Topic 5. Uncertainty

1. Lottery

• Choices facing consumers take the form of lotteries: $p\circ x\otimes (1-p)\circ y$$p \circ x \otimes(1-p) \circ y$, or "consumer receives prize $x$$x$ with probability $p$$p$ and prize $y$$y$ with probability $(1-p)$$(1-p)$ ".

• Assumptions about consumers' perceptions about lotteries:

  • receiving a prize with probability 1 is the same as getting the prize for certain: $1\circ x\otimes (1-1)\circ y\sim x$$1 \circ x \otimes(1-1) \circ y \sim x$

  • order of lotteries does not matter: $p\circ x\otimes (1-p)\circ y\sim (1-p)\circ y\otimes p\circ x$$p \circ x \otimes(1-p) \circ y \sim(1-p) \circ y \otimes p \circ x$

  • consumer's perception of a lottery depends only on the net probabilities of receiving various prizes $=$$=$ REDUCTION OF COMPOUND LOTTERIES:

• The fact that lotteries have only 2 outcomes is not restrictive because outcomes can be further lotteries. By compounding two-prize lotteries we can construct lotteries with arbitrary number of prizes:

• Preference over lotteries

  Analogous to consumer theory, here we assume that a decision maker has preference $\succsim$$\succsim$ over the set of single lotteries L summerized in the following axioms:

  • Completeness

  • Transitivity

  • Continuity

    For any lottery $l\in L$$l \in L$, there is some probability $\alpha \in [0,1]$$\alpha \in \left[0,1\right]$, such that $\ell \sim (\alpha \circ a_1\otimes (1-\alpha )\circ a_n)$$\ell \sim\left(\alpha \circ a_1 \otimes(1-\alpha) \circ a_n\right)$.

  • Independence

    The preference relation $\succsim$$\succsim$ on the space of simple lotteries $L$$L$ satisfies independence axiom if for all $\ell ,{\ell }^{\prime },{\ell }^{\prime \prime }\in L$$\ell, \ell^{\prime}, \ell^{\prime \prime} \in L$ and $\alpha \in (0,1)$$\alpha \in(0,1)$ we have: $\ell ⪰{\ell }^{\prime }$$\ell \succeq \ell^{\prime}$ iff $\alpha \ell +(1-\alpha ){\ell }^{\prime \prime }⪰\alpha {\ell }^{\prime }+(1-\alpha ){\ell }^{\prime \prime }$$\alpha \ell+(1-\alpha) \ell^{\prime \prime} \succeq \alpha \ell^{\prime}+(1-\alpha) \ell^{\prime \prime}$. This means that if we mix two lotteries with a third one, then the preference ordering of the two resulting mixtures is independent of the particular third lottery used.

    This means if we add a third lotteries as a linear combination of previous two, then preference relation is not affected.

    This assumption may be inrealistic.

2. Expected Utility Theorem

• vNM utility function

  If $(L,⪰)$$(L, \succeq)$ satisfy above axioms, there is a utility function $U:L\to \mathbb{R}$$U: L \rightarrow \mathbb{R}$ that satisfy the expected utility property: $U(p\circ x\otimes (1-p)\circ y)=pU(x)+(1-p)U(y)$$U(p \circ x \otimes(1-p) \circ y)=p U(x)+(1-p) U(y)$. Such a utility function is called von Neumann-Morgenstern (VNM) expected utility function.

   

• Expected utility functions are unique up to affine transformations. If $U(\cdot )$$U(\cdot)$ is expected utility function, so is $V(\cdot )=aU(\cdot )+c$$V(\cdot)=a U(\cdot)+c$, where $a>0$$a>0$.

  $$\begin{array}{}\text{(3)}& \begin{array}{rl}V[p\circ x\otimes (1-p)\circ y]& =aU[p\circ x\otimes (1-p)\circ y]+c\\ & =a[pU(x)+(1-p)U(y)]+c\\ & =p[aU(x)+c]+(1-p)[aU(y)+c]\\ & =pV(x)+(1-p)V(y).\end{array}\end{array}$$

  仿射变换（Affine transformation），又称仿射映射，是指在几何中，对一个向量空间进行一次线性变换并接上一个平移，变换为另一个向量空间。

3. Risk aversion

We say that an agent is risk averse if she prefers to receive the expected value of gamble for sure instead of receiving the risky gamble itself; i.e., the utility of the lottery is less than the utility of the expected value of that lottery.

• Certainty equivalent

  The amont of money $c(F,U)$$c(F,U)$ for which the individual is indifferent between the lottery and the certain amont.

  $$\begin{array}{}\text{(4)}& U[c(F,U)]=\int U(x)dF(x)\end{array}$$

• Risk premium ($\pi$$\pi$)

  The maximum amount of money that an agent is willing to pay to receive the expected value of the lottery instead of receiving the lottery itself.

  $$\begin{array}{}\text{(5)}& U(\int xdF(x)-\pi )=U(F)\equiv \int U(x)dF(x)\end{array}$$
  $$\begin{array}{}\text{(6)}& \begin{array}{rl}& \pi =\int xdF(x)-c(F,U)\\ & \pi =E(x)-C(F,U)\end{array}\end{array}$$

• Some equivalence

  • The decision maker is risk averse.

  • The utility function is concave $U^{{}^{″}}()\le 0$$U^{''}()\leq 0$.

  • $c(F,U)\le \int xdF(x)\mathrm{\forall }F(\cdot )$$c(F, U) \leq \int x d F(x) \quad \forall F(\cdot)$, which is equivalent to Jensen's inequality:

  $$\begin{array}{}\text{(7)}& \begin{array}{rl}c(F,U)\le \int xdF(x)& \iff U[c(F,U)]\le U(\int xdF(x))\\ & \iff \int U(x)dF(x)\le U(\int xdF(x))\end{array}\end{array}$$

  • $\pi \ge 0$$\pi \geq 0$.

• Measuring of risk aversion

• Arrow-Reatt coefficient of absolute risk aversion

  $$\begin{array}{}\text{(8)}& r_A(x)=-\frac{U^{\prime \prime }(x)}{U^{\prime }(x)}\end{array}$$

  The larger this ratio is, the larger risk averse is the agent.

  $R_a(W)$$R_a(W)$ is only a local measure of risk aversion, so it need not be the same at every level of wealth. Indeed, one expects that attitudes toward risk, and so the Arrow-Pratt measure, will ordinarily vary with wealth, and vary in 'sensible' ways. Arrow has proposed a simple classification of VNM utility functions (or utility function segments) according to how $R_a(w)$$R_a(w)$ varies with wealth. Quite straightforwardly, we say that a VNM utility function displays constant, decreasing, or increasing absolute risk aversion over some domain of wealth if, over that interval, $R_a(w)$$R_a(w)$ remains constant, decreases, or increases with an increase in wealth, respectively.

  Decreasing absolute risk aversion (DARA) is generally a sensible restriction to impose. Under constant absolute risk aversion, there would be no greater willingness to accept a small gamble at higher levels of wealth, and under increasing absolute risk aversion, we have rather perverse behavior: the greater the wealth, the more averse one becomes to accepting the same small gamble. DARA imposes the more plausible restriction that the individual be less averse to taking small risks at higher levels of wealth.

• Relative risk aversion