Transcript Slide 1
Writing Formal Proofs First points: • This is written for mathematical proofs. Unless you are doing math econ, formal game theory, or statistical/econometric development (not application) you may not do formal mathematical proofs. • Sometimes proofs in economics are more informal. They may be proved by logic or discussion of the derivations, for example, using assumed signs or magnitudes of derivatives to draw conclusions. • Even if you are not doing a formal mathematical proof, the steps are the same. The difference is in the (mathematical) notation and the presentation. Important steps: (from Cheng) 1. Plan your proof. 2. Decide how formal you will be. 3. Understand your path. Logical arguments are usually classified as either 'deductive' or 'inductive'. When you are writing a proof you are focusing on deduction, not induction. Deduction: • Begin with some statements, called 'premises', that are assumed to be true, • Then determine what else would have to be true if the premises are true. For example, you can begin by assuming that there is diminishing marginal utility, and then determine what would logically follow from such an assumption. • You provide absolute proof of your conclusions, given that your premises are correct. The premises themselves, however, remain unproven and unprovable, they must be accepted on face value, or by faith, or for the purpose of exploration. Induction: • You begin with some data, and then determine what general conclusion(s) can logically be derived from those data. • In other words, you determine what theory or theories could explain the data. For example, you note that people by less of a good as they have more of it, even if the price drops. From this you conclude that consumption demonstrates diminishing marginal utility. • Point is to provide a reasonable hypothesis given the data. • Induction does not prove that the theory is correct. There are often alternative theories that are also supported by the data. • What is important in induction is that the theory does indeed offer a logical explanation of the data. • Remember that most science, and especially economics, is inductive. • That doesn’t mean there are no deductive proofs in economics. Proofs in economics, like math, are deductive. • Induction might supply the premises. For example, you might find the coefficient in an estimation is negative. Taking that as a premise, you can then prove a conjecture about behavior. • Specify your premises or axioms. • Determine your approach (formal or informal). There are three general types of formal proofs: – Direct (Deductive) Proof – Indirect Proof or Proof by Contradiction. Assume something true. Show if it is true, your axioms must be false. – Inductive Proof. Prove statement P(1) is true. Prove is P(n)) is true then P(n+1) is true. Inductive proofs apply only to propositions about or indexed by integers. • Clearly fill in every step. • Check your logic. • Check your work. Specific Hints • Choose right mixture of mathematics and words. This is true for derivations or mathematical manipulations in theoretical or empirical papers. • Divide proofs into clearly identified steps or cases. • Provide all the conditions needed to reach a conclusion. • Don’t leave too many steps to the reader. What seems obvious to you may not to a reader. • Be clear when a proof ends. • Explore all possible variants of your results. Can you say something else? Have you said too much? Structure of a Proof • A proof is a series of statements, each of which follows logically from what precedes it. A proof has a beginning, a middle and an end. – Beginning: Definitions, premises, axioms. – Middle: statements, each following logically from the stuff before it – End: the thing we're trying to prove • Proofs are hard because we often know the end, and have to figure out the beginning and middle. – How the middle proceeds depends on what we assume at the beginning. Often when doing a proof you may find that you must add premises to make the steps needed to reach the conclusion. – Better proofs use fewer premises, because you are looking for generality. The more premises you impose, the less general and the more restrictive the proof is. • You will think of things in a different order than it logically follows. – you will almost always start with the end – what you are trying to prove. – If you are insightful you will understand (most of) the assumptions needed, and some idea of the path needed. But – Be ready to go back and reorder your steps and add to your assumptions to make sure your proof is complete. Types of things we prove • Two things are equal: Results from duality, so cost minimization gives the same output as profit maximization in a perfect competition. • One thing implies another: Profit maximization implies cost minimization; Giffen goods implies an inferior good. (xy) • Two things imply each other: This is “if and only if”. (xy) • Something has a particular characteristic: Gasoline demand is inelastic. • All of a certain type of agent behave in a particular way: All for-profit hospitals minimize costs. • There is a certain type of agent do something: There is a good with an upward sloping demand curve. • If more than one thing, another thing must be true: A forprofit hospital facing non-profit competition does not minimize cost. (xyzw) Some more about the structure of a proof • The end of a proof should come at the end. The proof should end with the thing you're trying to prove. • Before starting the formal proof, you can begin by announcing what the end is going to be. Other things • They are not tautologies. That is, they don’t begin and end at the same place. You don’t have a proof if you start with elastic goods have price elasticity greater than 1 and end with elasticity greater than 1 implies elastic goods. • They don’t begin at the end. If you goal is to prove that MR for elastic goods is positive, you don’t start with elastic goods have positive MR and prove backwards. • Skipping steps. Make sure the transitions are clear and obvious. Hand waving is way of skipping steps. • Dependent on incorrect, indefensible or assumptions, logic or definition. Assuming an inelastic good has positive marginal revenue. This includes assuming too much. An Example of an Informal Proof From my paper This proof shows that increasing the subsidy for low income people if they get sick later in life lowers their investment in healthy behavior early in life. Utility each period is U ( ) with U 0 and U 0. Hence we have risk aversion. Expected utility for poor people is Where Ctp is consumption endowment in period t, hp is investment in health, s and v are utility costs of being sick, (hp) with ’>0 and ”<0 is the probability of getting sick in the second period, and B is the subsidy if you get sick. The FONC with respect to hp is Hence and we have the proof graphically Example 2: Proofs following from proofs In my paper “Reimbursement and Investment: Prospective Payment and ForProfit Hospitals Market Share” written with Seugnchul Lee, we have a corollary 1 from two propositions: “The amount of quality enhancing technology adopted by the not-for-profit hospital under the average cost-based retrospective reimbursement system is greater when the payment includes investment expenditures than when the payment does not include investment expenditures.” The premises of this proof are propositions we already proved: 1. When the government fully pays treatment costs and hospitals are reimbursed their average cost excluding investment expenditures, the not-for-profit hospital invests only in quality enhancing technology while the for-profit hospital does not invest in either technology. 2. When the government fully pays treatment costs and hospitals are reimbursed their average cost plus investment expenditures, the not-forprofit hospital invests only in quality enhancing technology and obtain the globally maximal level of quality. Meanwhile the for-profit hospital again makes no investment of either type. 1. 2. 3. 4. 5. 6. 7. 8. The optimal level of quality enhancing technology under the average, costbased, retrospective reimbursement without investment expenditures and that with investment expenditures as Tn and Ty respectively. (definition) Government retrospectively fully pays average treatment costs. (premise) Utility depends on output, which depends of investment: U (q(T )) where q is quality, and the hospital maximizes utility. (premise) U / q 0 and q / T 0 U / T 0. (first two statements are premises, implying the last through transitivity. Hence a premise). AC with investment = ACy > ACn = AC without investment (mathematical statement which is actually a premise). A spending constraint is binding (proven earlier, in the paper, derived from utility maximization, hence a premise). Premises 5 and 6 imply that there is more spending on quality enhancing investment when investment costs are reimbursed than when they are not (5 6). 347 prove the proposition.