expected utility and jensen's inequality
TRANSCRIPT
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7/31/2019 Expected Utility and Jensen's Inequality
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Economic Rationality Qua
Expected Utility Theory
The math behind it via Jensens inequal
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If u is strictly concave and 0 < p < 1,
u(E(x)) > E(u(x)).
Jensens InequalityDerivation for the two-point support ca
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If u is strictly concave and 0 < p < 1,
u(E(x)) > E(u(x)).
Jensens InequalityDerivation for the two-point support ca
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If u is strictly concave and 0 < p E(u(x)).
Jensens InequalityDerivation for the two-point support ca
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x is a random variable with realizationsand x2, x1 < x2.
u is the utility function. u is increasing
strictly concave.p = prob{x = x1}, 1 p = prob{x = x2}
Model Setup
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x is a random variable with realizationsand x2, x1 < x2.
u is the utility function. u is increasing
strictly concave.p = prob{x = x1}, 1 p = prob{x = x2}
Model Setup
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x is a random variable with realizationsand x2, x1 < x2.
u is the utility function. u is increasing
strictly concave.p = prob{x = x1}, 1 p = prob{x = x2}
Model Setup
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E(x) = px1 + (1-p)x2E(u(x)) = pu(x1) + (1-p)u(x2)
Expectation of x and
Expectation of u(x)
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E(x) = px1 + (1-p)x2E(u(x)) = pu(x1) + (1-p)u(x2)
Expectation of x and
Expectation of u(x)
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Graph of Model Setup
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Larger D: Base = x2 - x1 Height = u(x2) u
Smaller D: Base = E(x) - x1
Height = ?
Base and Height
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Larger D: Base = x2 - x1 Height = u(x2) u
Smaller D: Base = E(x) - x1
Height = ?
Base and Height
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Larger D: Base = x2 - x1 Height = u(x2)
Smaller D: Base = E(x) - x1 Height = ?
E(x) - x1 = px1 + (1 p)x2 x1 = (1 p)(
A little bit of algebra
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Larger D: Base = x2 - x1 Height = u(x2)
Smaller D: Base = (1 p)(x2x1).
Smaller D: Height = (1 p)[u(x2) u(x1)].
Conclusion
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u(x1) + (1 p)[u(x2) u(x1)] =
u(x1) - (1 p)u(x1) + (1 p)u(x2) =
pu(x1) + (1 p)u(x2) = E(u(x))
More Algebra
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u(x1) + (1 p)[u(x2) u(x1)] =
u(x1) - (1 p)u(x1) + (1 p)u(x2) =
pu(x1) + (1 p)u(x2) = E(u(x))
More Algebra
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u(x1) + (1 p)[u(x2) u(x1)] =
u(x1) - (1 p)u(x1) + (1 p)u(x2) =
pu(x1) + (1 p)u(x2) = E(u(x))
More Algebra
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Justification of Labeling
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Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
Three Functional Forms for u
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Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
Three Functional Forms for u
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Constant Absolute Risk Aversion
Constant Relative Risk Aversion
Quadratic Utility
Three Functional Forms for u
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u(x) = -ke-gx
+ c; k, g, and c constants; k
u(x) = gke-gx > 0, u(x) = -g2ke-gx < 0.
-u(x)/u(x) = g.
Constant Absolute Risk Aversio
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u(x) = -ke-gx
+ c; k, g, and c constants; k
u(x) = gke-gx > 0, u(x) = -g2ke-gx < 0.
-u(x)/u(x) = g.
Constant Absolute Risk Aversio
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u(x) = -ke-gx
+ c; k, g, and c constants; k
u(x) = gke-gx > 0, u(x) = -g2ke-gx < 0.
-u(x)/u(x) = g.
Constant Absolute Risk Aversio
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u(x) = kxa
+ c; k, a, and c constants; k > 0, 0
u(x) = akxa-1 > 0, u(x) = a(a-1)kxa-2 < 0.
-xu(x)/u(x) = 1 - a.
Constant Relative Risk Aversio
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u(x) = kxa
+ c; k, a, and c constants; k > 0, 0
u(x) = akxa-1 > 0, u(x) = a(a-1)kxa-2 < 0.
-xu(x)/u(x) = 1 - a.
Constant Relative Risk Aversio
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u(x) = -ax2
+ bx + c; a, b, and c constants; a
u(x) = -2ax + b > 0 if x < b/2a, u(x) = -2a 0 if x < b/2a, u(x) = -2a