NAME:_____________________
ECON 6912 - Fall 2014
Assignment 1: 103 points. Due Monday 9/15
NOTE: Please show all your work and make sure answers are in the same order as the questions.
1. a.) (4) On two separate sets of axes, draw two separate budget constraints for goods X and Y,
one for which the price of X is increasing as consumption of X increases, and another for which
the price of X is decreasing as consumption of X increases.
b.) (4) There is a potential issue with finding a unique utility maximizing tangency point for one
of these cases. What is that issue? Hint: consider the tangency point for a typical indifference
curve.
2. Central Secondary School has $60,000 to spend on computers (C) and other stuff (Y), so its
budget equation is given by C + Y =60, 000, where C is expenditure on computers and Y is
expenditures on other things. C.H.S. currently plans to spend $20,000 on computers.
The State Education Commission wants to encourage “computer literacy” in the high schools
under its jurisdiction. The following plans have been proposed.
Plan A: This plan would give a grant of $10,000 to each high school in the state that the school
could spend as it wished.
Plan B: Plan B is a “matching grant.” For every dollar’s worth of computers that a high school
orders, the state will give the school 50 cents.
Plan C: This plan is like plan B, except that the maximum amount of matching funds that any
high school could get from the state would be limited to $10,000.
a) (9) Draw three graphs representing Central Secondary School’s budget set under each of the
above plans. Show the expenditure on computers in the horizontal axis, and the expenditure on
other things, Y, in the vertical axis. In each graph, also draw the Central secondary School’s
current budget constraint without any plan.
b.) (6) Suppose that the headmaster’s preferences for expenditures on computers and other things
are given by the utility function U=C*Y2.
How much would the headmaster choose to spend on computers and other goods if it does not
adopt any plans? Now calculate how much the headmaster will choose to spend on computers
under plan A and plan B.
c.) (4) Which would the headmaster prefer: No plan, Plan A or Plan B? Explain.
3. Scarlett likes to eat, eat, eat, apples (a) and bananas (b). Apples cost $4 per pound, bananas
cost $2 per pound, and she has $20 to spend.
a.) (3) If her utility function is given by: U = 3a + 7b, how many apples and bananas will she eat
and what is her total utility?
b.) (3) If her utility function is given by: U = 6a + 3b, how many apples and bananas will she eat
and what is her total utility?
c.) (3) If her utility function is given by: U = min (6a, 3b), how many apples and bananas will
she eat and what is her total utility?
4. In 2008, a typical 18-24 year old consumer bought 24 music tracks at a cost of $1.00/track, and spent a
an average of $36 on live music.
a.) (3) Estimate the typical consumer’s Cobb-Douglas utility function, where music tracks are q1 and q2.
b.) (3) What is the formula for the typical consumer’s marginal rate of substitution?
c.) (3) In 2012, the standard price on i-tunes increased from $1.00/track to $1.29/track. Suppose the
typical 18 – 24 year old consumer still spends $60 on all music, and the price of live music
remained constant. How many music tracks will the typical consumer buy after this increase in
price?
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5. Todd Odd enjoys goods X and Y according to the utility function:
U (x,y) = X2 + Y2
a. (5) Find the utility maximizing amount of X and Y in the standard way (by solving for the the
tangency point between his indifference curve and his budget constraint) if Px=$3, Py = $4 and he has
$50 to spend. What does his utility equal?
b. (6) Graph Todd’s budget constraint and the basic shape of his indifference curve (don’t worry if it is
exactly to scale). What would his utility equal if he spent all of his money on X? How does this
compare to part a. Explain why this is the case and how it relates to your graph.
6. Recent research by economists on Americans’ increasing obesity points to improved technology in
the preparation of tasty and more caloric foods as a possible explanation of weight gain (If you’re
curious see: Cutler, David M., Edward L. Glaeser and Jesse M. Shapiro. "Why Have Americans
Become More Obese?," Journal of Economic Perspectives, 2003.) For example, pre-1960 people
rarely ate french fries at home because of the significant of time amount required to peel, cut, and
cook them, whereas now they can just be purchased frozen and heated in a short time.
To show this through a utility framework, consider a representative consumer who consumes two
goods, potatoes and leisure, N. The amount of potatoes consumed doesn’t vary, but the amount of
time required to reach a level of tastiness, T, does. For each extra unit of tastiness, one must spend pt
hours in the kitchen. Thus, the time constraint is given by: N + pt * T =24. Suppose the
representative consumer’s utility function is given by: U = T*N1/2.
a.) (3) Calculate the marginal rate of substitution for this consumer (assume T is on the horizontal
axis) at the following points: 1) T=9; N=15 and 2) T=15; N=9.
b.) (4) Calculate the optimal amount of (T* , N*). NOTE: T* will be a function of pt.
c.) (4) Does a decrease in the price of tastiness contribute to weight gain? Does it influence the
amount of leisure consumed? Show this by calculating (T* , N*) at two hypothetical values to pt.
7. Consider the CES utility function: U = -X-1 – Y-1
a. (3) Solve for the marginal rate of substitution.
b. (3) If I = $2,000, Px = $36 and Py = $81, what are the utility maximizing amounts of X and Y?
c. (3) What does this consumer’s utility equal in part b.?
d. (3) What is this consumer’s marginal utility of income at the utility maximizing point?
e. (3) Suppose this consumer’s income increased to $4,000, the price of X increased to $72, and the
price of Y increased to $162. What are the new utility maximizing amounts of X and Y and what does
her utility equal?
8. Suppose Tara’s utility function is given by: U = X1/6 Y5/6.
a. (3) If I=$500, Px = $5 and Py = $10, how much X and Y will she consume?
b. (3) Solve for Tara’s expenditure function.
c. (3) What is the minimum amount necessary for Tara to achieve utility equal to 45?
d. (3) If the price of Y increases to $20, how much must be given to Tara to keep her utility constant at
45?
e. (3) Now suppose the price of Y is subsidized from $20 to $10. How much X and Y would Tara
consume? What would this cost the government? What does her utility equal?
f. (3) Now suppose Tara were given the amount in part e. directly in cash, and there is no subsidy.
How much X and Y would Tara consume? What does her utility equal? What principle does this
reflect?
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