#### Transcript Extensive Form - London School of Economics

Prerequisites Almost essential Welfare: Efficiency Adverse selection Frank Cowell: Microeconomics August 2006 Non-convexities MICROECONOMICS Principles and Analysis Frank Cowell Introduction Frank Cowell: Microeconomics What are non-convexities? Concerned with production… drop the convenient divisibility assumption potentially far-reaching consequences Approach: …awkward name …crucial concept 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: Other terms…? Frank Cowell: Microeconomics “Increasing returns” “Natural monopoly” but issue arises regardless of market form… … not essentially one of industrial structure “Public utilities” but increasing returns everywhere are not essential 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… Non-convexities Overview... Frank Cowell: Microeconomics The issues The nature of non-convexities Basic model Efficiency Implementation Issues: the individual firm Frank Cowell: Microeconomics Consider supply by competitive 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” upward-sloping portion of MC curve supply discontinuous if there is fixed cost 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? Issues: efficient allocations Frank Cowell: Microeconomics Related to the issue discussed for firm Concerns implementation through the market Relationship between CE and efficiency non-convexities seen an aspect of “market failure”? consider reason for this… …and a solution? fundamental to welfare economics examine key questions of implementation First a simple example of how it works… Implementation through the market Frank Cowell: Microeconomics Uh(xh) = Uh(x*h) f q2 h x2 Ff(qf) = 0 q*f p1 — p2 f q1`` all f and h optimise at these prices such that …for all pairs of goods MRS = MRT= price ratio Production possibilities Firm f max profits U contour h min expenditure p1 — p2 x*h x1`h` now for the two key questions. Frank Cowell: Microeconomics Efficiency and the market: key questions 1. Is a competitive equilibrium efficient? 2. Can an arbitrary Pareto-efficient allocation be supported by a competitive equilibrium? Yes if all consumers are greedy, there is no hidden information, and there are no externalities 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 Non-convexities Overview... Frank Cowell: Microeconomics The issues Back to the firm…. Basic model Efficiency Implementation A model of indivisibility (1) Frank Cowell: Microeconomics Take simplest model of production: The indivisibility: a single output (q)… …from a single input (z) 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 A model of indivisibility (2) Frank Cowell: Microeconomics q Case 1 z 0 k q Case 2 0 The minimum input requirement fz(∙) > 0, fzz(∙) < 0 Attainable set z k The minimum input requirement fz(∙) > 0, fzz(∙) = 0 Attainable set A model of indivisibility (3) Frank Cowell: Microeconomics 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 A model of indivisibility (4) Frank Cowell: Microeconomics p Case 1 q 0 p Case 2 0 Average cost Marginal cost Supply of competitive firm q Average cost Marginal cost Supply of competitive firm “Natural Monopoly” Frank Cowell: Microeconomics Subadditivity Natural monopoly C(w, q + q) < C(w, q) + C(w, q) 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 Non-convexity: the economy Frank Cowell: Microeconomics 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 Good 2. All other goods Assume: Rail network Gas supply system Electricity grid a given endowment of all good 2 good 1 is not essential for survival Consider consumption possibilities of two goods x1, x2 Fundamental non-convexity (1) Frank Cowell: Microeconomics Endowment of good 2 Fixed set-up cost to produce good 1 Possibilities once fixed-cost has been incurred x2 x° Attainable set is shaded area + “spike” Endowment point x° is technically efficient x1 0 Fundamental non-convexity (2) Frank Cowell: Microeconomics Endowment of good 2 Fixed set-up cost to produce good 1 Possibilities once fixed-cost has been incurred x2 x° MRT is everywhere constant Again endowment point x° is technically efficient x1 0 Non-convexities Overview... Frank Cowell: Microeconomics The issues An extension of the basic rules of thumb Basic model Efficiency Implementation Competitive “Failure” and Efficiency Frank Cowell: Microeconomics Characterisation problem: Implementation problem: Requires a modification of first-order conditions Involves intervention in, or replacement of, the market Usually achieved through some “public” institution or economic mechanism Efficiency: characterisation Frank Cowell: Microeconomics 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 Efficiency characterisation: case 1 Frank Cowell: Microeconomics Attainable set as before Reservation indifference curve Indifference map Point where MRS=MRT Efficient point x2 x° Attainable set is shaded x′ area + “spike” In this case MRS=MRT x1 0 is not sufficient Utility is higher if x1 = 0 Efficiency characterisation: case 2 Frank Cowell: Microeconomics Attainable set as before x2 Indifference map Consumption if none of good 1 is produced The efficient point x° x′ x1 0 Non-convexities Overview... Frank Cowell: Microeconomics The issues The market and alternatives Basic model Efficiency Implementation •Full information •Asymmetric information Efficiency: implementation Frank Cowell: Microeconomics Move on from describing the efficient allocation What mechanism could implement the allocation? Consider first the competitive market: Then consider a discriminating monopoly Assume given prices… …profit-maximising firm(s) Allow nonlinear fee schedule Then consider equivalent regulatory model Maximise social welfare… … by appropriate choice of regulatory régime Frank Cowell: Microeconomics Nonconvexity: effect of the competitive market x2 Efficient to produce where MRS=MRT Iso-profit-line x° Profit-maximisation over the attainable set x′ p1 — p2 x1 0 Frank Cowell: Microeconomics Nonconvexity: efficient fee schedule Efficient to produce at x' x2 MRS=MRT Fixed charge x° Variable charge x′ p1 — p2 x1 0 Implementation: problem Frank Cowell: Microeconomics Situation Competitive “solution”: Clearly inefficient… …monopoly would force price of good 1 above MC Discriminating monopoly Firms maximise profits by producing x1 = 0 at these prices. Goodbye Railways? Simple monopoly: U(x′) > U(x°) : x′ is optimal Prices at x′ given by MRS A combination of fixed charge… …plus linear variable charge How to implement this? Implementation: analysis Frank Cowell: Microeconomics Set up as a problem of regulating the firm Regulator can: observe quantity of output grant a subsidy of F F is raised from consumers through non-distortionary taxation? Criterion for regulator produces output q of good 1 values denominated in terms of good 2 a measure of consumer welfare the firm’s profits Take case where regulator is fully informed Regulation model: the firm Frank Cowell: Microeconomics There is a single firm – regulated monopoly Firm chooses output q, given The firm’s revenue is given by R = p(q) q + F Firm’s profits are price-per unit of output p(q) allowed by regulator fixed payment F costs C(q) P = R C(q) Firm seeks to maximise P subject to regime fixed by regulator Regulation model: the regulator Frank Cowell: Microeconomics Regulator can fix 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 price per unit p fixed payment to firms F firm’s output q firm’s revenue R So transform problem to one of regulator choosing (q, R) Regulation model: objectives Frank Cowell: Microeconomics 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 Note properties of CS(∙): CSq(q, R) = p(q) CSR(q, R) = 1 Social valuation taken a combination of welfare and profits: CS(q, R) = ∫0 p(x) dx R 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 Regulation model: solution Frank Cowell: Microeconomics Problem is choose (q, R) to max V (q, R) subject to R C(q) ≥ 0 Lagrangean is If “*” denote maximising values, first-order conditions are 1− b + l* = 0 p(q*) bCR(q*) − l* Cq(q*) = 0 Clearly l* = 1− b > 0 and from the FOCs Vq(q*, R*) − l*Cq(q*) = 0 VR(R*, q*) + l* = 0 l* [R C(q*)] = 0 Evaluate using the derivatives of V: V(q, R) + l [R C(q) ] R* = C(q*) p(q*) = CR(q*) So the (q* , R *) programme induces a zero-profit, efficient outcome Provisional summary Frank Cowell: Microeconomics Characterisation problem: Implementation problem: supplement the MRS = MRT rule by a "global search" rule for the optimum. Set user prices equal to marginal cost Cover losses (from fixed cost) with non-distortionary transfer Don't leave it to the unregulated market.... Non-convexities Overview... Frank Cowell: Microeconomics The issues Regulation… Basic model Efficiency Implementation •Full information •Asymmetric information The issue Frank Cowell: Microeconomics By hypothesis there is only room for only one firm The efficient payment schedule requires However, implementation of this is demanding A per-unit payment such that P = MC A fixed amount required to ensure break-even 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 Low-cost type Frank Cowell: Microeconomics Preferences x2 Efficient to produce where MRS=MRT Amount of good 1 produced x°F' Efficient payment schedule q =x1' is the amount that x' p 0 x1 x1' the regulator wants the low-cost type to produce F' is the (small) fixed charge allowed to the lowcost type by the regulator p is the variable charge allowed to low-cost type by the regulator (=MC) High-cost type Frank Cowell: Microeconomics 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 x'' high-cost type the large fixed charge F'' 0 x1 x1'' Misrepresentation Frank Cowell: Microeconomics 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 x2 High-cost type is allowed x'' higher fixed charge than low-cost type x' Low-cost type would like to get deal offered to highcost type 0 x1 x1'' x1' Second-best regulation: problem Frank Cowell: Microeconomics Regulator is faced with an informational problem Must take into account incentive compatibility Design the regime such that two constraints are satisfied Participation constraint Incentive compatibility constraint firm of either type will actually want to produce positive output must at least break even 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 Second-best regulation: solution Frank Cowell: Microeconomics 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. Lagrangean is Rb Cb(qb) ≥ 0 Ra Ca(qa) ≥ Rb Ca(qb) 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 Conclusion Frank Cowell: Microeconomics May give rise to inefficiency if we leave everything to the market So the goods may be produced in the public sector if there are non-convexities,,, …separation result does not apply 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