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The Core of Neoclassical EC 742: Handout 2: An Overview of the essential geometry and mathematics of Neoclassical Public Economics I. An Introduction to the Microeconomics of Public Economics. Part II: On the Geometry of Neoclassical Public Economics A. The first half of this handout provides a condensed overview of the basic geometry of public finance with some extensions to the theory of I. Rationality as Net Benefit Maximizing Choice regulation and public choice. A. Nearly all economic models can be developed from a fairly simple model of rational decision making that assume that individuals maximize i. It is intended to be a review for students who have had undergraduate courses in public finance and regulation, and as a quick overview for those who have not. their (expected) private net benefits. a. We will draw upon this “tool bag” repeatedly during this course. i. Consumers maximize consumer surplus: the difference between what a thing is b. These tools provide much of the neoclassical foundation for public economics, worth to them and what they have to pay for it. both as it was being worked out in the first half of the twentieth century and as CS(Q) = TB(Q) - TC(Q) it was applied and extended in the second half. ii. Firms maximize their profit:, the difference in what they receive in revenue from selling a product and its cost of production: B. The second half of the handout reviews some of the core mathematical tools of public economics. = TR(Q) - TC(Q) i. Most of these results and approaches are taken for granted in published research. B. The change in benefits, costs, etc. with respect to quantity consumed or produced is generally called Marginal benefit, or Marginal cost. ii. Other tools are now used, but most are grounded in those covered in this i. DEF: Marginal "X" is the change in Total "X" caused by a one unit change in handout or are “ideosyncratic” ones, used only in a handful of papers. quantity. It is the slope of the Total "X" curve. "X" {cost, benefit, profit, C. Part I begins with a somewhat novel net-benefit maximizing product, utility, revenue, etc.} representation of rational choice. ii. Important Geometric Property: Total "X" can be calculated from a Marginal "X" curve by finding the area under the Marginal "X" curve over the range of i. That model is used to derive supply and demand curves. interest (often from 0 to some quantity Q). This property allows us to ii. It also provides a useful logical foundation for a variety of geometric, algebraic, determine consumer surplus and/or profit from a diagram of marginal cost and and calculus representations of the burden of taxation. marginal revenue curves. iii. It can also be used for normative analysis, if we accept Pigou’s applied form of utilitarianism which economists tend to refer to as welfare economics. Figure 1 iv. The same geometric approach can also used to analyze (a) externality problems, $/Q and (b) public goods problems. In future lectures, this approach will also be used to analyze electoral and MC interest group--based politics D. Part II develops a more or less parallel calculus-based analysis of taxes I III and public goods. V II IV VI MB Q Q' Q* Q'' Page 1 EC 742: Handout 2: An Overview of the essential geometry and mathematics of Neoclassical Public Economics C. Examples: It is partly for this reason that most economic models begin with utility and/or production functions, even though sharp conclusions are rare from i. Given the marginal cost and marginal benefit curves in Figure 1, it is possible to those perspectives. calculate the total cost of Q' and the total benefit of Q' . These can be represented geometrically as areas under the curves of interest. TC(Q') = II ; ii. Net benefit maximizing models can be regarded as an operational counterpart to TB(Q') = I + II . utility function based analysis. ii. Similarly, one can calculate the net benefits by finding total benefit and total { Dollars unlike Utils are observable and measurable. cost for the quantity or activity level of interest, and subtracting them. Thus the iii. For many purposes the net benefit maximizing framework and geometry is more net benefit of output Q' is TB(Q') - TC(Q') = [I + II ] - [ II ] = I. tractable and more easily understood than its calculus based counter part. iii. Use Figure 1 to determine the areas that correspond to the total benefit, cost and This makes it a very useful tool for classroom analysis, especially in net benefit at output Q* and Q''. undergraduate classes, but also in graduate classes.. iv. Answers: F. The same geometry can be used to characterize ideal or optimal policies if a. TB(Q*) = I + II + III + IV , TC(Q*) = II + IV , NB(Q*) = I + III the “maximize social net benefits” normative theory is adopted and “all” b. TB(Q”) = I + II + III + IV + VI , TC (Q”) = II + IV + V + VI , NB(Q”) = relevant costs and benefits can be computed I + III - V This approach can be used to rationalize Benefit-Cost analysis D. If one attempts to maximize net benefits, it turns out that generally he or It is also widely used in essentially any diagram that represents dead weight she will want to consume or produce at the point where marginal cost loss or excess burden in terms of a numeraire good such as dollars or Euros. equals marginal benefit (at least in cases where Q is very divisible). Similarly, most partial equilibrium analyses of problems associated with i. There is a nice geometric proof of this. (The example above, C, nearly proves taxation, monopoly, externalities, and public goods are undertaken with this this. Note that NB(Q*) > NB(Q') and NB(Q*) > NB(Q").) basic geometry -- or at least can most easily be interpreted as such. ii. In the usual chase, a net-benefit maximizing decision maker chooses G. That each person maximizes their own net benefits does not imply that consumption levels (Q) such that their own marginal costs equal their own every person will agree about what the ideal level or output of a particular marginal benefits. good or service might be, nor does it imply that all rational persons are iii. They do this not because they care about "margins" but because this is how one selfish--although they may be self centered, e.g. focus on their own maximizes net benefits in most of the choice settings of interest to assessment of net benefits however calculated. economists. (Another common choice that maximizes net benefits is Q* = 0. Why?) i. Most individuals will have different marginal benefit or marginal cost curves, and so will differ about ideal service levels. iv. This characterization of net benefit maximizing decisions is quite general, and can be used to model the behavior of both firms and consumers in a wide range ii. To the extent that these differences can be predicted, they can be used to model of circumstances. both private and political behavior: a. (What types of persons will be most likely to lobby for subsidies for higher E. Notice that this approach yields sharper conclusions than the standard education? utility-based models, most of which are consistent with empirical work and economic intuition. b. What types of persons will prefer progressive taxation to regressive taxation? c. What industries will prefer a carbon tax to a corporate income tax?) Demand curves slope downward and supply curves upward, as shown iii. In cases in which an individual is not-selfish, benefits or costs that accrue to below. others will affect his or her own benefits. i. However, it does not generate comparative statics as well. To do this requires fleshing out the benefit and cost functions. H. One can use the consumer-surplus maximizing model to derive a consumer's demand curve for any good or service (given their marginal benefit curves) Page 2 EC 742: Handout 2: An Overview of the essential geometry and mathematics of Neoclassical Public Economics i. This is done by: (a) choosing a price, (b) finding the implied marginal cost curve II. Markets and Social Net Benefits for a consumer, (c) use MC and MB to find the CS maximizing quantity of the A. Market Demand can be determined by varying price of a single good and good or service, (d) plot the price and the CS maximizing Q*, and (e) repeat with other prices to trace out the individual's demand curve. adding up the amounts that consumers want to buy of that good at each price. ii. This method of deriving demand implies that demand curves always slope downward because the demand curve is composed of a subset of the downward i. Market Demand curves for ordinary private goods, thus, can be shown to be sloping sections of individual MB curves, namely all those that can maximize "horizontal" sums of individual demand curves social net benefits for given prices. ii. Similarly, Market Supply (for an industry with a fixed number of firms) can be If an individual’s MB curve is monotonicly downward sloping, then his or derived by varying price and adding up the amounts that each firm in the her demand curve is exactly the same as his or her MB curve. industry is willing to sell at each price. If an individual’s MB is not monotonicly downward sloping (e.g. has iii. Market Supply curves for ordinary private goods can be shown to be bumps), then only a subset of the points on the MB curve will turn up on "horizontal" sums of individual firm supply curves.
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