The Encyclopedia of Economic Principals

Chapter 1

Choice Under Risk and Across Time

Price risk with a certainty equivalent and time with an Euler equation.

Four of the chapter's worked examples, made interactive: insuring a risky crop, the Arrow-Pratt premium as wealth changes, two-period saving with and without a cap, and precautionary saving. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Insure the crop or bear the risk

When does a sure income below the lottery's expected income still win?

The chord between the two outcomes gives the lottery's expected utility at its expected income. Because the curve is concave, the chord lies below it, so the same utility is reached on the curve at a smaller sure income, the certainty equivalent. Any certain contract paying more than that beats the lottery.

Equation, written in LaTeX: u(c)=E[u(X)]

Equation, written in LaTeX: EU(A)=\tfrac12\sqrt{100}+\tfrac12\sqrt{25}=7.5.

Equation, written in LaTeX: EU(B)=\sqrt{58}\approx7.616

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Income x is in thousands of dollars and utility is u(x) = sqrt(x). Lottery A pays 100 or the bad-year income with probability one half each. Contract B pays a certain income after its premium. The certainty equivalent c is the sure income with the same expected utility as A.

Predict first. If bad-year income rises from 25 to 36, will the decision maker still pay for the 58 contract?

Your prediction

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Figure: Insure the crop or bear the risk. The square-root utility curve with a chord from bad-year income 25 to 100. On the chord at expected income 62.50 the utility is 7.500; the curve reaches that utility at the certainty equivalent 56.25. The insured point at 58 has utility 7.616.
Bad-year income ($ thousand): 25, Certain income with insurance ($ thousand): 58
Constructed example: the chapter's hypothetical crop lottery (100 or 25, insured 58); bad-year incomes 9, 16 and 36 and insured incomes 54 and 62 are added for comparison.

Calculated values

Expected income E[X] ($000)
62.50
EU(A)
7.500
Certainty equivalent ($000)
56.25
Risk premium ($000)
6.25
EU(B)
7.616
Choice
Insure (B)

EU(A) = 0.5 x sqrt(100) + 0.5 x sqrt(25) = 0.5 x 10 + 0.5 x 5 = 7.500. The certainty equivalent solves sqrt(c) = 7.500, so c = 56.25, and the risk premium is 62.50 - 56.25 = 6.25. EU(B) = sqrt(58) = 7.616. B gives higher expected utility, so the decision maker insures. Insurance gives up 4.50 thousand of expected income, less than the 6.25 thousand risk premium.

Worked steps

  1. E[X] = 0.5 x 100 + 0.5 x 25 = 62.50
  2. EU(A) = 0.5 x 10 + 0.5 x 5 = 7.500
  3. CE = 7.500 x 7.500 = 56.25
  4. Risk premium = 62.50 - 56.25 = 6.25
  5. EU(B) = sqrt(58) = 7.616, above EU(A) = 7.500

Use the idea

Before buying insurance, compute the certainty equivalent of going without it and compare it with the income the policy guarantees after its premium.

Where the conclusion applies

One period, two equally likely outcomes, square-root utility over final income and a contract that removes all risk. A different utility curvature changes the certainty equivalent and can reverse the choice.

Check your understanding: With bad-year income 36 and insured income 58, which contract is chosen, and what is the risk premium?
EU(A) = 0.5 x 10 + 0.5 x 6 = 8, so CE = 64 and E[X] = 68; the premium is 68 - 64 = 4. Since sqrt(58) = 7.616 is below 8, the uninsured lottery A is chosen.

Chapter 1 source: section "Expected-Utility Theorem".

Demonstration 2 of 4

Same gamble, richer person

How does the price of bearing a fixed dollar gamble change as wealth grows?

Log utility has absolute risk aversion 1/w, which falls as wealth rises. The same dollar risk therefore costs less in certainty-equivalent terms to a richer person. The dashed approximation uses only the curvature at w; the solid line is the exact premium.

Equation, written in LaTeX: A(w)=\frac{1}{w}

Equation, written in LaTeX: \pi\approx\frac{1}{2}A(w)\operatorname{Var}(\varepsilon)

Equation, written in LaTeX: CE=\exp[\frac{1}{2}\ln(11{,}000)+\frac{1}{2}\ln(9{,}000)]=\sqrt{99{,}000{,}000}\approx\$9{,}949.87

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w is initial wealth in dollars and utility is u(w) = ln w. The gamble adds or subtracts the stake with probability one half each, so its variance Var is the stake squared. A(w) is absolute risk aversion and pi is the risk premium.

Predict first. Doubling wealth from $10,000 to $20,000 with the same $1,000 gamble: does the premium halve, stay the same, or double?

Your prediction

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Figure: Same gamble, richer person. Risk premium against wealth for a gamble of plus or minus $1,000: the exact premium and the approximation both fall as wealth rises. At $10,000 the exact premium is $50.13 and the approximation $50.00.
Initial wealth ($): $10,000, Gamble size, plus or minus ($): $1,000
Constructed example: the chapter's hypothetical $10,000 wealth and $1,000 gamble, with the book's $20,000 and $2,000 cases; wealth of $5,000 and $40,000 and a $500 stake are added for comparison.

Calculated values

A(w) = 1/w
0.0001
Variance
1,000,000
Approximate premium
$50.00
Certainty equivalent
$9,949.87
Exact premium
$50.13
Approximation error
$0.13

A(w) = 1/10,000 = 0.0001, so the approximation is 0.5 x 0.0001 x 1,000,000 = $50.00. The exact certainty equivalent is sqrt(11,000 x 9,000) = $9,949.87, so the exact premium is $10,000 - $9,949.87 = $50.13. The approximation misses by $0.13.

Worked steps

  1. A(w) = 1 / 10,000 = 0.0001
  2. Var = 1,000 x 1,000 = 1,000,000
  3. pi approx = 0.5 x 0.0001 x 1,000,000 = $50.00
  4. CE = sqrt(11,000 x 9,000) = $9,949.87
  5. Exact premium = 10,000 - 9,949.87 = $50.13

Use the idea

Use (1/2) A(w) Var as a quick price for a small risk, and check it against the exact certainty equivalent when the stake is a large share of wealth.

Where the conclusion applies

Log utility over total wealth, a fair two-point gamble and no other risks. The approximation is local: it becomes less accurate as the stake grows relative to wealth.

Check your understanding: At wealth $40,000 and stake $1,000, what is the approximate premium?
A = 1/40,000 = 0.000025, so pi = 0.5 x 0.000025 x 1,000,000 = $12.50; the exact premium is also $12.50 to the cent.

Chapter 1 source: section "Arrow-Pratt Risk-Aversion Measure".

Demonstration 3 of 4

Saving when the interest rate moves

How much does a two-period consumer save, and what changes when saving is capped?

The Euler equation sets the slope of consumption growth, c1 = beta(1 + r) c0, and the lifetime budget fixes the level. The chosen point is where an indifference curve touches the budget line. A saving cap adds a wall: consumption today cannot fall below 120 minus the cap.

Equation, written in LaTeX: c_1=0.9984c_0

Equation, written in LaTeX: 120+\frac{100}{1.04}=216.1538

Equation, written in LaTeX: c_0+\frac{c_1}{1.04}=216.1538

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c0 and c1 are consumption today and next period, in thousands of dollars. Income is 120 today and 100 next period, beta = 0.96 is the discount factor and r is the interest rate. Utility is ln c0 + beta ln c1. The equations are the book's 4 percent case.

Predict first. Will saving rise or fall when r drops from 4 percent to 0?

Your prediction

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Figure: Saving when the interest rate moves. Two-period budget line at an interest rate of 4 percent through the income point (120, 100). The chosen point is (110.28, 110.11), where the indifference curve touches the budget line.
Interest rate: 4%, Saving cap: No cap
Constructed example: the chapter's hypothetical consumer (income 120 and 100, beta 0.96) at the book's rates 4 percent and 0 and its cap of 5; rates of 2 and 8 percent are added for comparison.

Calculated values

Lifetime resources
216.1538
c0 today
110.28
c1 next period
110.11
Saving
9.72
beta(1 + r)
0.9984
Saving cap
none

beta(1 + r) = 0.96 x 1.04 = 0.9984, so c1 = 0.9984 c0. Resources are 120 + 100 / 1.04 = 216.1538, so c0 = 216.1538 / (1 + 0.96) = 110.2826 and c1 = 0.9984 x 110.2826 = 110.11. Saving is 120 - 110.28 = 9.72. The book prints next-period consumption as 110.10 and the grown saving as about 10.10; the exact values are 110.106 and 10.106, which round to 110.11 and 10.11.

Worked steps

  1. beta(1 + r) = 0.96 x 1.04 = 0.9984
  2. Resources = 120 + 100 / 1.04 = 216.1538
  3. c0 = 216.1538 / 1.96 = 110.2826
  4. c1 = 0.9984 x 110.2826 = 110.11
  5. Saving = 120 - 110.2826 = 9.72

Use the idea

Before reading a consumption path as a sign of impatience, check whether a borrowing or saving limit could have produced it.

Where the conclusion applies

Two periods, log utility, a known interest rate and income, and free borrowing or saving up to any cap shown. The comparison across interest rates mixes substitution and wealth effects.

Check your understanding: With r = 0.04 and a cap of 5, what is next-period consumption?
c0 = 120 - 5 = 115 and c1 = 100 + 5 x 1.04 = 105.20.

Chapter 1 source: section "Intertemporal Choice and Discounting".

Demonstration 4 of 4

Precautionary saving and the third derivative

Does a riskier future make a consumer save more?

Saving stops where today's marginal utility equals expected marginal utility next period. When marginal utility is convex (u''' > 0), a spread around the same mean raises the expected value, so the crossing moves to the right.

Equation, written in LaTeX: \frac{1}{100-s}=\frac{1}{2}\frac{1}{50+s}+\frac{1}{2}\frac{1}{150+s}

Equation, written in LaTeX: s=\frac{-100+\sqrt{15{,}000}}{2}\approx 11.24

Equation, written in LaTeX: u(c)=200c-c^2/2

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s is saving carried from today (100 units) to next period at zero interest and no discounting. Future income is 100 plus or minus the spread, each with probability one half. u'(c0) is marginal utility today and expected u'(c1) is expected marginal utility next period.

Predict first. Under quadratic utility with the 50/150 lottery, is saving positive?

Your prediction

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Figure: Precautionary saving and the third derivative. Current and expected future marginal utility against saving. The curves cross at s = 11.24, with future income 50 or 150 and log utility.
Future income spread, plus or minus: 50, Utility: Log, u = ln c
Constructed example: the chapter's hypothetical consumer (spreads 0 and 50, log and quadratic utility); spreads of 25 and 75 are added for comparison.

Calculated values

Optimal saving s
11.24
c0 today
88.76
c1 in the low state
61.24
c1 in the high state
161.24
Future income
50 or 150

s = (-100 + sqrt(10,000 + 2 x 50 x 50)) / 2 = (-100 + 122.474) / 2 = 11.24. Log utility has u''' = 2/c^3 > 0, so risk raises expected future marginal utility and the consumer saves.

Worked steps

  1. Future income is 50 or 150 with probability one half each
  2. The Euler equation reduces to s^2 + 100 s - 1,250 = 0
  3. sqrt(10,000 + 5,000) = 122.474
  4. s = (-100 + 122.474) / 2 = 11.24
  5. c0 = 100 - 11.24 = 88.76; c1 = 61.24 or 161.24

Use the idea

When income becomes more uncertain, expect households with prudent preferences to build a buffer even if average income is unchanged.

Where the conclusion applies

Two periods, equal-probability outcomes, zero interest, no discounting and no borrowing limit. The quadratic case holds only over the range where 200 - c is positive.

Check your understanding: With log utility and spread 25 (future income 75 or 125), what is s?
s^2 + 100 s - 312.5 = 0, so s = (-100 + sqrt(11,250)) / 2 = (-100 + 106.066) / 2 = 3.03.

Chapter 1 source: section "Prudence and Precautionary Saving".