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14.13 Economics and Psychology (Lecture 4) Xavier Gabaix February 12, 2004 1 Prospect Theory value of the game Consider gambles with two outcomes: x with probability p, and y with probability 1 − p where x ≥ 0 ≥ y. The PT value of the game is V = π (p) u (x) + π (1 − p) u (y) • In prospect theory the probability weighting π is concave first and then convex, e.g. pβ π (p) = β p + (1 − p)β for some β ∈ (0, 1). In the figure below p is on the horizontal axis and π (p) on the vertical one. y 1 0.75 0.5 0.25 0 0 0.25 0.5 1 0.75 x • A useful parametrization of the PT value function is a power law function u (x) = |x|α for x ≥ 0 u (x) = −λ |x|α for x ≤ 0 y 2.5 0 -5 -2.5 0 2.5 5 x -2.5 -5 Meaning - Fourfold pattern of risk aversion u • Risk aversion in the domain of likely gains • Risk seeking in the domain of unlikely gains • Risk seeking in the domain of likely losses • Risk aversion in the domain of unlikely losses See tables on next page. Preferences between Positive and Negative Prospects Positive Prospects Problem 3: N = 95 Problem 4: N = 95 Problem 7: N = 66 Problem 8: N = 66 Negative Prospects (4,000, .80) < (3000). [20] [80]* (4,000, .20) > (3,000, .25). [65]* [35] (3,000, .90) > (6,000, .45). [86]* [14] (3,000, .002) < (6,000, .001). [27] [73]* Problem 3': (-4,000, .80) > (-3000). N = 95 [92]* [8] Problem 4': (-4,000, .20) < (-3,000, .25). N = 95 [42] [58] Problem 7': (-3,000, .90) < (-6,000, .45). N = 66 [8] [92]* Problem 8': (-3,000, .002) > (-6,000, .001). N = 66 [70]* [30] Percentage of Risk-Seeking Choices Gain Loss Subject p < .1 p > .5 p < .1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 100 85 100 71 83 100 100 87 16 83 100 100 87 100 66 60 100 100 60 100 100 100 100 71 100 38 33 10 0 0 5 10 0 0 0 26 30 100 16 0 21 0 5 15 22 10 5 0 0 31 0 0 20 0 30 20 0 30 10 80 0 0 10 10 30 30 10 20 10 60 0 0 0 0 80 10 75 93 58 100 100 86 100 100 93 100 100 94 100 100 100 100 93 63 81 100 92 100 100 87 10 2 88a 20 0 80a Risk seeking Risk neutral Risk averse 78a 12 10 a Values that correspond to the fourfold pattern. Note: The percentage of risk-seeking choices is given for low (p < . 1) and high (p > .5) probabilities of gain and loss for each subject (riskneutral choices were excluded). The overall percentage of risk-seeking, risk-neutral, and risk-averse choices for each type of prospect appears at the bottom of the table. p > .5 87a 7 6 Properties of power law PT value functions • they are scale invariant, i.e. for any k > 0 Consider a gamble and the same gamble scaled up by k: g= ( x y kg = ( kx ky with prob p with prob 1 − p with prob p with prob 1 − p then V P T (kg) = kαV P T (g) • if someone prefers g to g 0 then he will prefer kg to kg 0 for k > 0 • if x, y ≥ 0, V (−g) = −λV (g) • if x0, y 0 ≥ 0 and someone prefers g to g 0 then he will prefer −g 0 to −g 2 How robust are the results? • Very robust: loss aversion at the reference point, λ > 1 • Medium robust: convexity of u for x < 0 • Slightly robust: underweighting and overweighting of probabilities π (p) ≷ p 3 In applications we often use a simplified PT (prospect theory): π (p) = p and u (x) = x for x ≥ 0 u (x) = λx for x ≤ 0 4 Second order risk aversion of EU • Consider a gamble σ and −σ with 50 : 50 chances. • Question: what risk premium Π would people pay to avoid the small risk σ? ´ 2 • We will show that as σ → 0 this premium is O σ . This is called second order risk aversion. ³ • In fact we will show that for twice continuously diﬀerentiable utilities: ρ 2 Π (σ) ∼ = σ , 2 00 u where ρ is the curvature of u at 0 that is ρ = − u0 . • Let’s generalize and consider an agent starting with wealth x. The agent takes the gamble iﬀ: 1 1 B (Π) = u (x + Π + σ) + u (x + Π − σ) ≥ u (x) 2 2 i.e. Π ≥ Π∗ where: B (Π∗) = u (x) • Assume that u is twice diﬀerentiable and take the Taylor expansion of B(Π) for small σ and Π: 1 u(x + Π + σ) = u(x) + u0(x)(Π + σ) + u00(x)(Π + σ)2 + o(Π + σ)2 2 1 00 0 u(x + Π − σ) = u(x) + u (x)(Π − σ) + u (x)(Π − σ)2 + o(Π − σ)2 2 hence h i ³ ´ 1 00 0 2 2 2 2 B (Π) = u (x) + u (x) Π + u (x) σ + Π + o σ + Π 2 Then use the definition B(Π∗) = u(x) to get ³ ´ ∗2i ρh 2 ∗ 2 ∗2 Π = σ +Π +o σ +Π 2 h i ρ ∗ 2 ∗2 • to solve : Π = 2 σ + Π for small σ, call ρ0 = ρ/2. • Barbaric way : — find the roots of Π∗2 − ρ10 Π∗ + σ 2 = 0. ∗ compute the discriminant 1 ∆ = 02 − 4σ 2 ρ µ 1 ± 1 1 − 4σ 2 ∗ the roots are Π∗ = 2ρ 0 2 ρ02 1 1 ∗ Π = 0± 2ρ 2 Ã ¶1 2 1 2 − 4σ ρ02 !1 2 ∗ as when there is no risk, the risk premium should be 0, then the relevant root is: 1 1 Π∗ = 0 − 2ρ 2 Ã 1 2 − 4σ ρ02 !1 2 — take the Taylor expansion for small σ ³ ´1 1 1 Π∗ = 0 − 0 1 − 4ρ02σ 2 2 2ρ 2ρ µ ¶ 1 1 1 02 2 = 0 − 0 1 − 4ρ σ + o(σ 2) 2ρ 2ρ 2 = ρ0σ 2 — then remember that ρ0 = ρ/2: Π∗= ρ2 σ 2