Mera Arrow - Pratt

Arrow-Pratt measure - a measure of risk aversion used in economic theory .

Definition

The absolute measure of risk aversion to Arrow - Pratt is defined as follows:

R_{a}=-{\frac {u''(x)}{u'(x)}}=-{\frac {d\ln MU(x)}{dx}}

{\ displaystyle R_ {a} = - {\ frac {u '' (x)} {u '(x)}} = - {\ frac {d \ ln MU (x)} {dx}}}

,

that is, it is equal to the derivative of the logarithm of marginal utility in terms of consumption (with the opposite sign).

The relative measure of risk aversion to Arrow - Pratt is equal to the elasticity of marginal utility in terms of consumption (also with the opposite sign):

R=-{\frac {xu''(x)}{u'(x)}}=-{\frac {d\ln MU(x)}{d\ln x}}

{\ displaystyle R = - {\ frac {xu '' (x)} {u '(x)}} = - {\ frac {d \ ln MU (x)} {d \ ln x}}}

Pratt's Theorem

Pratt's theorem states the equivalence of the following three methods for ranking risk aversion.

The first way - according to Arrow - Pratt - the more, the greater the degree of risk aversion.

The second way - consumer 1 has a greater degree of risk aversion than consumer 2 if there is a strictly increasing strictly concave (convex up) function $G$ ${\ displaystyle G}$ such that $\forall x,u_{1}(x)=G(u(x))$ ${\ displaystyle \ forall x, u_ {1} (x) = G (u (x))}$ where $u_{1}(x),u_{2}(x)$ ${\ displaystyle u_ {1} (x), u_ {2} (x)}$ - the utility functions of the first and second consumers, respectively.

The third way - risk aversion is greater, the greater the so-called risk reward $\pi (x)$ ${\ displaystyle \ pi (x)}$ (for all $x$ ${\ displaystyle x}$ ), defined as a quantity such that $Eu(x)=u(Ex-\pi (x))$ ${\ displaystyle Eu (x) = u (Ex- \ pi (x))}$ , i.e. $Ex-\pi (x)$ ${\ displaystyle Ex- \ pi (x)}$ is the risk-free equivalent $x$ ${\ displaystyle x}$ .

The theorem assumes twice continuous differentiability of utility functions with standard conditions of positivity of the first derivative (marginal utility) and nonpositivity of the second (nonincreasing marginal utility, that is, concavity or convexity of utility functions upward).

It can be shown that the required risk reward in the first approximation is expressed through the Arrow-Pratt measure $r(x)$ ${\ displaystyle r (x)}$ in the following way $\pi (x)=r(x)\sigma ^{2}/2$ ${\ displaystyle \ pi (x) = r (x) \ sigma ^ {2} / 2}$ where $\sigma ^{2}$ ${\ displaystyle \ sigma ^ {2}}$ - variance of the lottery.

Utility Functions by Arrow Permanent Measures - Pratt

For a function with a constant absolute measure of risk aversion, Arrow - Pratt, the general form of the utility function is as follows:

u(x)=c_{1}-c_{2}e^{-\alpha x}

{\ displaystyle u (x) = c_ {1} -c_ {2} e ^ {- \ alpha x}}

.

Parameter $c_{1}$ ${\ displaystyle c_ {1}}$ here, it actually determines the maximum utility achieved asymptotically with growth $x$ ${\ displaystyle x}$ .

For a function with a constant relative measure of risk aversion, Arrow - Pratt, the general form of the utility function is as follows:

u(x)=c_{1}+c_{2}{\frac {x^{1-\theta }}{1-\theta }},\theta <>1

{\ displaystyle u (x) = c_ {1} + c_ {2} {\ frac {x ^ {1- \ theta}} {1- \ theta}}, \ theta <> 1}

.

In the particular (special) case of unit elasticity ( $\theta =1$ ${\ displaystyle \ theta = 1}$ ) the utility function has the form:

u(x)=c_{1}+c_{2}\ln x

{\ displaystyle u (x) = c_ {1} + c_ {2} \ ln x}

.

Mera Arrow - Pratt

Definition

Pratt's Theorem

Utility Functions by Arrow Permanent Measures - Pratt

Literature

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