ASSIGNMENT:
Instructions: ftis is an optional assignment whose purpose is to prepare you for the midterm. It consists of six problems. I strongly recommend that you attempt to prepare all questions. On Tuesday's Midterm I will pick four questions at random from this assignment which you will need to answer.
1. 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
UM (cM ; cS) = c1-αM cαS;
where cM and cS 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 pM = pS = 1).
(a) Argue that the exponent α can be taken as a measure of the degree of Michele's altruism by providing an interpretation of extreme values of α = 0 and α = 1. What value would make her a perfect altruist (regarding others the same as oneself)?
(b) Solve for Michele's optimal choices and demonstrate how they change with ?.
(c) Suppose that there is an income tax at rate i , i.e. net income now is just (1 - τ) I: Solve for Michele's optimal choices under the income tax rate.
(d) Now suppose that besides the income tax rate τ, there are charitable deductions, so that income spent on charitable deductions is not taxed. Argue that this amounts to changing the price pS from $1 to $(1 -τ). 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?
2. Suppose that a fast-food junkie derives utility from three goods-soft drinks (x), hamburgers (y), and ice cream sundaes (z)- according to the utility function
U (x; y; z) = x0.:5y0.5(1+ z)0.5.
Suppose also that the prices for these goods are given by px = 1; py = 4; and pz = 8 and that this consumer's income is given by I = 8:
(a) Show that, for z = 0, maximization of utility results in the same optimal choices as in the case of a Cobb-Douglas utility function U(x; y) = x0.5y0.5. Show also th at any choice that results in z > 0 (even for a fractional z) reduces utility from this optimum.
(b) How do you explain the fact that z D 0 is optimal here?
(c) How high would this individual's income have to be for any z to be purchased?
3. Consider the utility function
u(x, y) = (x + 2) (y + 3), x ≥ 0, y ≥ 0
with the accompanying budget constraint:
px x + py y ≤ I, px , py , I > 0:
(a) Fix a given utility level U > 0 and find an explicit expression for the indifference curve defined by the utility level U > 0. ften, derive an explicit expression for the marginal rate of substitution between good x and good y.
(b) Draw the indifference curve (for this associated level of utility U ) and carefully label the graph and its elements.
(c) Show that the utility function is strictly increasing (and hence monotone) in x and y.
(d) Now formally state the utility maximization problem and briefly describe its content, in par- ticular what constitutes the choice variables, and what constitutes parameters.
(e) Provide an argument why in the present utility maximization problem, we can restrict atten- tion to the case where the budget constraint holds as an equality. (fte argument should not involve the explicit computation of the optimal choices.)
4. Consider the utility function
U(x, y) = min(2x + y, x + 2y)
(a) Draw the indifference curve for U(x, y) = 20. Shade the area where U(x, y) ≥ 20.
(b) For what values of px/py will the unique optimum be x = 0?
(c) For what values of px/py will the unique optimum be y = 0?
(d) If neither x and y is equal to zero, and the optimum is unique, what must be the value of x/y ?
5. Consider a consumer with a utility function U(x, y) = e(ln(x) + y)1/3.
(a) What properties about utility functions will make th is problem easier to solve?
(b) Which of the non-negativity input demand constraints will bind for small I ?
(c) Derive the Marshallian demand functions and the indirect utility function (using the original utility function).
(d) Derive the expenditure function in terms of the original utils u.
6. ftere are two goods, food and clothing, whose quantities are denoted by x and y and prices px and py respectively. ftere is a consumer whose income is denoted by I and utility by U: His utility function is
U(x, y) = √xy:
(a) Find this consumer Marshallian demand functions. Find the indirect utility function and the expenditure function.
(b) Initially I = 100, px = 1 and py = 1. What quantities does the consumer buy, and what is his resulting utility?
(c) Now the price of food rises to px = 1.21, while income and the price of clothing are as before. What quantities does the consumer buy and what is his resulting utility?
(d) Suppose the increase in the price of food was caused by the government levying a tax of 21% on food. What is the government revenue from this tax? Hint: At the new prices, calculate the optimal consumption bundles (x*, y*). Then calculate (1.21 - 1)x*.
(e) If the government wants to compensate the consumer by giving him some extra income, how much extra income would be needed to restore him to the old utility level. (Hint: Use the expenditure function.) Is the government's revenue from the tax on good itself sufficient to provide this compensation? What is the economic intuition of your answer?
(f) If the government tries to compensate the consumer by giving him enough extra income to enable him to purchase the same quantities as he did at the original income and prices of part (b), how much extra income would the government have to give him ? With this income and the new prices, what quantities will the consumer actually buy? What will be his resulting utility?