Non-convexities

1. Prerequisites Almost essential Welfare: Efficiency Adverse selection Non-convexities MICROECONOMICS Principles and Analysis Frank Cowell August 2006

2. Introduction • What are non-convexities? • …awkward name • …crucial concept • Concerned with production… • drop the convenient divisibility assumption • potentially far-reaching consequences • Approach: • start with examination of economic issues • build a simple production model • examine efficiency implications • consider problems of implementation and policy • Terms other than “non-convexities” sometimes used… • …not always appropriately • but can give some insight on to the range of issues:

3. Other terms…? • “Increasing returns” • but increasing returns everywhere are not essential • “Natural monopoly” • but issue arises regardless of market form… • … not essentially one of industrial structure • “Public utilities” • but phenomenon is not necessarily in the public sector • None of these captures the concept exactly • We need to examine the economic issues more closely…

4. Non-convexities Overview... The issues Basic model The nature of non-convexities Efficiency Implementation

5. Issues: the individual firm • Consider supply by competitive firms • upward-sloping portion of MC curve • supply discontinuous if there is fixed cost • If there are lots of firms • average supply is approximately continuous • so we can get demand=supply at industry level • If there is in some sense a “natural monopoly” • perhaps very large fixed cost? • perhaps MC everywhere constant/falling? • no supply in competitive market? • In this case…. • how does the firm cover costs? • how “should” the firm behave? • how can it be induced to behave in the required manner?

6. Issues: efficient allocations • Related to the issue discussed for firm • Concerns implementation through the market • non-convexities seen an aspect of “market failure”? • consider reason for this… • …and a solution? • Relationship between CE and efficiency • fundamental to welfare economics • examine key questions of implementation • First a simple example of how it works…

7. f h q2 x2 f` h` q1` x1` Implementation through the market • Production possibilities Uh(xh) = Uh(x*h) • Firm f max profits • U contour • h min expenditure Ff(qf) = 0   p1 — p2 p1 — p2 q*f x*h now for the two key questions. • all f and h optimise at these prices such that • …for all pairs of goods MRS = MRT= price ratio

8. Efficiency and the market: key questions • Is a competitive equilibrium efficient? • Yes if all consumers are greedy, there is no hidden information, and there are no externalities • Can an arbitrary Pareto-efficient allocation be supported by a competitive equilibrium? • Yes if all consumers are greedy, there is no hidden information, there are no externalities and no non-convexities • If there are non-convexities the equilibrium price signals could take the economy away from the efficient allocation

9. Non-convexities Overview... The issues Basic model Back to the firm…. Efficiency Implementation

10. A model of indivisibility (1) • Take simplest model of production: • a single output (q)… • …from a single input (z) • The indivisibility: • A fixed amount of input required before you get any output • Otherwise production is conventional • q = f(z− k) , z≥ k • f(0) = 0, fz(∙) > 0, fzz(∙) ≤ 0 • q = 0, z< k • Given a required amount of output q > 0… • minimum amount of z required is: • f−1(q) + k

11. Case 1 Case 2 q q z z A model of indivisibility (2) • The minimum input requirement • fz(∙) > 0, fzz(∙) < 0 • Attainable set 0 k • The minimum input requirement • fz(∙) > 0, fzz(∙) = 0 • Attainable set 0 k

12. A model of indivisibility (3) • Suppose units of input can be bought for w • What is cost of output q? • clearly C(w, 0) = 0 and • C(w,q) = v(w,q) + C0, for q > 0, • where variable cost is v(w,q) = wf−1(q) • and fixed cost is C0 = wk • Therefore: • marginal cost: w / fz(f−1(q)) • average cost: wf−1(q) / q + C0 / q • In the case where is f a linear function • f−1(q) = aq • marginal cost: aw • average cost: aw+ C0 / q • Marginal cost is constant or increasing • Average cost is initially decreasing

13. Case 1 Case 2 p p q q A model of indivisibility (4) • Average cost • Marginal cost • Supply of competitive firm 0 • Average cost • Marginal cost • Supply of competitive firm 0

14. “Natural Monopoly” • Subadditivity • C(w, q + q) < C(w, q) + C(w, q) • Natural monopoly • Apply the above inequality… • C(w, 2q) < 2C(w, q) • And for any integer N > :1 • C(w, Nq) < NC(w, q) • Cheaper to produce in a single plant rather than two identical plants • But subadditivity consistent with U-shaped average cost • Does not imply IRTS

15. Non-convexity: the economy • Now transfer this idea to the economy as a whole • Use the same type of production model • An economy with two goods • Good 1. A good with substantial setup costs • Rail network • Gas supply system • Electricity grid • Good 2. All other goods • Assume: • a given endowment of all good 2 • good 1 is not essential for survival • Consider consumption possibilities of two goods x1, x2

16. x2 x1 Fundamental non-convexity (1) • Endowment of good 2 • Fixed set-up cost to produce good 1 • Possibilities once fixed-cost has been incurred • x° • Attainable set is shaded area + “spike” • Endowment point x° is technically efficient 0

17. x2 x1 Fundamental non-convexity (2) • Endowment of good 2 • Fixed set-up cost to produce good 1 • Possibilities once fixed-cost has been incurred • x° • MRT is everywhere constant • Again endowment point x° is technically efficient 0

18. Non-convexities Overview... The issues Basic model An extension of the basic rules of thumb Efficiency Implementation

19. Competitive “Failure” and Efficiency Requires a modification of first-order conditions • Characterisation problem: Implementation problem: Involves intervention in, or replacement of, the market Usually achieved through some “public” institution or economic mechanism

20. Efficiency: characterisation • Two basic questions: • Should good 1 be produced at all? • If so, how much should be produced? • The answer depends on agents’ preferences • assume… • …these represented by conventional utility function • …all consumers are identical • Method: • use the simple production model • examine efficiency in two cases… • …that differ only in representative agent’s preferences

21. x2 x1 Efficiency characterisation: case 1 • Attainable set as before • Reservation indifference curve • Indifference map • Point where MRS=MRT • x° • Efficient point • Attainable set is shaded area + “spike” • x′ • In this case MRS=MRT is not sufficient • Utility is higher if x1 = 0 0

22. x2 x1 Efficiency characterisation: case 2 • Attainable set as before • Indifference map • Consumption if none of good 1 is produced • x° • The efficient point • x′ 0

23. Non-convexities Overview... The issues Basic model The market and alternatives Efficiency • Full information • Asymmetric information Implementation

24. Efficiency: implementation • Move on from describing the efficient allocation • What mechanism could implement the allocation? • Consider first the competitive market: • Assume given prices… • …profit-maximising firm(s) • Then consider a discriminating monopoly • Allow nonlinear fee schedule • Then consider equivalent regulatory model • Maximise social welfare… • … by appropriate choice of regulatory régime

25. x2 • x° • x′ x1 0 Nonconvexity: effect of the competitive market • Efficient to produce where MRS=MRT • Iso-profit-line • Profit-maximisation over the attainable set p1 — p2

26. x2 x1 Nonconvexity: efficient fee schedule • Efficient to produce at x' • MRS=MRT • Fixed charge • x° • Variable charge • x′ p1 — p2 0

27. Implementation: problem • Situation • U(x′)> U(x°) : x′ is optimal • Prices at x′ given by MRS • Competitive “solution”: • Firms maximise profits by producing x1 = 0 at these prices. • Goodbye Railways? • Simple monopoly: • Clearly inefficient… • …monopoly would force price of good 1 above MC • Discriminating monopoly • A combination of fixed charge… • …plus linear variable charge • How to implement this?

28. Implementation: analysis • Set up as a problem of regulating the firm • produces output q of good 1 • values denominated in terms of good 2 • Regulator can: • observe quantity of output • grant a subsidy of F • F is raised from consumers • through non-distortionary taxation? • Criterion for regulator • a measure of consumer welfare • the firm’s profits • Take case where regulator is fully informed

29. Regulation model: the firm • There is a single firm – regulated monopoly • Firm chooses output q, given • price-per unit of output p(q) allowed by regulator • fixed payment F • costs C(q) • The firm’s revenue is given by • R = p(q) q+ F • Firm’s profits are • P= R C(q) • Firm seeks to maximise P subject to regime fixed by regulator

30. Regulation model: the regulator • Regulator can fix • price per unit p • fixed payment to firms F • But, given the action of the firm • revenue is R = p(q) q+ F • choosing q to max profits • …fixing p(∙) and F is equivalent to fixing • firm’s output q • firm’s revenue R • So transform problem to one of regulator choosing (q, R)

31. Regulation model: objectives • Assume consumers are identical • take a single representative consumer • consumes x1 = q • Assume zero income effects • so take consumer’s surplus (CS) as a measure of welfare q • CS(q, R) = ∫0 p(x) dx  R • Note properties of CS(∙): • CSq(q, R) = p(q) • CSR(q, R) =  1 • Social valuation taken a combination of welfare and profits: • V(R, q) = CS(q, R) + b [R C(q)] • b < 1 • Note derived properties of V(∙): • Vq(q, R) = p(q)  bCR(q) • VqR(q, R) = 1 + b

32. Regulation model: solution • Problem is choose (q, R) to max V (q, R) subject to R C(q) ≥ 0 • Lagrangean is • V(q, R) + l [R C(q) ] • If “*” denote maximising values, first-order conditions are • Vq(q*, R*) −l*Cq(q*) = 0 • VR(R*, q*) + l* = 0 • l* [R C(q*)] = 0 • Evaluate using the derivatives of V: • 1− b + l* = 0 • p(q*)  bCR(q*) −l*Cq(q*) = 0 • Clearly l* = 1− b > 0 and from the FOCs • R*= C(q*) • p(q*) = CR(q*) • So the (q* , R *)programme induces a zero-profit, efficient outcome

33. Provisional summary supplement the MRS = MRT rule by a "global search" rule for the optimum. • Characterisation problem: Implementation 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....

34. Non-convexities Overview... The issues Basic model Regulation… Efficiency • Full information • Asymmetric information Implementation

35. The issue • By hypothesis there is only room for only one firm • The efficient payment schedule requires • A per-unit payment such that P = MC • A fixed amount required to ensure break-even • However, implementation of this is demanding • requires detailed information about firm’s costs • by hypothesis, there isn't a pool of firms to provide estimates • To see the issues, let’s take a special case • Two possible types of firm • Known probability of high-cost/low-cost type

36. ' x1 Low-cost type • Preferences x2 • Efficient to produce where MRS=MRT • Amount of good 1 produced • x° F' • Efficient payment schedule • q =x1' is the amount that the regulator wants the low-cost type to produce • x' • F' is the (small) fixed charge allowed to the low-cost type by the regulator p • p is the variable charge allowed to low-cost type by the regulator (=MC) x1 0

37. '' x1 High-cost type • Preferences x2 • Efficient to produce where MRS=MRT • Amount of good 1 produced • x° F'' • Efficient payment schedule • Essentially same story as before • But regulator allows the high-cost type the large fixed charge F'' • x'' x1 0

38. '' ' x1 x1 Misrepresentation • Production possibilities and solution for low-cost type x2 • Production possibilities and solution for high-cost type • Outcome if low-cost type masquerades as high-cost type • High-cost type is allowed higher fixed charge than low-cost type • x' • x'' • Low-cost type would like to get deal offered to high-cost type x1 0

39. Second-best regulation: problem • Regulator is faced with an informational problem • Must take into account incentive compatibility • Design the regime such that two constraints are satisfied • Participation constraint • firm of either type will actually want to produce positive output • must at least break even • Incentive compatibility constraint • neither firm type should want to masquerade as the other… • …in order to profit from a more favourable treatment • each type must be allowed to make as much profit as if it were mimicking the other type • Requires a standard adaptation of the optimisation problem

40. Second-best regulation: solution • Model basics • low-cost firm is a-type – cost function Ca(∙) • high-cost firm is b-type – cost function Cb(∙) • probability of getting an a-type is p • objective is EV(q, R) = pV(qa, Ra) + [1−p]V(q, Rb) • Regulator chooses (qa, qb, Ra, Rb) to max EV(q, R) s.t. Rb Cb(qb) ≥ 0 Ra Ca(qa) ≥Rb Ca(qb) • Lagrangean is pV(qa, Ra) + [1−p]V(q, Rb) + l [Rb Cb(qb) ] + m [Ra Ca(qb)  Rb + Ca(qa) ] • Get standard second-best results: • type a: price = MC, makes positive profits • type b: price > MC, makes zero profits

41. Conclusion • May give rise to inefficiency if we leave everything to the market • if there are non-convexities,,, • …separation result does not apply • So the goods may be produced in the public sector • but they are not “public goods” in the conventional sense • public utilities? • Could private firms implement efficient allocation? • for certain goods – a monopoly with entrance fee • may be able to implement through pubic regulation • but may have to accept second-best outcome