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Stony ECO 303
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ASSIGNMENT: MIDTERM I PREPARATION
Due: Optional, but you should use it to prepare for the midterm.
Instructions: This 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
Tuesdayâs 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 Micheleâs preferences are represented by the
utility function
1 ? ?
UM .cM ; cS / D cM
cS ;
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where cM and cS are Michele and Sofiaâs consumption levels, appearing as goods in a standard
Cobb-Douglas utility function. Assume that Michele can spend her income either on her own
or Sofiaâs consumption (though charitable donations) and that $1 buys a unit of consumption for
either (thus, the âpricesâ of consumption are pM D pS D 1).
(a) Argue that the exponent ? can be taken as a measure of the degree of Micheleâs altruism by
providing an interpretation of extreme values of ? D 0 and ? D 1. What value would make
her a perfect altruist (regarding others the same as oneself)?
(b) Solve for Micheleâs optimal choices and demonstrate how they change with ? .
(c) Suppose that there is an income tax at rate , i.e. net income now is just .1
Micheleâs optimal choices under the income tax rate.
/I: Solve for
(d) Now suppose that besides the income tax rate , 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 /. 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 goodsâsoft drinks .x/, hamburgers .y/,
and ice cream sundaes .z/âaccording to the utility function
U.x; y; z/ D x 0:5 y 0:5 .1 C z/0:5 :
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Suppose also that the prices for these goods are given by px D 1; py D 4; and pz D 8 and that this
consumerâs income is given by I D 8:
(a) Show that, for z D 0, maximization of utility results in the same optimal choices as in the
case of a Cobb-Douglas utility function U.x; y/ D x 0:5 y 0:5 . Show also that 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 individualâs income have to be for any z to be purchased?
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3. Consider the utility function
u.x; y/ D .x C 2/.y C 3/;
x 0; y 0
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with the accompanying budget constraint:
px x C 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. Then, 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 particular what constitutes the choice variables, and what constitutes parameters.
(e) Provide an argument why in the present utility maximization problem, we can restrict attention to the case where the budget constraint holds as an equality. (The argument should not
involve the explicit computation of the optimal choices.)
4. Consider the utility function
U.x; y/ D min.2x C y; x C 2y/
(a) Draw the indifference curve for U.x; y/ D 20. Shade the area where U.x; y/ 20.
(b) For what values of ppyx will the unique optimum be x D 0?
(c) For what values of
px
py
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will the unique optimum be y D 0?
(d) If neither x and y is equal to zero, and the optimum is unique, what must be the value of yx ?
5. Consider a consumer with a utility function U.x; y/ D e .ln.x/Cy/
1=3
.
(a) What properties about utility functions will make th is problem easier to solve?
(b) Which of the non-negativity 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. There are two goods, food and clothing, whose quantities are denoted by x and y and prices px
and py respectively. There is a consumer whose income is denoted by I and utility by U: His utility
function is
p
U.x; y/ D xy:
(a) Find this consumer Marshallian demand functions. Find the indirect utility function and the
expenditure function.
(b) Initially I D 100; px D 1 and py D 1. What quantities does the consumer buy, and what is
his resulting utility?
(c) Now the price of food rises to px D 1:21, while income and the price of clothing are as before.
What quantities does the consumer buy and what is his resulting utility?
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(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 governmentâs revenue from the tax on the 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?
