microeconomics

Microeconomics : 

Microeconomics

Choice and risk : 

Choice and risk Choice is now not over consumption plans, but lotteries or gambles, which offer consumption plans as prizes with given probabilities. Rainy day,with umbrella Pleasant day,with umbrella 0.4 0.6 Gamble 1: Carry Umbrella Rainy day,without umbrella Pleasant day,without umbrella 0.4 0.6 Gamble 2: Don’t Carry Umbrella

Gambles : 

Gambles g=(p1◦a1, p2◦a2, …, pn◦an) is a gamble which offers prize a1 with probability p1, a2 with probability p2 and so on. We require p1+⋯pn = 1 Consumer has a complete and transitive preference ordering ≽ over gambles. This ordering depends on both the preference over prizes and the probabilities of getting each. Can we separate the two?

Expected utilities : 

Expected utilities An utility function u(·) defined over the set of prizes is an expected utility function if, for two gambles gx=(p1x◦a1, p2x◦a2, …, pnx◦an) and gy=(p1y◦a1, p2y◦a2, …, pny◦an) gx ≽ gy if and only if p1xu(a1) + ⋯ + pnxu(an) ≥ p1yu(a1)+ ⋯ + pnyu(an)

Two compound lotteries : 

Two compound lotteries As a doctor in a position of authority in the national government, you’ve been informed that a new flu epidemic will hit your country next winter and that this epidemic will result in the deaths of 600 people. There are two possible vaccination programs that you can undertake, and doing one precludes doing the other. The first will save 400 people with certainty. The second will save no one with probability 1/3 and 600 with probability 2/3. Which do you prefer?

Two other compound lotteries : 

Two other compound lotteries As a doctor in a position of authority in the national government, you’ve been informed that a new flu epidemic will hit your country next winter. To fight this epidemic, one of two possible vaccination programs is to be chosen, and undertaking one precludes the other. In the first program, 200 people will die with certainty. In the second, there is a 2/3 chance that no one will die and 1/3 chance that 600 will die. Which do you prefer?

Substitution Axiom : 

Substitution Axiom Suppose p and q are two lotteries such that p ≻ q. Suppose α is a number from the open interval (0,1), and r is some other lottery. Then, αp + (1 − α)r ≻ αq + (1 − α)r

Archimedian Axiom : 

Archimedian Axiom Suppose p, q and r are three lotteries such that p ≻ q ≻ r. Then numbers α and β exist, both from the open interval (0,1), such that, αp + (1 − α)r ≻ q ≻ βp + (1 − β)r

Expected Utility Theorem (von Neumann-Morgenstern) : 

Expected Utility Theorem (von Neumann-Morgenstern) If preferences over gambles is complete, transitive, satisfies the Substitution and Archimedian axioms then it has an expected utility representation.

Expected Utility of Money : 

Expected Utility of Money Utility Prize u2 u1 e a1 a2 E(g) CE(g)

Risk aversion : 

Risk aversion A consumer is Risk averse: if CE(g) ≤ E(g) Risk neutral: if CE(g) = E(g) Risk lover: if CE(g) ≥ E(g)