Michele, who has a relatively high income I, has altruistic feelings toward Sofia, who lives in such poverty that she essentially has no income. Suppose Michele's preferences are represented by the utility function U_M(c_M, c_S) = c^1 - alpha_M c^alpha_S where c_M and c_S are Michele and Sofia's consumption levels, appearing as goods in a standard Cobb-Douglas utility function. Assume that Michele can spend her income either on her own or Sofia's consumption (though charitable donations) and that $1 buys a unit of consumption for either (thus, the "prices" of consumption are p_M = p_S = 1). Argue that the exponent alpha can be taken as a measure of the degree of Michele's altruism by providing an interpretation of extreme values of alpha = 0 and alpha = 1. What value would make her a perfect altruist (regarding others the same as oneself)? Solve for Michele's optimal choices and demonstrate how they change with alpha. Suppose that there is an income tax at rate r, i. e. net income now is just (1 - r)I. Solve for Michele's optimal choices under the income tax rate. Now suppose that besides the income tax rate tau, there are charitable deductions, so that income spent on charitable deductions is not taxed. Argue that this amounts to changing the price p_S from $1 to $(1 - tau). Solve for the optimal choices under both the income tax rate and charitable deductions. Does the charitable deduction have a bigger incentive effect on more or less altruistic people?