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.
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.
Proofs are hard because we often know the end, and have to figure out the beginning and middle.
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 with and
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
and we have the proof graphically
In my paper “Reimbursement and Investment: Prospective Payment and For-Profit 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:
The optimal level of quality enhancing technology under the average, cost-based, retrospective reimbursement without investment expenditures and that with investment expenditures as Tn and Ty respectively. (definition)