. (2018: but this won’t be focal on the exam.)

HERE is a diagram depicting best-reponse functions with strategic complements.

What does SPNE predict (use backwards induction)?

Shepherd A: 120−2sA−sB

The payoff from staying Silent (cooperating) in each period is: −2×(1+g+g2+g3+...)−2×(1+g+g2+g3+...)-2 \times (1 + g + g^2 + g^3 + ... ) Here I get -2 in each period, starting today. Discounting this, we add up -2 (today), −2g−2g-2g (next period), −2g2−2g2-2g^2 (the period after next), etc, as represented above. The payoff from Confessing right away (after which both players Confess always) is: −1−3×(g+g2+g3+...)−1−3×(g+g2+g3+...) -1 -3 \times (g + g^2 + g^3 + ... ) Formula for a geometric series (where 0<g<10<g<10<g<1): g+g2+g3+g4...=g/(1−g)g+g2+g3+g4...=g/(1−g)g + g^2 + g^3 + g^4 ... = g/(1-g) Note on Maths: The standard derivation of this, which is pretty neat, is in the text. This formula is an important one in economics (and beyond), particularly for discounting a constant stream of payoffs, e.g., stock dividends Thus cooperation in a single period is ‘weakly preferred’ (at least as good) if (−2)×(1+g+g2+g3+...)≥(−1)+−3×(g+g2+g3+...)(−2)×(1+g+g2+g3+...)≥(−1)+−3×(g+g2+g3+...)(-2) \times (1 + g + g^2 + g^3 + ... ) \geq (-1) + -3 \times (g + g^2 + g^3 + ...) g+g2+g3+...≥1g+g2+g3+...≥1g + g^2 + g^3 + ... \geq 1 Note on the intuition for the second formula: the left side is loss of future payoffs (-3 vs -2 forever from next period, so a loss of 1 per period starting tomorrow). The right side is gain in ‘the present’ period (getting -1 rather than -2), so it is un-discounted. g/(1−g)≥1g/(1−g)≥1g/(1-g) \geq 1 g≥12

Prior exams with answers and ‘mock exams’

Bee2024 finishing insurance problem from sixth problem set

Shorter 'rooftop' video going through 3 cases of 'two-person Public goods' via powerpoint

Power rule

Individual optimisation solved problem

Nash proved that if we allow mixed strategies, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium.

More volatile underlying assets will translate to higher options premiums, because with volatility there is a greater probability that the options will end up in-the-money at expiration.

The option is European and can only be exercised at expiration.No dividends are paid out during the life of the option.Markets are efficient (i.e., market movements cannot be predicted).There are no transaction costs in buying the option.The risk-free rate and volatility of the underlying are known and constant.The returns on the underlying are normally distributed.

In low-income countries the vast majority are unwilling to pay for effective drugs simply because they are unable to pay. Low-income nations need more price discrimination—and vastly lower prices—if they are ever to afford the world's most effective medicines.

She found a German seller offering packs of the same nappies she buys in Luxembourg for the same price she normally pays. Looking more closely at the unit price, however, Nadine realised that the German packs contained 140 nappies, whereas the packs in Luxembourg had only 90, making them much more expensive. She switched straight away to buying all her nappies from the German shop.

Channel Tunnel

University strikes: Students demand tuition fee refunds for planned walkouts

3. Now draw the farmers’ ‘best response functions’ in a diagram.

Do parts A and B; part C is optional enrichment

This second assumption, called diminishing marginal utility, will imply ‘risk aversion’!

Two statistics about reducing your risk of an early death made headlines around the world recently. The first seems to be a great reason to add a four-legged friend to your life. It suggests that owning a dog is tied to lowering your chance of dying early by nearly a quarter. The second statistic claims that even a minimal amount of running is linked to reducing your risk of premature death by up to 30%. Ruth Alexander finds out what’s behind these numbers and we hear from epidemiologist, Gideon Meyerowitz-Katz.

Should we help companies tailor prices to your wage packet?