Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Non-convexities

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

Almost essential Welfare: EfficiencyAdverse selection

Prerequisites

August 2006 August 2006

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Introduction What are non-convexities?What are non-convexities?

……awkward nameawkward name ……crucial conceptcrucial concept

Concerned with production…Concerned with production… drop the convenient divisibility assumptiondrop the convenient divisibility assumption potentially far-reaching consequences potentially far-reaching consequences

Approach:Approach: start with examination of economic issuesstart with examination of economic issues build a simple production modelbuild a simple production model examine efficiency implicationsexamine efficiency implications consider problems of implementation and policyconsider problems of implementation and policy

Terms other than “non-convexities” sometimes used…Terms other than “non-convexities” sometimes used… ……not always appropriatelynot always appropriately but can give some insight on to the range of issues:but can give some insight on to the range of issues:

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Other terms…? ““Increasing returns” Increasing returns”

but increasing returns everywhere are not essentialbut increasing returns everywhere are not essential ““Natural monopoly”Natural monopoly”

but issue arises regardless of market form…but issue arises regardless of market form… … … notnot essentially essentially one of industrial structure one of industrial structure

““Public utilities” Public utilities” but phenomenon is not but phenomenon is not necessarilynecessarily in the public sector in the public sector

None of these captures the concept exactlyNone of these captures the concept exactly We need to examine the economic issues more closely…We need to examine the economic issues more closely…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The issues

Basic model

Efficiency

Implementation

Non-convexities

The nature of non-convexities

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Issues: the individual firm Consider supply by competitive firmsConsider supply by competitive firms

upward-sloping portion of MC curveupward-sloping portion of MC curve supply discontinuous if there is fixed costsupply discontinuous if there is fixed cost

If there are lots of firmsIf there are lots of firms average supply is approximately continuousaverage supply is approximately continuous so we can get demand=supply at industry levelso we can get demand=supply at industry level

If there is in some sense a “natural monopoly”If there is in some sense a “natural monopoly” perhaps very large fixed cost?perhaps very large fixed cost? perhaps MC everywhere constant/falling?perhaps MC everywhere constant/falling? no supply in competitive market?no supply in competitive market?

In this case….In this case…. how does the firm cover costs?how does the firm cover costs? how “should” the firm behave?how “should” the firm behave? how can it be induced to behave in the required manner?how can it be induced to behave in the required manner?

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Issues: efficient allocations Related to the issue discussed for firmRelated to the issue discussed for firm Concerns implementation through the marketConcerns implementation through the market

non-convexities seen an aspect of “market failure”?non-convexities seen an aspect of “market failure”? consider reason for this…consider reason for this… ……and a solution?and a solution?

Relationship between CE and efficiencyRelationship between CE and efficiency fundamental to welfare economics fundamental to welfare economics examine key questions of implementationexamine key questions of implementation

First a simple example of how it works…First a simple example of how it works…

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Implementation through the market

p1

—p2

q*f p1

—p2

x*h

q2

q1`

f

f`

x2

x1`

h

h`

f(qf) = 0

Uh(xh) = Uh(x*h)Production possibilitiesFirm f max profitsU contourh min expenditure

all f and h optimise at these prices such that…for all pairs of goods MRS = MRT= price ratio

now for the two key questions.

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency and the market: key questions1.1. Is a competitive equilibrium efficient?Is a competitive equilibrium efficient?

Yes if all consumers are greedy, there is no hidden information, Yes if all consumers are greedy, there is no hidden information, and there are no externalitiesand there are no externalities

2.2. Can an arbitrary Pareto-efficient allocation be supported Can an arbitrary Pareto-efficient allocation be supported by a competitive equilibrium? by a competitive equilibrium? Yes if all consumers are greedy, there is no hidden information, Yes if all consumers are greedy, there is no hidden information,

there are no externalities and no non-convexitiesthere are no externalities and no non-convexities

If there are non-convexities the equilibrium price signals If there are non-convexities the equilibrium price signals could take the economy away from the efficient allocationcould take the economy away from the efficient allocation

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The issues

Basic model

Efficiency

Implementation

Non-convexities

Back to the firm….

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

A model of indivisibility (1) Take simplest model of production:Take simplest model of production:

a single output (a single output (qq)… )… ……from a single input (from a single input (zz))

The indivisibility:The indivisibility: A fixed amount of input required before you get any outputA fixed amount of input required before you get any output Otherwise production is conventionalOtherwise production is conventional qq = = ((zz − − kk) ) , , zz ≥ ≥ kk ((00) = 0, ) = 0, zz(∙) > 0, (∙) > 0, zzzz(∙) ≤ 0(∙) ≤ 0 qq = 0, = 0, zz < < kk

Given a required amount of output Given a required amount of output q q > 0…> 0… minimum amount of minimum amount of zz required is: required is: −−11((qq) + ) + kk

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Case 1Case 1

z

q

A model of indivisibility (2) The minimum input The minimum input

requirementrequirement zz(∙) > 0, (∙) > 0, zzzz(∙) < 0(∙) < 0 Attainable setAttainable set

0 k

Case 2Case 2

z

q The minimum input The minimum input requirementrequirement

zz(∙) > 0, (∙) > 0, zzzz(∙) = 0(∙) = 0 Attainable setAttainable set

0 k

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

A model of indivisibility (3) Suppose units of input can be bought for Suppose units of input can be bought for ww What is cost of output What is cost of output qq??

clearly clearly CC((ww, 0) = 0 and, 0) = 0 and CC((ww,,qq) = ) = vv((ww,,qq) + ) + CC00, for , for q q > 0, > 0, where variable cost iswhere variable cost is v v((ww,,qq) = ) = ww−−11((qq) ) and fixed cost is and fixed cost is CC00 = = wkwk

Therefore: Therefore: marginal cost: marginal cost: ww / / zz((−−11((qq)) )) average cost: average cost: ww−−11((qq) / ) / q q + + CC00 / / qq

In the case where is In the case where is a linear function a linear function −−11((qq) = ) = qq marginal cost: marginal cost: ww average cost: average cost: ww + + CC00 / / qq

Marginal cost is constant or increasingMarginal cost is constant or increasing Average cost is initially decreasingAverage cost is initially decreasing

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Case 1Case 1

q

p

A model of indivisibility (4) Average costAverage cost Marginal costMarginal cost Supply of competitive firmSupply of competitive firm

0

Case 2Case 2

q

p Average costAverage cost Marginal costMarginal cost Supply of competitive firmSupply of competitive firm

0

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

“Natural Monopoly” SubadditivitySubadditivity

CC((ww, , q q ++ q q) < ) < CC((ww, , qq) + ) + CC((ww, , qq)) Natural monopolyNatural monopoly

Apply the above inequality…Apply the above inequality… CC((ww, 2, 2qq) < 2) < 2CC((ww, , qq) ) And for any integer And for any integer NN > :1 > :1 CC((ww, , NqNq) < ) < NCNC((ww, , qq))

Cheaper to produce in a single plant rather than two Cheaper to produce in a single plant rather than two identical plantsidentical plants

But subadditivity consistent with U-shaped average costBut subadditivity consistent with U-shaped average cost Does not imply IRTSDoes not imply IRTS

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Non-convexity: the economy Now transfer this idea to the economy as a wholeNow transfer this idea to the economy as a whole Use the same type of production modelUse the same type of production model An economy with two goodsAn economy with two goods

Good 1. A good with substantial setup costsGood 1. A good with substantial setup costs Rail networkRail network Gas supply systemGas supply system Electricity gridElectricity grid

Good 2. All other goodsGood 2. All other goods Assume: Assume:

a given endowment of all good 2a given endowment of all good 2 good 1 is not essential for survivalgood 1 is not essential for survival

Consider consumption possibilities of two goods Consider consumption possibilities of two goods xx11, , xx22

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

x1

x2

Fundamental non-convexity (1)

0

Endowment of good 2Fixed set-up cost to produce good 1Possibilities once fixed-cost has been incurred x°

Attainable set is shaded area + “spike”Endowment point x° is technically efficient

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Fundamental non-convexity (2)

x1

x2

0

Endowment of good 2Fixed set-up cost to produce good 1Possibilities once fixed-cost has been incurred x°

MRT is everywhere constantAgain endowment point x° is technically efficient

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The issues

Basic model

Efficiency

Implementation

Non-convexities

An extension of the basic rules of thumb

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Competitive “Failure” and Efficiency Characterisation Characterisation

problem: problem:

Requires a modification of

first-order conditions

Implementation Implementation problem: problem:

Involves intervention in, or replacement of, the market

Usually achieved through some “public” institution or economic mechanism

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency: characterisation Two basic questions: Two basic questions: Should good 1 be produced at all?Should good 1 be produced at all? If so, how much should be produced?If so, how much should be produced? The answer depends on agents’ preferencesThe answer depends on agents’ preferences

assume…assume… ……these represented by conventional utility functionthese represented by conventional utility function ……all consumers are identicalall consumers are identical

Method:Method: use the simple production modeluse the simple production model examine efficiency in two cases…examine efficiency in two cases… ……that differ only in representative agent’s preferencesthat differ only in representative agent’s preferences

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

x1

x2

Efficiency characterisation: case 1

0

x′

Reservation indifference curveIndifference mapPoint where MRS=MRTEfficient point

x°

Attainable set is shaded area + “spike” In this case MRS=MRT is not sufficient Utility is higher if x1 = 0

Attainable set as before

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency characterisation: case 2

0

x°

x1

x2

Attainable set as before Indifference map

x′

Consumption if none of good 1 is produced The efficient point

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The issues

Basic model

Efficiency

Implementation

Non-convexities

The market and alternatives

•Full information•Asymmetric information

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Efficiency: implementation Move on from Move on from describingdescribing the efficient allocation the efficient allocation What mechanism could implement the allocation?What mechanism could implement the allocation? Consider first the competitive market:Consider first the competitive market:

Assume given prices…Assume given prices… ……profit-maximising firm(s)profit-maximising firm(s)

Then consider a discriminating monopolyThen consider a discriminating monopoly Allow nonlinear fee scheduleAllow nonlinear fee schedule

Then consider equivalent regulatory modelThen consider equivalent regulatory model Maximise social welfare…Maximise social welfare… … … by appropriate choice of regulatory rby appropriate choice of regulatory réégimegime

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Nonconvexity: effect of the competitive market

Efficient to produce where MRS=MRT

0

x°

x′

x1

x2

p1

—p2

Iso-profit-lineProfit-maximisation over the attainable set

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Nonconvexity: efficient fee schedule

Efficient to produce at x'

0

x°

x′

x1

x2

p1

—p2

MRS=MRTFixed chargeVariable charge

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

SituationSituation U(x′) > U(x°) : : x′ is optimal is optimal Prices at Prices at x′ given by MRS given by MRS

Competitive “solution”:Competitive “solution”: Firms maximise profits by producing Firms maximise profits by producing xx11 = 0 at these prices. = 0 at these prices. Goodbye Railways?Goodbye Railways?

Simple monopoly:Simple monopoly: Clearly inefficient…Clearly inefficient… ……monopoly would force price of good 1 above MCmonopoly would force price of good 1 above MC

Discriminating monopolyDiscriminating monopoly A combination of fixed charge…A combination of fixed charge… ……plus linear variable chargeplus linear variable charge

How to implement this?How to implement this?

Implementation: problem

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Implementation: analysis Set up as a problem of regulating the firmSet up as a problem of regulating the firm

produces output produces output qq of good 1 of good 1 values denominated in terms of good 2values denominated in terms of good 2

Regulator can: Regulator can: observe quantity of output observe quantity of output grant a subsidy of grant a subsidy of FF FF is raised from consumers is raised from consumers through non-distortionary taxation?through non-distortionary taxation?

Criterion for regulatorCriterion for regulator a measure of consumer welfarea measure of consumer welfare the firm’s profitsthe firm’s profits

Take case where regulator is fully informedTake case where regulator is fully informed

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Regulation model: the firm There is a single firm – regulated monopolyThere is a single firm – regulated monopoly Firm chooses output Firm chooses output qq, given, given

price-per unit of output price-per unit of output pp((qq) allowed by regulator) allowed by regulator fixed payment fixed payment FF costs costs CC((qq))

The firm’s revenue is given byThe firm’s revenue is given by R = pR = p((qq)) q q + F+ F

Firm’s profits areFirm’s profits are = R= R C C((qq))

Firm seeks to maximise Firm seeks to maximise subject to regime fixed by subject to regime fixed by regulatorregulator

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Regulation model: the regulator Regulator can fixRegulator can fix

price per unit price per unit pp fixed payment to firms fixed payment to firms FF

But, given the action of the firmBut, given the action of the firm revenue isrevenue is R = p R = p((qq)) q q + F+ F choosing choosing qq to max profits to max profits

……fixing fixing pp((∙∙) and ) and FF is equivalent to fixing is equivalent to fixing firm’s output firm’s output qq firm’s revenue firm’s revenue RR

So transform problem to one of regulator choosing So transform problem to one of regulator choosing ((qq, , RR))

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Regulation model: objectives Assume consumers are identical Assume consumers are identical

take a single representative consumertake a single representative consumer consumes consumes xx11 = q = q

Assume zero income effectsAssume zero income effects so take consumer’s surplus (CS) as a measure of welfareso take consumer’s surplus (CS) as a measure of welfare qq CSCS((qq, , RR)) = ∫ = ∫0 0 pp((xx) d) dx x R R

Note properties of CS(Note properties of CS(∙∙):): CSCSqq((qq, , RR)) = = pp((qq)) CSCSRR((qq, , RR)) = = 11

Social valuation taken a combination of welfare and profits:Social valuation taken a combination of welfare and profits: VV((RR, , qq) = ) = CSCS((qq, , RR) + ) + [[RR C C((qq)])] < 1< 1

Note derived properties of Note derived properties of VV((∙∙):): VVqq((qq, , RR)) = = pp((qq) ) CCRR((qq)) VVqRqR((qq, , RR)) = = 1 + 1 +

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Regulation model: solution Problem is choose (Problem is choose (qq, , RR)) to to max max V V ((qq, , RR) subject to ) subject to RR C C((qq) ) ≥≥ 0 0 Lagrangean isLagrangean is

VV((qq, , RR) + ) + [ [RR C C((qq) ]) ] If “*” denote maximising values, first-order conditions areIf “*” denote maximising values, first-order conditions are

VVqq((qq**, , RR**) ) −− **CCqq((qq**)) VVRR((RR**, , qq**) + ) + ** **RR C C((qq**))

Evaluate using the derivatives of Evaluate using the derivatives of VV:: − − + + ** pp((qq**) ) CCRR((qq**) ) −− **CCqq((qq**))

Clearly Clearly **− − and from the FOCs and from the FOCs RR** C C((qq**)) pp((qq**) ) = C= CRR((qq**) )

So the (So the (qq* * , , R R **))programme induces a zero-profit, efficient outcomeprogramme induces a zero-profit, efficient outcome

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Characterisation Characterisation problem: problem:

supplement the MRS = MRT

rule by a "global search" rule for the optimum.

Implementation Implementation problem: problem:

Set user prices equal to marginal cost

Cover losses (from fixed cost) with non-distortionary transfer

Don't leave it to the unregulated market....

Provisional summary

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Overview...The issues

Basic model

Efficiency

Implementation

Non-convexities

Regulation…

•Full information•Asymmetric information

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

The issue By hypothesis there is only room for only one firmBy hypothesis there is only room for only one firm The efficient payment schedule requires The efficient payment schedule requires

A per-unit payment such that P = MCA per-unit payment such that P = MC A fixed amount required to ensure break-evenA fixed amount required to ensure break-even

However, implementation of this is demandingHowever, implementation of this is demanding requires detailed information about firm’s costsrequires detailed information about firm’s costs by hypothesis, there isn't a pool of firms to provide estimatesby hypothesis, there isn't a pool of firms to provide estimates

To see the issues, let’s take a special caseTo see the issues, let’s take a special case Two possible types of firmTwo possible types of firm Known probability of high-cost/low-cost typeKnown probability of high-cost/low-cost type

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Low-cost type

0 x1

x2

x'

x1'

x°

Efficient to produce where MRS=MRTAmount of good 1 producedEfficient payment schedule

Preferences

p

F'

F' is the (small) fixed charge allowed to the low-cost type by the regulator

q =x1' is the amount that the regulator wants the low-cost type to produce

p is the variable charge allowed to low-cost type by the regulator (=MC)

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

High-cost type

0 x1

x2

x''

x1''

x°

Efficient to produce where MRS=MRTAmount of good 1 producedEfficient payment schedule

Preferences

Essentially same story as before

F''

But regulator allows the high-cost type the large fixed charge F''

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Misrepresentation

0 x1

x2

x''

x1'x1

''

Production possibilities and solution for low-cost typeProduction possibilities and solution for high-cost typeOutcome if low-cost type masquerades as high-cost type

High-cost type is allowed higher fixed charge than low-cost type x'Low-cost type would like to get deal offered to high-cost type

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Second-best regulation: problem Regulator is faced with an informational problemRegulator is faced with an informational problem Must take into account incentive compatibilityMust take into account incentive compatibility Design the regime such that two constraints are satisfiedDesign the regime such that two constraints are satisfied Participation constraint Participation constraint

firm of either type will actually want to produce positive outputfirm of either type will actually want to produce positive output must at least break evenmust at least break even

Incentive compatibility constraint Incentive compatibility constraint neither firm type should want to masquerade as the other…neither firm type should want to masquerade as the other… ……in order to profit from a more favourable treatmentin order to profit from a more favourable treatment each type must be allowed to make as much profit as if it were each type must be allowed to make as much profit as if it were

mimicking the other typemimicking the other type Requires a standard adaptation of the optimisation problemRequires a standard adaptation of the optimisation problem

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Second-best regulation: solution Model basics Model basics

low-cost firm is low-cost firm is aa-type – cost function -type – cost function CCaa((∙∙)) high-cost firm is high-cost firm is bb-type – cost function -type – cost function CCbb((∙∙)) probability of getting an probability of getting an aa-type is -type is objective is objective is EEVV((qq, , RR) = ) = VV((qqaa, , RRaa) + [1) + [1−−]]VV((qq, , RRbb) )

Regulator chooses (Regulator chooses (qqaa, , qqbb, , RRaa, , RRbb)) to to max max EEVV((qq, , RR) s.t.) s.t.RRbb C Cbb((qqbb) ) ≥≥ 0 0RRaa C Caa((qqaa) ) ≥≥ RRbb C Caa((qqbb))

Lagrangean isLagrangean isVV((qqaa, , RRaa) + [1) + [1−−]]VV((qq, , RRbb) )

+ + [ [RRbb C Cbb((qqbb)) ] ] + + [ [RRaa C Caa((qqbb) ) R Rbb + + CCaa((qqaa)) ] ]

Get standard second-best results:Get standard second-best results: type type aa: price = MC, makes positive profits: price = MC, makes positive profits type type bb: price > MC, makes zero profits: price > MC, makes zero profits

Frank Cow

ell: Frank C

owell: M

icroeconomics

Microeconom

ics

Conclusion May give rise to inefficiency if we leave everything to the May give rise to inefficiency if we leave everything to the

marketmarket if there are non-convexities,,,if there are non-convexities,,, ……separation result does not applyseparation result does not apply

So the goods may be produced in the public sectorSo the goods may be produced in the public sector but they are not “public goods” in the conventional sensebut they are not “public goods” in the conventional sense public utilities?public utilities?

Could private firms implement efficient allocation? Could private firms implement efficient allocation? for certain goods – a monopoly with entrance feefor certain goods – a monopoly with entrance fee may be able to implement through pubic regulationmay be able to implement through pubic regulation but may have to accept second-best outcomebut may have to accept second-best outcome