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Public Goods Microeconomics 2 Public goods Microeconomics 2 Bernard Caillaud Master APE - Paris School of Economics February 2, 2017 (Lecture 4) Bernard Caillaud Public goods I. The nature of goods { I.1. Examples and definitions Suppose I grow rare flowers.... ... I can sell them to you: rivalry and exclusion ... I can open a flower exhibition and charge you an entry fee for the delightful view: non-rivalry but exclusion ... I can keep them but it improves the chances that these rare seeds continue to exist, i.e. I contribute to biodiversity: non-rivalry and non-exclusion In the first case, there is a market that "works" probably well enough In the second case, there is some sort of a market that works differently In the third case, there is (yet) no market: if I stop incurring the cost, we (on earth) will all become "poorer" ! Bernard Caillaud Public goods I.1. Examples and definitions Definitions - A good is rival (in consumption) if the same unit of the good cannot be consumed by more than one person at the same time. - A good is excludable if it is technologically or/and institution- ally feasible to prevent some people to consume the good. Rival and excludable goods: private consumption goods... we know that ! Rival and non-excludable goods: common resources, e.g. red tuna in the sea Non-rival and excludable goods: pay-TV, computer soft- ware, patented knowledge - ideas Non-rival and non-excludable goods: pure public goods, e.g. national defense, scientific knowledge - ideas, public TV Bernard Caillaud Public goods I.1. Examples and definitions In fact, a matter of degree of rivalry and exclusion A public good makes collective consumption possible But the satisfaction from consuming it may depend on others consuming it (e.g. network effects, congestion,...) Subtle difference: reduction of value vs destruction by con- sumption! Strong link between public goods and externalities "Public" goods are not necessarily supplied by the govern- ment: e.g. TF1, research in private universities "Private" goods may be supplied by public firms / organi- zations: health services, mail delivery Bernard Caillaud Public goods I.1. Examples and definitions The intriguing example of roads A non-toll road with fluid traffic is a public good: I can drive without bothering others and I cannot be prevented from driving on this road Toll highways are not pure public goods, they are excludable. Roads may also be forbidden for heavy trucks. Paris' circular highway (Boulevard P´eriph´erique)is packed almost always: one additional driver prevents the others from using this facility: the good become (almost) rival due to extreme congestion externalities. Bernard Caillaud Public goods I.2. Road map for today Objectives: analysis of economy with a pure public good Markets tend not to provide public goods efficiently Foundations for public / market intervention Introduction to public economics and environmental eco- nomics Precise roadmap: The basic market failure in a simple example: BLS condi- tions for efficiency, inefficient private provision Remedies: quotas, taxes, Lindhal equilibria, voting on public good provision Link between externalities and public goods Why asymmetric information is a major concern Bernard Caillaud Public goods II. The basic market failure { II.1. A simple economy with public good We will investigate the case of a pure public good, that is of a non-rival, non-excludable good, in a simple environment An economy with I consumers, and H + 1 goods Goods h = 1; :::; H are standard private goods; good h = 1 is normalized as the num´eraire Good h = 0 is a public good: when x0 is available in the economy, all consumers benefit from x0 i i Consumers' preferences are represented by: u (x0; x ) in which i i i i i x is the bundle of private good consumption x = (x1; x2; :::; xH ) ui(:) is assumed differentiable, increasing and concave Bernard Caillaud Public goods II.1. A simple economy with public good Production of the public good through a firm (or equiva- lently a sector of identical firms) with technology: y0 ≤ f(y) with y = (y1; y2; :::; yH ) the vector of input (counted posi- tively) f(:) is assumed differentiable increasing and concave Available total initial endowments in private goods ! = (!1;!2; :::; !H ) Bernard Caillaud Public goods II.2. Optimal provision of public good We look for Pareto optima in this economy 1 1 max u (x0; x )[µ1 = 1] i i 8i 6= 1; u (x0; x ) ≥ vi [µi] y0 ≤ f(y)[ν] P i y0 = x0 and 8h > 0; i xh + yh = !h [λh] Usual FOC across private goods: 8(h; k) non-null and 8i i λh i @hu @hf = MRSh;k = i = = MRTh;k λk @ku @kf Bowen-Lindhal-Samuelson conditions (BLS): 8h; i I I i ν X X @0u 1 = MRSi = = = MRT λ 0;h @ ui @ f 0;h h i=1 i=1 h h Bernard Caillaud Public goods II.2. Optimal provision of public good dx0 i dx0 requires of input h and increases by @0u · dx0 any agent @hf i's utility; maintaining i's utility (i 6= 1) constant by reducing i's j i consumption of h by dxh = −MRS0;h · dx0; overall, for agent 1: 1 1 dx0 X i @0u · dx0 + @hu · [− + MRS0;h · dx0] = 0 , BLS @hf i6=1 Remark: Given FOC wrt private goods, BLS for h = 1 is suffi- cient Particular case: for separable utilities without revenue effect, I X 1 @ ui(xOpt) = = mc(xOpt) 0 @ f(xOpt) 0 i=1 1 sum of marginal benefits = marginal cost of public good Bernard Caillaud Public goods II.2. Optimal provision of public good Linear quadratic example Only two goods, the public good and the num´eraire i γi 2 u (x0; xi) = xi − 2 (1 − x0) , with γ1 ≥ γ2 ≥ ::: ≥ γI 1 2 1 2 f(y) = (2y) , so that c(y0) = 2 y0 and mc(y0) = y0 BLS conditions: I I I X i X i X @0u (x) = mb (x0) = ( γi)(1 − x0) = x0 = mc(x0) i=1 i=1 i=1 the social marginal benefit of increasing public good is the sum of private marginal benefits (one unit benefits all!), it must be equal to the (social) marginal cost PI Opt i=1 γi Pareto optimum: x0 = PI 1+ i=1 γi Opt Iγ Symmetric case: x0 = 1+Iγ which goes to 1 when I ! 1 Bernard Caillaud Public goods II.3. Private provision for the public good Do market allocation mechanisms attain Pareto optimality ? Suppose total quantity of public good = sum of all quantities purchased individually by consumers i Each consumer i chooses how much of the public good x0 to buy, taking as given the price system AND the amount of public good purchased by other consumers Subscription equilibrium, i.e. private provision of public good i∗ i∗ I ∗ ∗ ∗ ∗ (x0 ; x )i=1; (y0; y ); (p0; p ) such that: i∗ i∗ i i P j∗ i (x ; x ) = arg max i i u (x + x ; x ) under budget 0 x0;x 0 j6=i 0 ∗ i ∗ i ∗ ∗ i constraint p0x0 + p · x ≤ B(p0; p ) and x0 ≥ 0; non- negativity not trivial if others make purchases! ∗ ∗ ∗ ∗ (y0; y ) = arg maxy0;y(p0y0 − p · y) with y0 ≤ f(y) P i∗ ∗ P i∗ ∗ P i Markets clear: i x0 = y0 and 8h > 0; i xh +yh = i !h Bernard Caillaud Public goods II.3. Private provision for the public good FOC: 8(h; k) non-null and 8i i ∗ i @hu ph @hf MRSh;k = i = ∗ = = MRTh;k @ku pk @kf i i ∗ i @0u + ξ p0 1 MRS0;h = i = ∗ = = MRT0;h @hu ph @hf with ξi ≥ 0 multiplier associated to non-negativity constraint ∗ ^ If at the equilibrium, y0 > 0, then 9i with a positive demand ∗ ^i ^i p0 for public good, hence ξ = 0 and MRS = ∗ = MRT0;h 0;h ph i If the public good is always a "good", positive MRS0;h for PI i all i and therefore: i=1 MRS0;h > MRT0;h Bernard Caillaud Public goods II.3. Private provision for the public good The sum of MRS across agents not equal to MRT ! Inefficiency. The free-rider problem: agent i only takes into account his own private marginal benefit of purchasing the public good, and not positive effects on others. With concavity: demand for the public good too low, compared to optimum Particular separable linear case: i ∗ i∗ ∗ @0u (x0; x ) = p0, for i with positive demand, so: I X j ∗ j∗ ∗ @0u (x0; x ) > p0 j=1 Under-provision of public "good", over-provision if public "bad" No obvious conclusion in general equilibrium (prices move) Bernard Caillaud Public goods II.3. Private provision for the public good Linear quadratic example For any i, i ∗ cons: ∗ prod: ∗ ξ + γi(1 − x0) = p0 = x0 Only one agent can have a positive demand at equilibrium, agent 1 who has the largest private marginal benefit: hence, x1 = γ1 = x∗ while for i > 1, xi = 0 0 1+γ1 0 0 PI γ Under-provision: x∗ = γ1 < i=1 i = xOpt 0 1+γ1 PI 0 1+ i=1 γi ∗ γ In the symmetric case, equilibrium is x0 = 1+γ , shared in any away among agents: insensitive to I, while Pareto opti- Opt mum x0 ! 1 when I increases: large inefficiency in large economies, the free-rider problems become more serious Bernard Caillaud Public goods II.3. Private provision for the public good prices p0 Social willingness marg. cost to pay for public good mb1 mb2 Demand curve mb3 * Opt x0 x0 1 Public good x0 Bernard Caillaud Public goods II.3. Private provision for the public good Graphical analysis in the separable linear case with L = 1: To determine the aggregate demand for a private good, we sum marginal benefit curves horizontally To determine the social marginal benefit for the public good, which then dictates Pareto optimality characterization, we have to sum marginal benefit curves vertically! Yet, in the equilibrium of private provision of the public good, only the private marginal benefit curve for the highest marginal benefit agent matters.
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