I have an economics question but it more so interested me mathematically; related to Arrow-Pratt theorem - risk aversion; hence I wanted to ask the following.
Given $p \in \mathbb{R^s}_{++}$ $\pi_s \geq 0$, $\sum_{s=1}^{S}\pi_s = 1$, $x \in B(p,w) := \{ y \in \mathbb{R}^S_+ \; | \; p.y = w \}$, for $t = 1,2$, $u: \mathbb{R} \rightarrow \mathbb{R} $ differentiable once, $u_t'(.) > 0$ where $u_1(.) = T(u_0(.))$ for $T(.)$ strictly concave (if not working, can be assumed to be strictly increasing).
Let $x^t$ be the maximizers of the following:
$$ max_{x \in B(p,w)} \sum_{s=1}^S \pi_su_t(x_s)$$
For $S=2, $If $x^0 = (x_1^0, x_2^0)$ is such that $x_1^0 < x_2^0$,
Please help me prove that $x_1^1 \geq x_1^0$. The intuition is that strictly concave transformations moves optimizers to more balanced bundles.
The original question:
- An economy has one physical good and two states of the world. Each consumer has consumption set $X=\mathbb{R}_{+}^2$ and endowment $\omega=\left(\omega_1, \omega_2\right) \gg 0$, where $\omega_s$ is the consumer's endowment of the good in state $s$ for $s=1,2$. The price of the good is in state $s$ is $p_s>0$. Each consumer's budget constraint is $p_1 c_1+p_2 c_2 \leq p_1 \omega_1+p_2 \omega_2=w$. There are two possible consumers, $t=0,1$. Both consumers have subjective-expected-utility preferences with belief $\pi=\left(\pi_1, \pi_2\right) \gg 0$ about the states, and they have the same endowment. They only differ in their von Neumann - Morgenstern (vN-M) utilities. Consumer $t$ 's vNM utility is $u_t$, a continuously differentiable function with $u_t^{\prime}>0$. (Neither is necessarily concave.) In addition, for every $z \geq 0, u_1(z)=T(u(z))$, where $u=u_0$ and $T: \mathbb{R} \rightarrow \mathbb{R}$ is a strictly concave function. In words, consumer 1 is strictly more risk averse than consumer 0 . Let $c^t$ solve consumer $t$ 's problem, and, if necessary, relabel the states so that $c_1^0 \leq c_2^0$. Prove that $c_1^1 \geq c_1^0$ in two steps.
(a) First suppose that $c_1^0=c_2^0$; prove that $c_1^1 \geq c_1^0$.
(b) Next suppose that $c_1^0<c_2^0$; prove that $c_1^1 \geq c_1^0$. (Hint: consider, as an preliminary step, what consumer $t=1$ would chose if it were constrained to choose $c_1 \leq c_1^0$.) You might find this notation useful: $$ \bar{c}=\frac{w}{p_1+p_2} $$ it is the just-affordable consumption in each state when consumption in each state is equal.
