# COMPETITIVE MARKETS. (Partial Equilibrium Analysis) Partial Equilibrium Analysis. Analysis of market

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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 +PJ j=1 ylj, l = 1, ....L.

Pareto Optimality.

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

The allocation is feasible if

IX i=1

xli ≤ ωl + JX

j=1

ylj for l = 1, ...L.

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

ui(x 0 i) ≥ ui(xi) for all i = 1, ...I,

and

ui(x 0 i) > ui(xi) for some i = 1, ...I.

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

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

IX i=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 .

IX i=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 a price 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

max yj∈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,

IX i=1

x∗li = ωl + JX

j=1

y∗lj.

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

If goods are "desirable", for example if marginal utility is always 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 price vector 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) holds for l = k).

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

p[ IX

i=1

(xi − ωi − ( JX

j=1

θijyj))] = 0

so that

LX l=1

pl( IX

i=1

xli − ωl − IX

i=1

JX j=1

θijylj) = 0

i.e., LX l=1

pl( IX

i=1

xli − ωl − JX

j=1

ylj) = 0

or,

X l 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 and since 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 a small part of the economy.

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

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

Substitution effect of change in the price of the good is dispersed among all goods and so prices of other goods are approximately unaffected.

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

Expenditure on all other goods taken to be a composite commodity - 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 in terms of the numeraire good

.

Firm j = 1, ...J, produces qj units of good l using (at least) 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 minimization problem with fixed input prices.]

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

Also continuity of cj at 0 rules out any fixed cost that is not 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 the economy.

Competitive Equilibrium:

Profit max.

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

max qj≥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:

max mi∈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 to the above problem.

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

max xi∈R+

[ωmi + JX

j=1

θij(p ∗q∗j − cj(q∗j ))− p∗xi + φi(xi)]

or equivalently, x∗i must solve

max xi≥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 a price 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:

IX i=1

x∗i = JX

j=1

q∗j . (6)

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

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

IX i=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 entirely independent of the distribution of initial endowments and ownership shares of firms.

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

Assume:

max i

φ0i(0) > min j

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

max i

φ0i(0) ≤ min j

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