3. Decisions under Uncertainty

Chapter theme: decisions under uncertainty. Enlarge the commodity space to include states (\(m_1\) states × \(m_2\) physical goods, \(m=m_1 m_2\)) and impose expected-utility preferences. §3.1 Risk aversion: expected utility \(u^i(\mathbf x)=\sum_s v^i(\cdot)\pi^i_s\) (3.1); Definition 3.1 risk aversion (\(v^i\) concave); Proposition 3.1 concave subutility \(v^i\) \(\Rightarrow\) concave \(u^i\) (Jensen). §3.2 More on risk aversion: Arrow-Pratt absolute risk aversion \(ra(x)=-u''/u'\) (Def 3.2), relative \(rra=ra\cdot x\) (Def 3.3); linear/log/CRRA/CARA examples; absolute premium \(p\) (Def 3.4) small-risk approximation \(p\approx\frac12(-u''/u')\sigma^2\) (Prop 3.2), relative premium \(\rho\) (Def 3.5, Prop 3.3); certainty equivalence \(c_e\) (Def 3.6); large-risk premia (quadratic utility, CARA+normal via Lemma 3.1 \(\mathbb E[e^{aX}]=e^{a\mu+a^2\sigma^2/2}\), CRRA+log-normal); Theorem 3.1 Arrow-Pratt three equivalent statements. §3.3 Equilibrium risk sharing: under common beliefs, Theorem 3.2 the Pareto allocation \(x^i_s=g^i(\bar e_s)\) depends only on the aggregate endowment. §3.4 Arrow-Debreu state price: two periods, state price \(p_s\); household/planner FOCs; \(p_s\) increasing in \(\pi_s\), decreasing in \(\bar e_s\) (3.10). §3.5 Security markets: \(K\) securities, payoff matrix \(D\), two budget constraints (3.11–3.14). §3.6 A-D vs security-market economy: price consistency \(\mathbf q=D\mathbf p\) (Def 3.7), endowment equivalence (Def 3.8); Propositions 3.4/3.5 the correspondence of feasibility and market clearing (\(D\) full rank = complete markets). §3.7 Asset prices and equity premium: risk-free bond + aggregate stock, \(RP_k=\frac{\mathbb E[\partial v^i]\mathbb E[d_k]}{\mathbb E[\partial v^i]\mathbb E[d_k]+\text{Cov}(\partial v^i,d_k)}\), positive covariance \(\Rightarrow r_k

Definition 3.1 (Risk averse) Agent \(i\) is risk averse if his utility function \(u^i\) satisfies $$v^i\left(\sum_{s=1}^m x_s\pi^i_s\right)>\sum_{s=1}^m v^i(x_s)\pi^i_s\tag{3.2}$$ for any random variable \(\mathbf x\).

Proposition 3.1 If subutility function \(v^i\) is concave, then utility function \(u^i\) is concave.

Proof Given that \(v^i\) is concave, we have $$v^i(\alpha x_s+(1-\alpha)y_s)\ge(1-\alpha)v^i(x_s)+\alpha v^i(y_s)$$ for any \(\alpha\in[0,1]\) and \(x_s,y_s\in\mathbb R\). Then consider \(u^i\), by definition, for \(\mathbf x=(x_1,\dots,x_m)\) and \(\mathbf y=(y_1,\dots,y_m)\), we have $$\begin{aligned}u^i(\alpha\mathbf x+(1-\alpha)\mathbf y)&=\sum_{s=1}^m v^i(\alpha x_s+(1-\alpha)y_s)\pi^i_s\\&\ge\sum_{s=1}^m\left[(1-\alpha)v^i(x_s)+\alpha v^i(y_s)\right]\pi^i_s\\&\ge(1-\alpha)\sum_{s=1}^m v^i(x_s)\pi^i_s+\alpha\sum_{s=1}^m v^i(y_s)\pi^i_s\\&=(1-\alpha)u^i(\mathbf x)+\alpha u^i(\mathbf y)\end{aligned}$$ which implies that \(u^i\) is concave. \(\blacksquare\)

Remark 3.1 Recall that Jensen's inequality states that if a function \(f:\mathbb R\to\mathbb R\) is strictly concave, then for a random variable \(x\in\mathbb R\) we have that $$f(\mathbb E[x])>\mathbb E[f(x)]$$ So the definition of risk averse simply means that \(v^i\) and \(u^i\) are concave.

Definition 3.2 (Arrow-Pratt Coefficient of Absolute Risk Aversion) Absolute risk aversion coefficient measures the curvature of the utility function around each point of wealth, i.e. $$ra(x)=-\frac{u''(x)}{u'(x)}$$ Higher the coefficient implies greater curvature and more risk aversion.

Definition 3.3 (Coefficient of Relative Risk Aversion) Relative risk aversion coefficient is defined as $$rra(x)=ra(x)\cdot x=-\frac{u''(x)}{u'(x)}x$$

Example 3.1 (Specific utility functions' \(ra(x)\) and \(rra(x)\)) - Linear utility \(u(x)=ax+b\): \(ra(x)=-\frac{0}{a}=0\), \(rra(x)=0\). - Log utility \(u(x)=\ln x\): \(ra(x)=-\frac{-1/x^2}{1/x}=\frac1x\), \(rra(x)=1\). - Constant Relative Risk Aversion (CRRA) utility \(u(x)=\frac{x^{1-\gamma}-1}{1-\gamma}\): \(ra(x)=-\frac{-\gamma x^{-\gamma-1}}{x^{-\gamma}}=\frac\gamma x\), \(rra(x)=\gamma\). Note that \(u(x)=\frac{x^{1-\gamma}-1}{1-\gamma}\) is a generalization of \(u(x)=\ln x\): $$\lim_{\gamma\to1}\frac{x^{1-\gamma}-1}{1-\gamma}\overset{\text{L'Hopital}}{=}\lim_{\gamma\to1}\frac{-x^{1-\gamma}\cdot\ln x}{-1}=\ln x$$ - Constant Absolute Risk Aversion (CARA) utility \(u(x)=-\frac1a e^{-ax}\): \(ra(x)=-\frac{-ae^{-ax}}{e^{-ax}}=a\), \(rra(x)=ax\).

Definition 3.4 (Absolute insurance premium) Absolute insurance premium \(p\) is defined as $$u(\mathbb E[\tilde x]-p)=\mathbb E[u(\tilde x)]\tag{3.3}$$

Proposition 3.2 Suppose that the risk is small. Then, the absolute risk premium \(p\) is approximated by $$p=\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\tag{3.4}$$ where \(\bar x=\mathbb E[\tilde x]\) and \(\sigma^2\) is the variance of \(\tilde x\).

Proof Since we assume the risk is very small, \(p\) and \(\tilde x-\bar x\) are both very small. So we can do the following Taylor's expansions: $$u(\bar x-p)\approx u(\bar x)+u'(\bar x)(\bar x-p-\bar x)=u(\bar x)-pu'(\bar x)$$ and $$\begin{aligned}u(\tilde x)&\approx u(\bar x)+u'(\bar x)(\tilde x-\bar x)+\frac{u''(\bar x)}{2}(\tilde x-\bar x)^2\\\Rightarrow\mathbb E[u(\tilde x)]&\approx u(\bar x)+u'(\bar x)\mathbb E[\tilde x-\bar x]+\frac{u''(\bar x)}{2}\mathbb E[(\tilde x-\bar x)^2]\\\Rightarrow\mathbb E[u(\tilde x)]&\approx u(\bar x)+\frac{u''(\bar x)}{2}\sigma^2\end{aligned}$$ Combine these results: $$\begin{aligned}u(\bar x)-pu'(\bar x)&\approx\mathbb E[u(\tilde x)]\approx u(\bar x)+\frac{u''(\bar x)}{2}\sigma^2\\\Rightarrow-pu'(\bar x)&\approx\frac{u''(\bar x)}{2}\sigma^2\\\Rightarrow p&\approx\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\end{aligned}\tag{3.5}$$ \(\blacksquare\)

Definition 3.5 (Relative insurance premium) Relative insurance premium is defined as $$u((1-\rho)\bar x)=\mathbb E[u(\bar x(1+\varepsilon))]\tag{3.6}$$ such that \(\mathbb E[\varepsilon]=0\) and \(\mathbb E[\varepsilon^2]=\sigma^2_\varepsilon\).

Proposition 3.3 Suppose that the risk is small. Then, the relative insurance premium \(\rho\) is approximated by $$\rho=\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\bar x\right)\sigma^2_\varepsilon\tag{3.7}$$

Proof The relationship between \(p\) and \(\rho\) is \(\bar x-p=(1-\rho)\bar x\), which implies that \(p=\bar x\rho\). Then, consider \(\tilde x\) and \(\varepsilon\): \(\text{Var}(\tilde x)=\text{Var}(\bar x(1+\varepsilon))\), or equivalently \(\sigma^2=\bar x^2\sigma^2_\varepsilon\). We can thus use the result \(p\approx\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\) obtained previously to derive the approximation of \(\rho\): $$\bar x\rho\approx\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\right)\bar x^2\sigma^2_\varepsilon\Rightarrow\rho\approx\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\bar x\right)\sigma^2_\varepsilon$$ \(\blacksquare\)

Definition 3.6 (Certainty equivalence) The certainty equivalence of risk \(\tilde x\) is denoted by \(c_e(\tilde x)\) such that \(u(c_e(\tilde x))=\mathbb E[u(\tilde x)]\). Therefore, \(c_e(\tilde x)=\bar x-p=\bar x(1-\rho)\).

Example 3.2 (Quadratic utility) Assume that the wealth \(x\) is a random variable satisfying the moment conditions \(\mathbb E[\tilde x]=\mu\) and \(\text{Var}(\tilde x)=\sigma^2\). Also assume that the utility function is \(u(x)=x-\frac\alpha2 x^2\). By the definition of absolute insurance premium \(p\): $$\begin{aligned}&u(\mu-p)=\mathbb E[u(\tilde x)]\\\Rightarrow{}&(\mu-p)-\frac\alpha2(\mu-p)^2=\mathbb E\left[\tilde x-\frac\alpha2\tilde x^2\right]\\\Rightarrow{}&(\mu-p)-\frac\alpha2(\mu-p)^2=\mu-\frac\alpha2\left(\text{Var}(\tilde x)+\mathbb E[\tilde x]^2\right)\\\Rightarrow{}&(\mu-p)-\frac\alpha2(\mu-p)^2=\mu-\frac\alpha2(\sigma^2+\mu^2)\\\Rightarrow{}&-p-\frac\alpha2(\mu^2-2\mu p+p^2)=-\frac\alpha2(\sigma^2+\mu^2)\\\Rightarrow{}&-\frac\alpha2 p^2-(1-\alpha\mu)p+\frac\alpha2\sigma^2=0\end{aligned}$$ We can solve for the roots: $$p=\frac{(1-\alpha\mu)\pm\sqrt{(1-\alpha\mu)^2+\alpha^2\sigma^2}}{-\alpha}$$ in which the positive root is the correct solution for \(p\), i.e. $$p=\frac{(1-\alpha\mu)-\sqrt{(1-\alpha\mu)^2+\alpha^2\sigma^2}}{-\alpha}$$

Lemma 3.1 If \(X\sim N(\mu,\sigma^2)\), then \(\mathbb E[e^{aX}]=e^{a\mu+\frac{a^2\sigma^2}{2}}\).

Proof $$\begin{aligned}\mathbb E[e^{aX}]&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2}\cdot e^{ax}\,dx\\&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{x-\mu}{\sigma}\right)^2+ax}\,dx\\&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{x^2+\mu^2-2\mu x}{\sigma^2}\right)+ax}\,dx\\&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{x^2+\mu^2-2\mu x-2a\sigma^2 x}{\sigma^2}\right)}\,dx\\&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{\left(x-\mu-a\sigma^2\right)^2-a^2\sigma^4-2\mu a\sigma^2}{\sigma^2}\right)}\,dx\\&=\int_{\mathbb R}\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac12\left(\frac{(x-\mu-a\sigma^2)^2}{\sigma^2}\right)+\frac{a^2\sigma^2}{2}+\mu a}\,dx\\&=e^{a\mu+\frac{a^2\sigma^2}{2}}\end{aligned}$$ where the last step uses the fact that the normalizing integral equals 1. \(\blacksquare\)

Example 3.3 (CARA utility and risk with normal distribution) Assume that we have \(\tilde x\sim N(\mu,\sigma^2)\) and the utility function is $$u(x)=-\frac1\lambda\exp(-\lambda x),\ \lambda>0$$ Then, example 3.1 tells us that \(ra(x)=-\frac{u''(x)}{u'(x)}=\lambda\) for \(\forall x\). By the definition of absolute insurance premium \(p\): $$\begin{aligned}&u(\mu-p)=\mathbb E[u(\tilde x)]\\\Rightarrow{}&-\frac1\lambda\exp(-\lambda(\mu-p))=-\frac1\lambda\mathbb E[\exp(-\lambda\tilde x)]\\\Rightarrow{}&\exp(-\lambda(\mu-p))=\mathbb E[\exp(-\lambda\tilde x)]=\exp\left(-\mu\lambda+\frac12\lambda^2\sigma^2\right)\end{aligned}$$ where the last line is true because of \(\tilde x\sim N(\mu,\sigma^2)\) and lemma 3.1. Then, $$-\lambda(\mu-p)=-\mu\lambda+\frac{\sigma^2\lambda^2}{2}\Rightarrow-(\mu-p)=-\mu+\frac{\sigma^2\lambda}{2}\Rightarrow p=\frac12\lambda\sigma^2$$

Example 3.4 (CRRA utility and log-normal risk) Assume that the state follows \(\log\tilde x\sim N(\mu,\sigma^2)\). Also assume that the utility function is $$u(x)=\frac{x^{1-\gamma}}{1-\gamma}$$ for \(\gamma>0\). Example 3.1 informs us that \(rra(x)=-\frac{u''(x)}{u'(x)}x=\gamma\). By the definition of relative insurance premium \(\rho\): $$\begin{aligned}&u((1-\rho)\bar x)=\mathbb E[u(\tilde x)]\\\Rightarrow{}&\frac{((1-\rho)\bar x)^{1-\gamma}}{1-\gamma}=\mathbb E\left[\frac{\tilde x^{1-\gamma}}{1-\gamma}\right]\\\Rightarrow{}&((1-\rho)\bar x)^{1-\gamma}=\mathbb E[\tilde x^{1-\gamma}]\end{aligned}$$ In order to use lemma 3.1, introduce the log of \(\tilde x\): \(x=\log\tilde x\), where \(x\sim N(\mu,\sigma^2)\). Then, $$\begin{aligned}\mathbb E[\tilde x]&=\mathbb E[e^x]=e^{\mu+\frac{\sigma^2}{2}}\\\Rightarrow\mathbb E[\tilde x^{1-\gamma}]&=\mathbb E[e^{(1-\gamma)x}]=e^{(1-\gamma)\mu+\frac{(1-\gamma)^2\sigma^2}{2}}\\\Rightarrow((1-\rho)\bar x)^{1-\gamma}&=e^{(1-\gamma)\mu+\frac{(1-\gamma)^2\sigma^2}{2}}\\\Rightarrow(1-\rho)\bar x&=e^{\mu+\frac{(1-\gamma)\sigma^2}{2}}\\\Rightarrow(1-\rho)e^{\mu+\frac{\sigma^2}{2}}&=e^{\mu+\frac{(1-\gamma)\sigma^2}{2}}\\\Rightarrow1-\rho&=e^{\frac{-\gamma\sigma^2}{2}}\\\Rightarrow\rho&=1-e^{\frac{-\gamma\sigma^2}{2}}\end{aligned}$$

Theorem 3.1 (Arrow-Pratt) Suppose that \(u\) and \(v\) are both utility functions. Then, we have the following statements that are equivalent. 1. \(u\) is an increasing and concave transformation of \(v\), i.e. there exists a function \(f\) such that \(f'(\cdot)>0\), \(f''(\cdot)<0\) and \(u(x)=f(v(x))\). 2. The insurance premium of \(u\) is higher than the insurance premium of \(v\), i.e. for all random variables \(\tilde x\), i.e. \(p_u(\tilde x)>p_v(\tilde x)\). 3. The absolute risk aversion coefficient of \(u\) is higher than the absolute risk aversion coefficient of \(v\) at any point, i.e. \(-\frac{u''(x)}{u'(x)}>-\frac{v''(x)}{v'(x)}\).

Proof In order to show the equivalence of statement 1, 2 and 3, we will show \(1\Rightarrow2\), \(2\Rightarrow3\), and \(3\Rightarrow1\).

First, let's show \(1\Rightarrow2\). Since \(f\) is strictly concave and increasing, by Jensen's Inequality, $$\mathbb E[u(\tilde x)]=\mathbb E[f(v(\tilde x))]p_v(\tilde x)\).

Then, let's show \(2\Rightarrow3\). Since there is no simple characterization of absolute risk premium of large risk, let's consider the small risk case. From the result in (3.5), we can obtain \(p_u(\tilde x)\approx\frac12\left(-\frac{u''(\bar x)}{u'(\bar x)}\right)\sigma^2\) and \(p_v(\tilde x)\approx\frac12\left(-\frac{v''(\bar x)}{v'(\bar x)}\right)\sigma^2\). So \(p_u(\tilde x)>p_v(\tilde x)\) implies \(-\frac{u''(\bar x)}{u'(\bar x)}>-\frac{v''(\bar x)}{v'(\bar x)}\).

Finally, let's show \(3\Rightarrow1\). Define \(f(w)=u(v^{-1}(w))\), and let \(w=v(x)\), then $$f(v(x))=u(v^{-1}(v(x)))=u(x)$$ Now we only need to show that \(f\) is strictly increasing and strictly concave. Differentiate \(f\) with respect to \(w\): $$f'(w)=u'v^{-1\prime}(w)>0$$ which is true because \(u',v'>0\). Differentiate again: $$f''(w)=\underbrace{u''(v^{-1}(w))}_{<0}\left(v^{-1\prime}(w)\right)^2-\underbrace{u'(v^{-1}(w))}_{>0}\underbrace{v^{-1\prime\prime}(w)}_{>0}<0$$ So we have shown that \(f\) is strictly increasing and strictly concave. \(\blacksquare\)

Theorem 3.2 For any arbitrary vector of weights \(\boldsymbol\lambda\), the corresponding Pareto optimal allocation can be characterized by a set of strictly increasing functions \(g^i\) of the aggregate endowment, which means that the optimal allocation can be represented by \(x^i_s=g^i(\bar e_s)\) for \(\forall i\in\mathbf I\).

Remark 3.2 Note that the functional form \(g^i\) is the same across all states. So we can use this function to only consider the change in the state variable instead of considering each state separately.

Proof Pick and fix a weight vector \(\boldsymbol\lambda\) for the social planner's problem (also the Pareto optimal allocation problem with weight \(\boldsymbol\lambda\)). By the definition of the agent \(i\)'s utility function \(u^i\) and subutility function \(v^i\), we can rewrite the objective function of the social planner's problem: $$\begin{aligned}\sum_{i\in\mathbf I}\lambda_i u^i(\mathbf x^i)&=\sum_{i\in\mathbf I}\lambda_i\left(\sum_{s=1}^m v^i(x^i_s)\pi_s\right)\\&=\sum_{s=1}^m\left(\sum_{i\in\mathbf I}\lambda_i v^i(x^i_s)\right)\pi_s\end{aligned}$$ Based on this objective function, the social planner's maximization problem becomes $$\max_{\{x^i_s\}_{s\in\{1,\dots,m\},i\in\mathbf I}}\sum_{s=1}^m\left(\sum_{i\in\mathbf I}\lambda_i v^i(x^i_s)\right)\pi_s\quad\text{s.t.}\quad\sum_{i\in\mathbf I}x^i_s=\bar e_s\ \text{for }\forall s=1,\dots,m$$ The problem above can be solved by solving the following sub-problem for each state \(s\in\{1,\dots,m\}\): $$\max_{\{x^i_s\}_{i\in\mathbf I}}\sum_{i\in\mathbf I}\lambda_i v^i(x^i_s)\quad\text{s.t.}\quad\sum_{i\in\mathbf I}x^i_s=\bar e_s$$ where we don't have the probability \(\pi_s\) entering the sub-problem because of our assumption of common beliefs. For each sub-problem, the only difference is the level of the aggregate endowment \(\bar e_s\), which makes the solution to any of the sub-problem takes the form \(x^i_s=g^i(\bar e_s)\).

Now it remains to show that \(g^i\) are strictly increasing functions. For any two states \(s\) and \(s'\) with \(\bar e_s>\bar e_{s'}\), at least for some agent \(i\) the allocation is strictly larger in \(s\), i.e. \(x^i_s>x^i_{s'}\), to make the feasibility constraint bind. So we have shown that for this agent, \(g^i\) is increasing in \(\bar e_s\). Now let's prove \(g^j\) for any other agent \(j\) is also increasing. Since we assumed that \(v^i\)'s are all differentiable and concave, the following f.o.c. is necessary and sufficient for the sub-problem to be solved in state \(s\) and \(s'\): $$\lambda_i\frac{\partial v^i(x^i_s)}{\partial x^i_s}=\lambda_j\frac{\partial v^j(x^j_s)}{\partial x^j_s}\tag{3.8}$$ $$\lambda_i\frac{\partial v^i(x^i_{s'})}{\partial x^i_{s'}}=\lambda_j\frac{\partial v^j(x^j_{s'})}{\partial x^j_{s'}}\tag{3.9}$$ Note that \(x^i_s>x^i_{s'}\) and \(v^i\) is strictly concave, so we have $$\lambda_i\frac{\partial v^i(x^i_s)}{\partial x^i_s}<\lambda_i\frac{\partial v^i(x^i_{s'})}{\partial x^i_{s'}}$$ then we can conclude from equation (3.8) and (3.9) that $$\lambda_j\frac{\partial v^j(x^j_s)}{\partial x^j_s}<\lambda_j\frac{\partial v^j(x^j_{s'})}{\partial x^j_{s'}}$$ Since \(v^j\) is also concave, we have that \(x^j_s>x^j_{s'}\). Therefore, \(g^j\) for any other agent \(j\) is also strictly increasing. \(\blacksquare\)

Remark 3.3 This result shows risk sharing because for a given set of weights, all the agents in the economy have their consumption determined by the aggregate endowment level rather than their own endowment level. If the aggregate endowment goes up, then everyone will benefit from it. If the aggregate endowment goes down, then everyone loses.

Remark 3.4 Is it possible for agent \(i\) to have his own realized endowment \(e^i_s\) go down but still consume more as aggregate endowment \(\bar e_s\) goes up? Yes, this is exactly what we have proved to be risk sharing. It is because we assumed the weight vector \(\boldsymbol\lambda\) to be fixed, which equivalently assumed that the distribution rule (not the shares themselves, but the rule that agent \(i\) with higher \(\lambda_i\) ends up consuming more) of the total endowment is fixed. So even agent \(i\) with higher \(\lambda_i\) have a very low realized endowment \(e^i_s\), he will still be compensated in some way to make him wealthy enough to buy from others and thus have a higher share of the realized aggregate endowment \(\bar e_s\). In a word, in order to have the risk sharing, we need to fix the weight vector (the importance of each agent in the social planner's eyes), or equivalently, we need to fix the distribution rule of the aggregate endowment among all agents.

Remark 3.5 This result is very intuitive. The higher the probability for a state, the more likely that the A-D security for that state will realize its payoff and thus the higher its price will be. And the larger \(\bar e_s\) means state \(s\) is more abundant in resources, so the price should be lower for that state.

Definition 3.7 (Price consistency) The prices \(\mathbf q\) and payoffs \(D\) are consistent with state prices \(\mathbf p\) if $$q_k=\sum_{s=1}^m p_s d_{ks},\ \text{for }\forall k=1,2,\dots,K$$ or equivalently \(\mathbf q=D\mathbf p\) (3.15).

Definition 3.8 (Endowment equivalence) The endowment \(\mathbf e^i\) and \((\hat{\mathbf e}^i,\boldsymbol\theta^i)\) are equivalent if $$\hat e^i_s+\sum_{k=1}^K d_{ks}\theta^i_k=e^i_s,\ \text{for }\forall s=1,2,\dots,m$$ or equivalently \(\hat{\mathbf e}^i+D'\boldsymbol\theta^i=\mathbf e^i\) (3.16).

Proposition 3.4 Assume that prices \(\mathbf q\) and payoffs \(D\) are consistent with state prices \(\mathbf p\) and that the endowments are equivalent, then we have 1. If \((\mathbf x^i,\mathbf h^i)\) is budget feasible in the security market economy, then \(\mathbf x^i\) is also budget feasible in the A-D economy, i.e. $$\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i\text{ and }\mathbf q'\mathbf h^i=\mathbf q'\boldsymbol\theta^i\Rightarrow\mathbf p'\mathbf x^i=\mathbf p'\mathbf e^i$$ 2. If \(\mathbf x^i\) is budget feasible in the A-D economy, then it must be budget feasible in the security market economy when \(D\) has full rank, i.e. $$\mathbf p'\mathbf x^i=\mathbf p'\mathbf e^i\text{ and }D\text{ has full rank}\Rightarrow\exists\mathbf h^i:\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i\text{ and }\mathbf q'\mathbf h^i=\mathbf q'\boldsymbol\theta^i$$ (If the payoff matrix \(D\) does not have full rank, the economy has incomplete markets. If \(D\) has full rank, the economy has complete markets.)

Remark 3.6 Price consistency means that there is no arbitrage between the A-D security market and the security market in the second economy. So we can use securities in either market to construct the securities in the other market with same costs and returns.

Remark 3.7 The full rank requirement in 2 of proposition 3.4 is actually requiring that we can use securities in the second economy's security market to construct any state security in A-D economy. This is a very natural requirement because state securities can be used to construct any payoff patterns but the reverse is not necessarily true, so we need to impose the full rank condition of \(D\) under which each state security in A-D economy can be constructed by the securities in the second economy.

Proof (Proposition 3.4, statement 1) Assume that \((\mathbf x^i,\mathbf h^i)\) is budget feasible in the security market economy, then $$\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i,\qquad\mathbf q'\mathbf h^i=\mathbf q'\boldsymbol\theta^i\tag{3.17}$$ Multiply through (3.17) by \(\mathbf p'\): \(\mathbf p'\mathbf x^i=\mathbf p'D'\mathbf h^i+\mathbf p'\hat{\mathbf e}^i\). Since the two endowments are equivalent, \(\hat{\mathbf e}^i\) can be replaced by \(\mathbf e^i-D'\boldsymbol\theta^i\): $$\begin{aligned}\mathbf p'\mathbf x^i&=\mathbf p'D'\mathbf h^i+\mathbf p'(\mathbf e^i-D'\boldsymbol\theta^i)\\&=\mathbf p'D'(\mathbf h^i-\boldsymbol\theta^i)+\mathbf p'\mathbf e^i\end{aligned}$$ Since the prices are equivalent, \(\mathbf q=D\mathbf p\Rightarrow\mathbf q'=\mathbf p'D'\), therefore $$\mathbf p'\mathbf x^i=\underbrace{\mathbf q'(\mathbf h^i-\boldsymbol\theta^i)}_{=0}+\mathbf p'\mathbf e^i=\mathbf p'\mathbf e^i$$

Proof (Proposition 3.4, statement 2) Since \(D\) has full rank, \(D\) is invertible, and \(D'\) is also invertible. Starting from the feasible conditions for the A-D economy \(\mathbf p'\mathbf x^i=\mathbf p'\mathbf e^i\). Since the prices are equivalent, \(\mathbf q=D\mathbf p\) implies \(\mathbf p=D^{-1}\mathbf q\), therefore $$\mathbf q'(D^{-1})'\mathbf x^i=\mathbf q'(D^{-1})'\mathbf e^i$$ Since the two endowments are equivalent, \(\mathbf e^i\) can be replaced by \(\hat{\mathbf e}^i+D'\boldsymbol\theta^i\): $$\begin{aligned}\mathbf q'(D^{-1})'\mathbf x^i&=\mathbf q'(D^{-1})'(\hat{\mathbf e}^i+D'\boldsymbol\theta^i)\\&=\mathbf q'(D^{-1})'\hat{\mathbf e}^i+\mathbf q'\boldsymbol\theta^i\end{aligned}$$ Then we want to show that if consumption in the security market economy is budget feasible, then the security transactions are also budget feasible, i.e. use \(\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i\) and the condition derived above to prove that \(\mathbf q'\mathbf h^i=\mathbf q'\boldsymbol\theta^i\). Rewrite \(\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i\) as \(\mathbf h^i=(D')^{-1}\mathbf x^i-(D')^{-1}\hat{\mathbf e}^i\). Then, $$\mathbf q'\mathbf h^i=\mathbf q'(D')^{-1}\mathbf x^i-\mathbf q'(D')^{-1}\hat{\mathbf e}^i=\mathbf q'\boldsymbol\theta^i$$ \(\blacksquare\)

Proposition 3.5 Suppose that the endowments \(\mathbf e^i\) and \((\hat{\mathbf e}^i,\boldsymbol\theta^i)\) are equivalent. Then, 1. If \((\mathbf x^i,\mathbf h^i)\) clears the security markets in the security market economy, then \(\mathbf x^i\) clears the markets in the A-D economy. 2. Suppose \(D\) has full rank. If \(\mathbf x^i\) clears the market in the A-D economy, then \((\mathbf x^i,\mathbf h^i)\) clears the security market in the security market economy.

Proof (Proposition 3.5) First, statement 1. Suppose \((\mathbf x^i,\mathbf h^i)\) clears the market, then \(\sum_{i\in\mathbf I}\mathbf x^i=\sum_{i\in\mathbf I}(\hat{\mathbf e}^i+D'\boldsymbol\theta^i)\). Since the two economies have exactly equivalent endowments, $$\sum_{i\in\mathbf I}\mathbf x^i=\sum_{i\in\mathbf I}(\hat{\mathbf e}^i+D'\boldsymbol\theta^i)=\sum_{i\in\mathbf I}\mathbf e^i$$ So \(\mathbf x^i\) clears the market in the A-D economy.

Then, statement 2. Assume the goods market in the security market economy clears to show that the security market also clears. Since the goods market in the security market economy clears, \(\mathbf x^i=D'\mathbf h^i+\hat{\mathbf e}^i\). Use the fact that \(D\) is invertible to rewrite: $$\begin{aligned}\mathbf h^i&=(D')^{-1}(\mathbf x^i-\hat{\mathbf e}^i)\\&=(D')^{-1}(\mathbf x^i-(\mathbf e^i-D'\boldsymbol\theta^i))\\&=(D')^{-1}(\mathbf x^i-\mathbf e^i)+\boldsymbol\theta^i\end{aligned}\tag{3.18}$$ which is true because endowments are equivalent and \(\hat{\mathbf e}^i\) can be replaced by \(\mathbf e^i-D'\boldsymbol\theta^i\). Sum (3.18) across all agents: $$\sum_{i\in\mathbf I}\mathbf h^i=(D')^{-1}\sum_{i\in\mathbf I}(\mathbf x^i-\mathbf e^i)+\sum_{i\in\mathbf I}\boldsymbol\theta^i$$ Market clearing condition of \(\mathbf x^i\) in the A-D economy implies that \(\sum_{i\in\mathbf I}(\mathbf x^i-\mathbf e^i)=0\), i.e. $$\sum_{i\in\mathbf I}\mathbf h^i=\sum_{i\in\mathbf I}\boldsymbol\theta^i$$ \(\blacksquare\)

Intuition The marginal utility \(\frac{\partial v^i}{\partial x}\) decreases in consumption; when a security pays a lot in states where consumption is high and marginal utility is low (i.e. payoff is negatively correlated with marginal utility, negative covariance), it is a poor hedge and requires a higher expected return as compensation, so \(r_k\ge r_1\) — which is the source of the equity premium.