9. Lionel, who has a relatively high income $I$, has an altruistic feeling toward Cristiano, who lives in such poverty that he essentially has no income. Suppose Lionel's preferences are represented by the utility function

$$U_1(c_1, c_2) = c_1^\alpha c_2^{1-\alpha},$$

where $c_1$ and $c_2$ are Lionel and Cristiano's consumption levels, appearing as goods in a standard Cobb-Douglas utility function. Assume that Lionel can spend his income either on his own or Cristiano's consumption (through charitable donations) and that $1 buys a unit of consumption for either (thus, the "prices" of consumption are $p_1 = p_2 = 1$).

a. Solve for Lionel's optimal choices and demonstrate how they change with \(\alpha\). (6%)