← 9. Risk and Uncertainty

Exercises

Problems to check your understanding of this chapter. The solutions are folded away — try them yourself first.

Medium

1Medium

You are handed, for free, a lottery ticket that pays 180180 or 2020 with equal probability (expected value 100100).

With utility u(x)=xu(x) = \sqrt{x}, what is the smallest sure amount you would trade the ticket for?

Read as insurance against the loss instead, what are the fair premium and the maximum premium?

Wealth 180180 or 2020 — the certainty equivalent 8080 is marked on the curve.
Show solution

  1. Expected utility is EU=0.5180+0.520≈8.94EU = 0.5\sqrt{180} + 0.5\sqrt{20} \approx 8.94.

    The certainty equivalent is CE=EU2=14(180+20+2180⋅20)=14(200+120)=80CE = EU^2 = \frac{1}{4}(180 + 20 + 2\sqrt{180 \cdot 20}) = \frac{1}{4}(200 + 120) = 80.

    Sell for any sure 8080 or more; below that, keep the ticket.

  2. The risk premium is 100−80=20100 - 80 = 20: a gamble worth 100100 on average feels like a sure 8080. Read as insurance, a loss of 160160 hits with probability 0.50.5, so the fair premium is 0.5⋅160=800.5 \cdot 160 = 80 and the maximum premium is 180−80=100180 - 80 = 100 — exactly the fair premium plus the risk premium of 2020.

Hard

2Hard

The job-loss probability has risen to 50%50\%.

(1) How much does the scene's household (income 100100 today, 6060 or 2020 tomorrow, β=0.9\beta = 0.9) save, and what is its consumption today? Compare with the no-risk case.

(2) For the scene's factory investment (cost 100100, value tomorrow 110±12110 \pm 12), does investing now or waiting win, and what is the option value of waiting?

Job-loss risk 50%50\%, spread 1212 — saving jumped, and waiting has just taken the lead.
Show solution

  1. (1) With no risk, saving is about 11.511.5 and consumption today is 88.588.5.

    At a job-loss probability of 0.50.5, optimal saving jumps to 25.2525.25 and consumption today falls to 74.7574.75.

    Average income never changed, yet the household saves 13.7513.75 more — precautionary saving made purely of fear.

  2. (2) Investing now earns expected profit 110−100=10110 - 100 = 10.

    Waiting skips the bad state (value 98<10098 < 100), giving 0.5⋅(122−100)+0.5⋅0=110.5 \cdot (122 - 100) + 0.5 \cdot 0 = 11, and even after the one-period discount of 0.950.95 that is 10.45>1010.45 > 10, so waiting wins.

    The pure option value is 11−10=111 - 10 = 1. A spread of 1212 has just crossed the flip threshold of about 11.111.1.

  3. Both decisions reacted to uncertainty, not to any average. When households and firms pull back at the same time, aggregate demand and investment fall together — the path by which an uncertainty shock becomes a recession.