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

2. Frank Cowell: Microeconomics Introduction What are non-convexities? What are non-convexities?  …awkward name  …crucial concept Concerned with production… Concerned with production…  drop the convenient divisibility assumption  potentially far-reaching consequences Approach: 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… Terms other than “non-convexities” sometimes used…  …not always appropriately  but can give some insight on to the range of issues:

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

4. Frank Cowell: Microeconomics Overview... The issues Basic model Efficiency Implementation Non-convexities The nature of non-convexities

5. Frank Cowell: Microeconomics Issues: the individual firm Consider supply by competitive firms Consider supply by competitive firms  upward-sloping portion of MC curve  supply discontinuous if there is fixed cost If there are lots of firms 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” 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…. 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. Frank Cowell: Microeconomics Issues: efficient allocations Related to the issue discussed for firm Related to the issue discussed for firm Concerns implementation through the market 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 Relationship between CE and efficiency  fundamental to welfare economics  examine key questions of implementation First a simple example of how it works… First a simple example of how it works…

7. Frank Cowell: Microeconomics Implementation through the market  p1—p2p1—p2 q *f  p1—p2p1—p2 x *h q 2 q 1` f f ` x 2 x 1` h h `  f (q f ) = 0 U h (x h ) = U h (x *h )   Production possibilities   Firm f max profits   U contour   h 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.

8. Frank Cowell: Microeconomics Efficiency and the market: key questions 1. Is a competitive equilibrium efficient?  Yes if all consumers are greedy, there is no hidden information, and there are no externalities 2. 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 If there are non-convexities the equilibrium price signals could take the economy away from the efficient allocation

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

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

11. Frank Cowell: Microeconomics Case 1 z q A model of indivisibility (2) The minimum input requirement The minimum input requirement  z (∙) > 0,  zz (∙) 0,  zz (∙) < 0 Attainable set Attainable set 0 k Case 2 z q The minimum input requirement The minimum input requirement  z (∙) > 0,  zz (∙) = 0  z (∙) > 0,  zz (∙) = 0 Attainable set Attainable set 0 k

12. Frank Cowell: Microeconomics A model of indivisibility (3) Suppose units of input can be bought for w Suppose units of input can be bought for w What is cost of output q? What is cost of output q?  clearly C(w, 0) = 0 and  C(w,q) = v(w,q) + C 0, for q > 0,  where variable cost is v(w,q) = w  −1 (q)  and fixed cost is C 0 = wk Therefore: Therefore:  marginal cost: w /  z (  −1 (q))  average cost: w  −1 (q) / q + C 0 / q In the case where is  a linear function In the case where is  a linear function   −1 (q) =  q  marginal cost:  w  average cost:  w + C 0 / q Marginal cost is constant or increasing Marginal cost is constant or increasing Average cost is initially decreasing Average cost is initially decreasing

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

14. Frank Cowell: Microeconomics “Natural Monopoly” Subadditivity Subadditivity  C(w, q + q) < C(w, q) + C(w, q) Natural monopoly 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 Cheaper to produce in a single plant rather than two identical plants But subadditivity consistent with U-shaped average cost But subadditivity consistent with U-shaped average cost  Does not imply IRTS

15. Frank Cowell: Microeconomics Non-convexity: the economy Now transfer this idea to the economy as a whole Now transfer this idea to the economy as a whole Use the same type of production model Use the same type of production model An economy with two goods 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: Assume:  a given endowment of all good 2  good 1 is not essential for survival Consider consumption possibilities of two goods x 1, x 2 Consider consumption possibilities of two goods x 1, x 2

16. Frank Cowell: Microeconomics x1x1 x2x2 ll ll Fundamental non-convexity (1) 0   Endowment of good 2   Fixed set-up cost to produce good 1   Possibilities once fixed-cost has been incurred l l x°   Attainable set is shaded area + “spike”   Endowment point x° is technically efficient

17. Frank Cowell: Microeconomics Fundamental non-convexity (2) x1x1 x2x2 ll ll 0   Endowment of good 2   Fixed set-up cost to produce good 1   Possibilities once fixed-cost has been incurred l l x°   MRT is everywhere constant   Again endowment point x° is technically efficient

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

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

20. Frank Cowell: Microeconomics 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’ preferences The answer depends on agents’ preferences  assume…  …these represented by conventional utility function  …all consumers are identical Method: Method:  use the simple production model  examine efficiency in two cases…  …that differ only in representative agent’s preferences

21. Frank Cowell: Microeconomics x1x1 x2x2 ll ll Efficiency characterisation: case 1 0 l l x′   Reservation indifference curve   Indifference map   Point where MRS=MRT   Efficient point l l x°   Attainable set is shaded area + “spike”   In this case MRS=MRT is not sufficient   Utility is higher if x 1 = 0   Attainable set as before

22. Frank Cowell: Microeconomics Efficiency characterisation: case 2 0 l l x° ll ll x1x1 x2x2   Attainable set as before   Indifference map l l x′   Consumption if none of good 1 is produced   The efficient point

23. Frank Cowell: Microeconomics Overview... The issues Basic model Efficiency Implementation Non-convexities The market and alternatives Full information Asymmetric information

24. Frank Cowell: Microeconomics Efficiency: implementation Move on from describing the efficient allocation Move on from describing 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…  …profit-maximising firm(s) Then consider a discriminating monopoly Then consider a discriminating monopoly  Allow nonlinear fee schedule Then consider equivalent regulatory model Then consider equivalent regulatory model  Maximise social welfare…  … by appropriate choice of regulatory régime

25. Frank Cowell: Microeconomics Nonconvexity: effect of the competitive market   Efficient to produce where MRS=MRT 0 l l x° ll ll l l x′ x1x1 x2x2 p1—p2p1—p2 p1—p2p1—p2   Iso-profit-line   Profit-maximisation over the attainable set

26. Frank Cowell: Microeconomics Nonconvexity: efficient fee schedule   Efficient to produce at x' 0 l l x° ll ll l l x′ x1x1 x2x2 p1—p2p1—p2 p1—p2p1—p2   MRS=MRT   Fixed charge   Variable charge

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

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

29. Frank Cowell: Microeconomics Regulation model: the firm There is a single firm – regulated monopoly There is a single firm – regulated monopoly Firm chooses output q, given 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 The firm’s revenue is given by  R = p(q) q + F Firm’s profits are Firm’s profits are   = R  C(q) Firm seeks to maximise  subject to regime fixed by regulator Firm seeks to maximise  subject to regime fixed by regulator

30. Frank Cowell: Microeconomics Regulation model: the regulator Regulator can fix Regulator can fix  price per unit p  fixed payment to firms F But, given the action of the firm 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 …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) So transform problem to one of regulator choosing (q, R)

31. Frank Cowell: Microeconomics Regulation model: objectives Assume consumers are identical Assume consumers are identical  take a single representative consumer  consumes x 1 = q Assume zero income effects 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(∙): Note properties of CS(∙):  CS q (q, R) = p(q)  CS R (q, R) =  1 Social valuation taken a combination of welfare and profits: Social valuation taken a combination of welfare and profits:  V(R, q) = CS(q, R) +  [R  C(q)]   < 1 Note derived properties of V(∙): Note derived properties of V(∙):  V q (q, R) = p(q)   C R (q)  V qR (q, R) =  1 + 

32. Frank Cowell: Microeconomics Regulation model: solution Problem is choose (q, R) to max V (q, R) subject to R  C(q) ≥ 0 Problem is choose (q, R) to max V (q, R) subject to R  C(q) ≥ 0 Lagrangean is Lagrangean is  V(q, R) + [R  C(q) ] If “*” denote maximising values, first-order conditions are If “*” denote maximising values, first-order conditions are  V q (q *, R * ) − * C q (q * )   V R (R *, q * ) + *   *  R  C(q * )  Evaluate using the derivatives of V: Evaluate using the derivatives of V:   −  + *   p(q * )   C R (q * ) − *  C q (q * )  Clearly *  −  and from the FOCs Clearly *  −  and from the FOCs R*  C(q*)R*  C(q*)R*  C(q*)R*  C(q*)  p(q * ) = C R (q * ) So the (q *, R * )  programme induces a zero-profit, efficient outcome So the (q *, R * )  programme induces a zero-profit, efficient outcome

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

34. Frank Cowell: Microeconomics Overview... The issues Basic model Efficiency Implementation Non-convexities Regulation… Full information Asymmetric information

35. Frank Cowell: Microeconomics The issue By hypothesis there is only room for only one firm By 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 = MC  A fixed amount required to ensure break-even However, implementation of this is demanding 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 To see the issues, let’s take a special case  Two possible types of firm  Known probability of high-cost/low-cost type

36. Frank Cowell: Microeconomics Low-cost type 0 x1x1 x2x2 l l x' x1x1 ' l l x°   Efficient to produce where MRS=MRT   Amount of good 1 produced   Efficient payment schedule   Preferences p p F'F' F'F'   F' is the (small) fixed charge allowed to the low- cost type by the regulator   q =x 1 ' 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)

37. Frank Cowell: Microeconomics High-cost type 0 x1x1 x2x2 l l x'' x1x1 '' l l x°   Efficient to produce where MRS=MRT   Amount of good 1 produced   Efficient payment schedule   Preferences   Essentially same story as before F''   But regulator allows the high-cost type the large fixed charge F''

38. Frank Cowell: Microeconomics Misrepresentation 0 x1x1 x2x2 l l x'' x1x1 ' x1x1 ''   Production possibilities and solution for low-cost type   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 l l x'   Low-cost type would like to get deal offered to high- cost type

39. Frank Cowell: Microeconomics Second-best regulation: problem Regulator is faced with an informational problem Regulator is faced with an informational problem Must take into account incentive compatibility Must take into account incentive compatibility Design the regime such that two constraints are satisfied Design the regime such that two constraints are satisfied Participation constraint Participation constraint  firm of either type will actually want to produce positive output  must at least break even Incentive compatibility constraint 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 Requires a standard adaptation of the optimisation problem

40. Frank Cowell: Microeconomics Second-best regulation: solution Model basics Model basics  low-cost firm is a-type – cost function C a (∙)  high-cost firm is b-type – cost function C b (∙)  probability of getting an a-type is   objective is E V(q, R) =  V(q a, R a ) + [1−  ]V(q, R b ) Regulator chooses (q a, q b, R a, R b ) to max E V(q, R) s.t. Regulator chooses (q a, q b, R a, R b ) to max E V(q, R) s.t. R b  C b (q b ) ≥ 0 R a  C a (q a ) ≥ R b  C a (q b ) Lagrangean is Lagrangean is  V(q a, R a ) + [1−  ]V(q, R b ) + [ R b  C b (q b ) ] +  [ R a  C a (q b )  R b + C a (q a ) ] Get standard second-best results: Get standard second-best results:  type a: price = MC, makes positive profits  type b: price > MC, makes zero profits

41. Frank Cowell: Microeconomics Conclusion May give rise to inefficiency if we leave everything to the market 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 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? 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