# Arrow-Pratt Theorem - More risk aversion leads to more balanced bundles

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:

1. 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 $$c_1^0; prove that $$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.