmi +JX

j=1

θij(p∗q∗j − cj(q

∗j ))− p∗xi + φi(xi)]

or equivalently, x∗i must solve

maxxi≥0

[φi(xi)− p∗xi]

and m∗i is determined by

m∗i = ωmi +JX

j=1

θij(p∗q∗j − cj(q

∗j ))− p∗x∗i .

A necessary and sufficient first order condition:

p∗ ≥ φ0i(x∗i ), if x∗i = 0, (4)

= φ0i(x∗i ), if x∗i > 0. (5)

Thus, an equilibrium allocation is characterized fully by aprice p∗ of good l and the vector (x∗1, ...., x∗I, q∗1, ..., q∗J)of consumption and production of good l.

Finally, market clearing for good l requires:

IXi=1

x∗i =JX

j=1

q∗j . (6)

Proposition: The allocation (x∗1, ...., x∗I, q∗1, ..., q∗J) andprice p∗ constitutes a competitive equilibrium if and onlyif.

p∗ ≤ c0j(q∗j ), if q∗j = 0= c0j(q∗j ), if q∗j > 0, j = 1, ....J

p∗ ≥ φ0i(x∗i ), if x∗i = 0,= φ0i(x∗i ), if x∗i > 0, .i = 1, ...I

IXi=1

x∗i=JX

j=1

q∗j .

The above (I+J+1) conditions determine the (I+J+1)equilibrium values (x∗1, ...., x∗I, q∗1, ..., q∗J , p∗).

The equilibrium allocation and price of good l are entirelyindependent of the distribution of initial endowments andownership shares of firms.

Observe: since φ0i(xi) > 0,∀xi ≥ 0, it follows that theequilibrium price p∗ > 0.

Assume:

maxi

φ0i(0) > minj

c0j(0).

Then, in equilibrium, total consumption and production

of good l :IX

i=1

x∗i=JX

j=1

q∗j > 0.

[If all consumers consume 0 and all firms produce 0, then

c0j(0) ≥ p∗ ≥ φ0i(0), i = 1, ...I, j = 1, ...J,

so that

maxi

φ0i(0) ≤ minj

c0j(0),

a contradiction.]

One can derive the equilibrium through traditional Mar-shallian demand-supply analysis.

Demand :

For any p, each consumer’s first order condition:

p ≥ φ0i(xi), if xi = 0,= φ0i(xi), if xi > 0.

Note φ0i is a continuous and strictly decreasing functionon R+ with range [0, φ0i(0)].

Individual Walrasian demand function of consumer i :

xi(p) = 0, p ≥ φ0i(0),= φ0−1i (p), p ∈ (0, φ0i(0).

Note individual demand xi(p) is independent of wealth,continuous & non-increasing in p and strictly decreasingin p on (0, φ0i(0)).

Aggregate (market) demand for good l:

x(p) =IX

i=1

xi(p)

- independent of endowment & the distribution of endow-ments,

- continuous & non-increasing in p and

- strictly decreasing in p on (0,maxi φ0i(0)).

Note that individual and aggregate demand is infinite atzero price.

Also, aggregate demand is zero for p ≥ maxi φ0i(0).

Supply:

For any p, firm j0s profit max yields the following firstorder condition:

p ≤ c0j(qj), if qj = 0= c0j(qj), if qj > 0.

If p < c0j(0), firm’s supply is zero.

Suppose cj is strictly convex (upward sloping marginalcost) and c0j(q)→∞ as q →∞.

Then, for each p > c0j(0), there is a unique qj such thatp = c0j(qj).

The firm’s supply curve in that case:

qj(p) = 0, p ≤ c0j(0),= c0−1j (p), p > c0j(0).

qj(p) is

- continuous and non-decreasing in p &

- strictly increasing for p > c0j(0).

The aggregate (market) supply curve is given by:

q(p) =JX

j=1

qj(p).

Note that q(p) = 0 for p ≤ minj c0j(0).

For p > minj c0j(0), q(p) is strictly positive, strictly in-

creasing and continuous.

The market equilibrium price p∗ is given by the pointwhere aggregate demand and supply intersect i.e.,

x(p∗)− q(p∗) = 0 (7)

Let z(p) = x(p)− q(p). Then,

z(p) = x(p) > 0, p ≤ minj

c0j(0)

= −q(p) < 0, p ≥ maxi

φ0i(0).

z(p) is continuous and strictly decreasing in p on (minj c0j(0),max

Unique p∗ ∈ (minj c0j(0),maxi φ0i(0)), such that z(p∗) =0 i.e.,(7) holds.

The equilibrium allocation is given by setting x∗i = xi(p∗), i =

1, ...I, q∗j = qj(p∗), j = 1, ...J.

If cj is convex but not strictly convex (for example, lin-ear), there may not be unique solution to the profit maxproblem and so qj(p) is a correspondence (upper hemi-continuous, using Maximum Theorem).

Similar analysis goes through with more technical argu-ments.

Important case: Constant returns to scale. cj(qj) = cjqjwhere cj > 0 is the constant average as well as marginalcost ("unit cost").

Firm j’s supply function

qj(p) = 0, p < cj

∈ [0,∞), p = cj

= ∞, p > cj.

The aggregate supply is infinite (not well defined as a realnumber) for p > cj.

If all firms have constant returns to scale technology withcost functions cj(qj) = cjqj,j = 1, ...J,

then the unique equilibrium price p∗ = minj cj.

Only firms with the minimum unit cost can produce inequilibrium (such firms are indifferent between all levelsof output at that price).

The total quantity of good l produced and consumedis given by the aggregate demand function and equalsx(p∗).

If there are multiple firms with the minimum unit cost,the way the total quantity demanded x(p∗) is producedacross these firms is not uniquely determined.

Recall, industry supply:

q(p) =JX

j=1

qj(p)

where

qj(p) = c0−1j (p),∀j such that qj(p) > 0.

For any y > 0, the inverse of the aggregate supply func-tion given by q−1(y) indicates the equalized marginalcost of all firms that produce this output: q−1(y) is theindustry’s marginal cost curve.

Define the industry’s aggregate cost of producing anylevel of total output y by:

C(y) = minqj,j=1,...J

JXj=1

cj(qj)

s.t.JX

j=1

qj = y, qj ≥ 0, j = 1, ..J.

Lagrangean:

L(q1, ....qJ, λ) =JX

j=1

cj(qj) + λ(y −JX

j=1

qj)

First order necessary and sufficient conditions:

c0j(bqj) = λ, bqj > 0

≥ λ, bqj = 0

- all firms that produce strictly positive output, marginalcost is equalized to λ

-all firms that produce zero output, marginal cost at zerois no larger than λ

For any p, letting λ = p, we can see that bqj = qj(p), j =

1, ...J, must minimize industry’s aggregate cost of pro-

ducing y =JX

j=1

qj(p).

Further, using envelope theorem:

C0(y) = λ

= c0j(bqj),∀j such that bqj > 0.

Thus, industry’s marginal cost of producing y =JX

j=1

qj(p)

is given by c0j(qj(p)), for all j such that qj(p) > 0 i.e.,q−1(y), the inverse aggregate supply curve.

Therefore,

C(q) = C(0) +

qZ0

C0(y)dy

= C(0) +

qZ0

q−1(y)dy

so that

qZ0

q−1(y)dy = C(q)− C(0)

The area under the aggregate supply curve is equal to the(minimized) total variable cost of the industry (i.e., thetotal cost excluding the sunk cost).

The area under individual supply curve is the total vari-able cost of the firm.

Recall,

x(p) =IX

i=1

xi(p)

where

xi(p) = φ0−1i (p),∀i such that xi(p) > 0.

Let P (y) = x−1(y) be the inverse aggregate demandfunction.

Then, for any y, P (y) = φ0i(xi(P (y))), ∀i such thatxi(p) > 0.

In other words, P (y) represents the marginal benefit fromconsumption of good l to a consumer that consumesstrictly positive quantity when the price is P (y) and totalquantity y is consumed.

So, P (y) represents the marginal benefit to society fromtotal consumption of amount y.

For any y > 0, define the (maximum) social benefit fromtotal consumption of amount y :

B(y) = maxxi,i−1,..I

[φi(xi)]

s.t.IX

i=1

xi = y, xi ≥ 0, i = 1, ..I.

Define Lagrangean

L(x1, ....xI, μ) =IX

i=1

(φi(xi)) + μ(y −IX

i=1

xi)

First order necessary and sufficient conditions:

φ0i(bxi) = μ, bxi > 0

≤ μ, bxi = 0so that for all consumers that consume strictly positiveamount of good l, marginal utility is equalized to μ andfor all consumers that consume zero amount, marginalutility at zero does not exceed μ.

For any p, letting μ = p, we can see that bxi = xi(p), i =

1, ...I, must maximize society’s benefit from consuming

y =IX

j=1

xi(p).

Further, using envelope theorem:

B0(y) = μ

= φ0i(bxi),∀i such that bxi > 0.

Thus, society’s marginal benefit from consuming y =IX

j=1

xi(p) is given by φ0i(xi(p)),∀i such that xi(p) > 0

(the height of the individual demand curves at xi(p)) i.e.,P (y), the inverse aggregate demand curve.

Therefore, (using B(0) = 0)

B(x) =

xZ0

B0(y)dy

=

xZ0

P (y)dy

The area under the aggregate demand curve is equal tothe (maximum) social benefit from consumption of goodl.

The area under individual demand curve is the total ben-efit to the individual consumer.

To sum up:

1. Profit maximization by price taking firms ensures thatthe total output produced by the industry at any priceminimizes the industry’s cost of producing this amount(i.e., market distributes total output across firms opti-mally).

2. Utility maximization by price taking consumers en-sures that the total consumption in society is distributedacross consumers so as to maximize the total benefit tosociety (optimal distribution of consumption).

3. The height of the aggregate supply curve indicates theindustry’s marginal cost of production.

The area under the aggregate (inverse) supply curve in-dicates the industry’s total variable cost.

4. The height of the aggregate demand curve indicatesthe society’s marginal benefit from consumption.

The area under the aggregate (inverse) demand curve in-dicates society’s total benefit from consumption of goodl.

Pareto Optimality & Competitive Equilibrium.

Fix the consumption and the production levels of good lat (x1, ...., xI, q1, ..., qJ) where

IXi=1

xi=JX

j=1

qj.

The total amount of the numeraire available for distrib-ution amount consumers is

ωm −JX

j=1

cj(qj).

Quasilinear utility: transferable utility.

Transferring numeraire good across consumers in variousways one can generate a utility possibility set:

{(u1, u2, ..., uI) :IX

i=1

ui ≤IX

i=1

φi(xi)+ωm−JX

j=1

cj(qj)

The right hand side of the inequality defining the set is aconstant (given (x1, ...., xI, q1, ..., qJ)).

So, the frontier of this utility possibility set is a hyperplanewith normal vector (1, 1, ..., 1).

Changing the consumption and production levels of goodl i.e., the vector (x1, ...., xI, q1, ..., qJ) shifts the frontierof the utility possibility set in a parallel fashion.

The frontier moves outward or inward according to whether

IXi=1

φi(xi) + ωm −JX

j=1

cj(qj)

increases or decreases when we change (x1, ...., xI, q1, ..., qJ).

As long as the frontier can be shifted outwards by changein the vector (x1, ...., xI, q1, ..., qJ) the original situationis not Pareto optimal.

Thus, every Pareto optimal allocation must involve con-sumption and production profile (ex1, .....exI, ey1, ..., eyJ)for good l so as to shift the frontier as far out as possiblei.e., (ex1, .....exI, ey1, ..., eyJ) solves

max(x1,...,xI)≥0(q1,....qJ)≥0

[IX

i=1

φi(xi)−JX

j=1

cj(qj)]

s.t.IX

i=1

xi =JX

j=1

qj

The maximand

[IX

i=1

φi(xi)−JX

j=1

cj(qj)]

is often called the Marshallian aggregate surplus (or totalsurplus).

It measures the net benefit to society from producing andconsuming good l.

There exists a solution to the surplus maximization prob-lem.

While there are a continuum of Pareto optimal allocationscorresponding to points on the highest utility possibilityfrontier, they all must involve a consumption and pro-duction vector for good l such as (ex1, .....exI, ey1, ..., eyJ)which solves the surplus max problem.

In particular, if the solution to the surplus max problemis unique (for example, when cj is strictly convex):

- all Pareto optimal allocations must involve exactly thesame production and consumption vector for good l

- the only difference in various Pareto optimal alloca-tions would arise from differences in distribution of thenumeraire good

(that can transfer utility from one agent to another unitfor unit).

Lagrangean:

L =IX

i=1

φi(xi)−JX

j=1

cj(qj) + μ[JX

j=1

qj −IX

i=1

xi]

First order necessary and sufficient condition (maximandis concave, feasible set is convex):

μ ≤ c0j(eqj), if eqj = 0= c0j(eqj), if eqj > 0, j = 1, ...J.

φ0i(exi) ≤ μ, if exi = 0,= μ, if exi = 0, i = 1, ...I

JXj=1

eqj = IXi=1

exiSetting μ = p∗, we see that these conditions are satisfiedby the production and consumption profile for good l inany competitive equilibrium allocation.

The First Fundamental Theorem of welfare Eco-nomics.

Proposition.

If the price p∗ and the allocation (x∗1, ...., x∗I, q∗1, ..., q∗J)constitutes a competitive equilibrium, then this allocationis Pareto optimal.

Conversely, consider any Pareto optimal allocation.

The production and consumption levels of good l in anysuch allocation must solve the surplus max problem andsetting p = μ we can check that the solution to surplusmax (ex1, .....exI, ey1, ..., eyJ) is a competitive equilibrium.

Consider this competitive equilibrium.

The price, the equilibrium consumption and productionof good l and the profits of firms are unaffected by anytransfer of the numeraire good from one agent to another.

Transferring the numeraire good from one agent to an-other changes the utility of the agents by exactly theamount of the transfer.

Therefore, can always generate the exact profile of util-ities and numeraire good consumption in the candidatePareto optimal allocation by transferring numeraire goodfrom one agent to another.

Can attain any point on the Pareto optimal boundary ofthe utility possibility set by transferring numeraire goodacross consumers.

The Second Fundamental Theorem of Welfare Eco-nomics.

Proposition.

For any Pareto optimal levels of utility (u∗1, ..., u∗I), thereare transfers of the numeraire commodity (T1, ....TI) sat-isfying

PIi=1 Ti = 0 such that a competitive equilibrium

reached from the endowments (ωm1+T1, ...., ωmI+Ti)

yields precisely the utilities (u∗1, ..., u∗I).

Convex structure important for this result.

Welfare Analysis in Partial Equilibrium.

Social welfare function: assigns social welfare value (realnumber) to each profile of utility levels (u1, u2, ...uI) :

W (u1, u2, ...uI)

(Utilitarian welfare).

Assume: W is strongly monotonic in its arguments.

For any given consumption and production levels of goodl, (x1, ...xI, q1, ..., qJ), where

IXi=1

xi=JX

j=1

qj,

the utility vectors that are attainable are given by:

{(u1, u2, ..., uI) :IX

i=1

ui ≤IX

i=1

φi(xi)+ωm−JX

j=1

cj(qj)}.

As the boundary of this set expands, the maximum socialwelfare W attainable on this set (through redistributionof the numeraire good) increases (strictly).

Thus,

*For any strongly monotonic social welfare function W,a change in the consumption and production of good l

leads to an increase in (the maximum attainable) socialwelfare if and only if it increases the Marshallian surplus:

S(x1, ...xI, q1, ...qJ) = [IX

i=1

φi(xi)−JX

j=1

cj(qj)].

Thus, social welfare analysis of changes in the consump-tion and production of good l can be carried out exclu-sively in terms of the Marshallian surplus.

Indeed, as we have seen, Pareto efficiency also requiresthat the consumption and production of good l mustsatisfy

max(x1,...,xI)≥0(q1,....qJ)≥0

S(x1, ...xI, q1, ...qJ)

s.t.IX

i=1

xi=JX

j=1

qj.

Consider a consumption and production vector of good

l, (bx1, ...bxI, bq1, ...bqJ) such that for by = IXi=1

bxi(i) (bx1, ...bxI) solves:

maxxi,i−1,..I

[IX

i=1

φi(xi)]

s.t.IX

i=1

xi = by, xi ≥ 0, i = 1, ..I.

(ii) (bq1, ...bqJ) solvesmin

qj,j=1,...J

JXj=1

cj(qj)

s.t.JX

j=1

qj = by, qj ≥ 0, j = 1, ..J.

We have seen that:

φ0i(bxi) = P (by) = B0(by), ∀i such that bxi > 0

c0j(bqj) = C0(by),∀j such that bqj > 0,

where P is the inverse aggregate demand function, B0(.)is the industry marginal benefit and C0(.) is the industrymarginal cost (or the aggregate inverse supply function).

S(bx1, ...bxI, bq1, ...bqJ) = [IX

i=1

φi(bxi)− JXj=1

cj(bqj)]= B(by)− C(by)=

byZ0

B0(y)dy − C(0)−byZ0

C0(y)dy

=

byZ0

P (y)dy −byZ0

C0(y)dy − C(0)

= [

byZ0

[P (y)− C0(y)]dy]− S(0)

Note:

[

byZ0

[P (y)− C0(y)]dy]

is the area between the aggregate demand and supplysurves and can be written as :

[

byZ0

[P (y)− C0(y)]dy]

= [

byZ0

[P (y)− byP (by)] + [byP (by)− (C(by)− C(0))]

= CS(P (by)) + PS(P (by))where CS(p) and PS(p) denote the aggregate consumerand producer surplus generated in a (hypothetical) mar-ket with price taking consumers and producers at pricemarket price p.

Therefore, in partial equilibrium analysis, social welfaremaximization, Marshallian surplus maximization and Paretoefficiency are roughly equivalent in their implication forthe production and consumption of "the good" and even-tually reduce to maximization of CS + PS.

It is easy to see that [

byZ0

[P (y)−C0(y)]dy] is maximized

at the output where:

P (y∗) = C0(y∗)

i.e., social marginal benefit equates industry’s marginalcost.

As C0(y) is inverse aggregate supply curve, this is alsothe aggregate output consumed and produced in a com-petitive equilibrium (supply=demand).

Thus, competitive equilibrium outcome is equivalent toMarshallian surplus maximization.

All of this assumes no externalities or other distortions(taxes, subsidies etc).

Welfare loss due to distortions is measured by the changein CS +PS i.e., the area between aggregate demand andthe supply (or industry MC curve).

Sometmes, called deadweight loss.

Example. Welfare loss due to a distortionary tax (in acompetitive market).

Sales tax on good l: t per unit paid by consumers.

Tax revenue returned to consumers through lump sumtransfer (non distortionary spending).

Let (x∗1(t), ..., x∗I(t), q∗1(t), ..., q∗J(t)) and p∗(t) be thecompetitive equilibrium allocation and price given tax ratet.

FOC:

φ0i(x∗i (t)) = p∗(t) + t, for all i such that x∗i (t) > 0.

c0j(q∗j (t)) = p∗(t), for all j such that x∗j(t) > 0.

Let

x∗(t) = x(p∗(t) + t) =IX

i=1

x∗i (t).

Market clearing:

x(p∗(t) + t) = q(p∗(t))

* Easy to check that (over the range where a strictlypositive quantity is traded) p∗(t) is strictly decreasing int and that (p∗(t) + t) is strictly increasing in t.

x∗(t) is strictly decreasing in t (as long as it is strictlypositice) and

x∗(t) < x∗(0), t > 0.

Let S∗(t) = S(x∗1(t), ..., x∗I(t), q∗1(t), ..., q∗J(t)).

We have that

S∗(t) = [x∗(t)Z0

[P (y)− C0(y)]dy]− S(0)

Welfare change

= S∗(t)− S∗(0)

=

x∗(t)Zx∗(0)

[P (y)− C0(y)]dy

which is negative since x∗(t) < x∗(0) and P (y) >C0(y) for all y ∈ [0, x∗(0)).