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RHIT / Department of Humanities & Social Sciences / K. Christ 

Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 

1. You have an income of $40 to spend on two commodities. Commodity 1 costs $10 per unit and commodity 2 costs 

$5 per unit. 

a. Write down your budget equation: ______________________________________________. 

b. If you spend all of your income on commodity 1, how much of it could you buy? _________. 

c. If you spend all of your income on commodity 2, how much of it could you buy? _________. 

d. Use blue ink to draw your budget line in the graph below: 

x 2 

8 

6 

4 

2 

x 1 

0 2 4 6 8 

e. Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write down your new 

budget equation: _______________________________________________. On the graph above, use red ink 

to draw your new budget line. 

f. Suppose that the amount you are allowed to spend falls to $30, while the prices of both commodities remain at 

$5. Write down your budget equation: ________________________. Use black ink to draw this budget line. 

2. Alex was consuming 100 units of X and 50 units of Y. The price of X rose from 2 to 3. The price of Y remained at 

4. How much would Alex's income have to rise so that he can still exactly afford 100 units of X and 50 units of Y? 

_____________________________. 

3. Louis Foulmont consumes two commodities, garbage and old computer games. He doesn't actually consume the 

former, but dumps it in his back yard where it is eaten by mountain goats while Louis sits in his hovel and plays 

games on his computer. People pay him $4 per sack for taking the garbage off their hands. Louis can accept as



much garbage as he wishes at that price, and has no other source of income. Old computer games cost him $12 

each. 

a. If Louis accepts zero sacks of garbage, how many computer games can he buy? __________. 

b. If Louis accepts 15 sacks of garbage, how many computer games can he buy? ___________. 

c. Write down an equation for his budget constraint: __________________________________. 

d. Draw his budget line and shade in his feasible budget set. 

Garbage 

20 

15 

10 

5 

0 5 10 15 20 

Computer 

Games 

4. Coach Steroid likes his football players to be big, fast, and obedient. If player A is better than player B in two of 

these three characteristics, then he prefers A to B. Otherwise, he is indifferent between them. Casey Christopher 

weighs 340 pounds, runs very slowly, and is fairly obedient. Kenny Kim weighs 240 pounds, runs very fast, and 

very rarely follows instructions. Bobby Bremmer weighs 150 pounds, has average running speed, and is extremely 

obedient. 

a. Given a choice between Christopher and Kim, who does Coach Steroid prefer? ____________. 

b. Given a choice between Kim and Bremmer, who does Coach Steroid prefer? ____________. 

c. Given a choice between Christopher and Bremmer, who does Coach Steroid prefer? __________. 

d. Does Coach Steroid have transitive preferences? __________.



5. Marcel consumes only nuts and berries. Both commodities are economic goods for Marcel. The set of 

consumption bundles (x 1 , x 2 ) such that Marcel is indifferent between (x 1 , x 2 ) and (1, 16) is the set of bundles such 

x 

2 

4 

20 x 

that x 1 , x 2 > 0 and 

1 

. The set of consumption bundles (x 1 , x 2 ) such that Marcel is indifferent 

between (x 1 , x 2 ) and (36, 0) is the set of bundles such that x 1 , x 2 > 0 and x2 24 4 x1 

. 

a. On the diagram below, plot several points that lie on the indifference curve that passes through the point 

(1, 16) and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve 

passing through the point (36, 0). 

b. Is the set of bundles that Marcel prefers to (1, 16) a convex set? 

c. What is the slope of Marcel's indifference curve at the point (9, 8)? 

d. What is the slope of Marcel's indifference curve at the point (4, 12)? _________. 

e. What is the slope of Marcel's indifference curve at the point (9, 12)? _________. What is the slope of 

Marcel's indifference curve at the point (4, 16)? __________. 

f. Do the indifference curves you have drawn exhibit diminishing marginal rate of substitution? _________. 

g. Does Marcel have convex preferences? __________. 

x 2 

40 

, Berries 

30 

20 

10 

0 10 20 30 40 

x 1 

, Nuts



6. Brittany Arrow likes to spend some time studying calculus and some time dating Rose-Hulman hunks. Her 

indifference curves between hours per week spent studying and hours per week spent dating are concentric 

circles around her favorite combination, which is 20 hours of studying and 15 hours of dating. The closer she 

is to her favorite combination, the happier she is. 

a. Suppose that Brittany is currently studying 25 hours per week and dating 3 hours per week. Would she 

prefer to be studying 30 hours per week and dating 8 hours per week? _________. 

b. On the diagram below, draw a few of Britanny's indifference curves, and then illustrate on the diagram 

which of the two time allocations discussed above Brittany would prefer. 

Dating 

40 

30 

20 

10 

0 10 20 30 40 

Studying 

7. For each of the following expressions, state the formal assumption that is being made about the individual utility 

function. You may find it helpful to illustrate your response with a diagram. 

a. Margarine is just as good as butter. 

b. Peanut butter and jelly go together like a horse and carriage. 

c. Things go better with Coke. 

d. Pringles: “Once you pop, you just can’t stop.” 

e. Double the bubbles are twice as nice.



8. Adrian's utility function is U(x, y) = (x + 2)(y + 1). 

a. Write an equation, y = f(x), for Adrian's indifference curve that goes through the point (x, y) = 2, 8). 

_________________________________. On the diagram below, sketch Adrian's indifference curve for U = 36. 

b. Suppose that the price of each good is 1 and that Adrian has an income of 11. Write out an expression for the 

budget constraint: ___________________________. Based on this information, add an appropriate budget 

line to the diagram. Can Adrian achieve U = 36 with this budget? ___________. 

c. Write out a mathematical expression for Adrian's marginal rate of substitution in terms of x and y: 

_____________________. (Remember, 

MRS 

MU 

x 

.) 

MU 

y 

d. Using this information, what is Adrian's optimal consumption bundle? x = _____; y = ______. 

16 

y 

12 

8 

4 

0 4 8 12 16 

x



9. Consider a consumer with the utility function U(x 1 , x 2 ) = (x 1 + 2)(x 2 + 10). Derive general demand expressions for 

x 

1 

and x 

2 

as functions of prices and income, m. 

10. If a consumer’s utility function is 

U 

x 3 

x 5 

1 2 

goods? _____________________________________________. 

, what proportion of her income will be spent on each of the two 

11. Consider the following utility function: 

( ) 

a. Derive the indirect utility function. 

b. Derive demand functions for x and y (in terms of prices and income). 

c. What are the price elasticities of demand for goods x and y? For each good, is demand elastic, unit elastic, or 

inelastic? 

12. Suppose that an individual’s utility function is 

U . 

x 2 

x 8 

1 2 

a. Write an expression for this individual’s marginal rate of substitution: ___________________. 

b. Making use of your result in part (a) and of the general formulation of the budget equation ( 

p x1 

p2x2 

1 

m ), derive a general statement for this individual’s demand for good x 

1 

as a function of 

prices and income. ____________________________________________________. 

c. Now, if p 1 

2 , p 4 2 

, and 240 

m , how much of good x 

1 

will the individual demand and what 

portion of his income will he spend on it? __________________________.



d If p1 

doubles (to 4), p 

2 

increases by 50% (to 6), and income remains unchanged, how much of good x 

1 

will 

the individual demand and what portion of his income will he spend on it? ___________________________. 

e. Compare the portion of income spent on good x1 

in parts (c) and (d). How could you have predicted this result 

before you even began calculating demand functions? 

13. Suppose that Phil likes to consume only apples and oranges. He has $3.00 per week to spend on these goods, the 

price of apples is $0.20 each, and the most oranges he could consume is 10. 

a. Write out Phil’s budget constraint and graph it on the diagram below. _____________________. 

b. Assume that Phil is indifferent between the following bundles, where the first number in parenthesis is a 

quantity of apples, while the second number is a quantity of oranges: (5,10); (7,6); (9,4); (12,3). Plot these 

bundles and draw Phil’s indifference curve. Are Phil’s preferences convex? 

c. What is Phil’s marginal rate of substitution at the bundle (9,4)? _________________. 

d. If someone offered Phil 2 oranges for 1 apple, would he accept the trade? _________________. 

Oranges 

Apples 

14. Suppose a consumer's utility function is given by U(x 1 , x 2 ) = x 1 x 2 , that p 

1 

=1, p 

2 

=2, and m = 40, and that p 

2 

suddenly falls to 1. 

a. Before the price change, this consumer will consume ______ units of x 

1 

and ______ units of x 

2 

. On the 

diagram below draw an indifference curve/budget diagram that illustrates this initial choice point, labeling the 

bundle that the consumer chooses under these conditions with the letter A. 

b. If, after the price change, the consumer's income had changed so that she could exactly afford her original 

consumption bundle, her new income would have to be __________. With this new income and at the new



prices, however, this consumer would consumer ______ units of x 

1 


2 

. On the diagram 

below, use red ink to draw the budget line corresponding to this income and these prices. Label the bundle that 

the consumer chooses with the letter B. 

c. Does the substitution effect of the fall in p 

2 

cause the consumer to buy more or less of x 

2 

? _______. How 

much more or less? ______________________________________________. 

d. After the price change, the consumer actually buys ______ units of x 

1 


2 

. Use blue ink 

to draw the actual budget line after the price change. Put the label C on the bundle that she actually chooses 

after the price change. 

e. Draw three horizontal lines on the diagram, one from point A to the vertical axis, one from B to the vertical 

axis, and one from C to the vertical axis. Along the vertical axis, label the income and substitution effects on 

the demand for x 

2 

. 

e. Draw three vertical lines on the diagram, one from point A to the horizontal axis, one from B to the horizontal 

axis, and one from C to the horizontal axis. Along the horizontal axis, label the income and substitution effects 

on the demand for x 

1. 

x 2 

40 

Diagram on next page. 

30 

20 

10 

0 10 20 30 40 

x 1 

15. A certain economist spends his free time consuming fine French wine. When the prices of all other goods are fixed 

at current levels, his demand function for rich, red bordeaux wine is q 0. 

005m 

2p 

, where m is his income, p 

is the price of the wine in dollars, and q is the number of bottles that he demands. His income is $30,000 

(obviously he is overpaid), and the dollar price of a bottle of suitable wine is $30. 

a. At his current level of income and with current prices, how many bottles will he demand? _________. 

b. If the depreciation of the dollar against the euro causes the dollar price of a bottle of bordeaux to rise to $50, 

what would his income need to be in order to be exactly able to afford the amount of bordeaux and the amount



of other goods that he bought before the price change? ____________. At this income and the price of $50, 

how many bottles would he demand? ______________. 

c. At his original income of $30,000 and a price of $50, how many bottles would he demand? _________. 

d. When the dollar price of bordeaux rises from $30 to $50, the number of bottles that this economist demanded 

decreased by _________. The substitution effect [increased, decreased] his demand by _________ bottles, and 

the income effect [increased, decreased] his demand by _________. 

16. Given U(x 1 , x 2 ) = x 1 x 2 2 and p 1 = 10, p 2 = 10, and m = 3,000 … 

a. Use this information and your knowledge of demand functions for Cobb-Douglass utility functions to calculate 

the total demand for x and y and the total expenditures on each good (p 1 x 1 and p 2 x 2 ). You do not need to 

show the derivation of the demand functions. 

b. Suppose the price of x 1 increases to 25. Calculate the new demands for x 1 and x 2 . 

c. Decompose the change in demand for x 1 due to this price change into a Slutsky substitution and income 

effect. In other words, calculate the portion of the total change in demand for x 1 that is attributable to a 

substitution effect and the portion of the total change in demand for x 1 that is attributable to an income 

effect. For full credit, show the steps in your work, including computation of m’, compensated income. 

d. Carefully and neatly construct a diagram illustrating your solution, indicating all key values along the 

horizontal and vertical axes. 

x 2 

x 1



17. Suppose that a new political party, The Pigovians, gains control of Congress and their first move is to 

impose a steep carbon tax on gasoline to encourage conservation and “do something about climate 

change.” For simplicity, assume that the new tax on gasoline causes the retail price to increase from 

$3 per gallon to $5 per gallon (yikes!). Now assume that a typical consumer with annual income of 

$40,000 and convex preferences consumes 1,200 gallons of gas before imposition of the tax 900 

gallons of gas after imposition of the tax. 

a. Carefully and neatly construct a diagram that illustrates this consumer's equilibrium before and 

after imposition of the tax. Specify equilibrium amounts (before and after the tax) on both axes. 

Your diagram need not be drawn to scale -- in fact it will be easier if you do not attempt to draw to 

scale along the vertical axis. 

Everything Else 

Gas 

b. It took a while for the Pigovians to figure out why everyone was so mad at them after the 

imposition of the tax. But then you showed them your diagram and explained it to them. What 

did you say? 

c. After talking with you, the Pigovians have a new idea: they will send everyone a rebate for the 

amount of new tax on gasoline they spent. Their argument is very simple: "We didn't want to 

make people worse off, we just wanted them to use less gas. So if we rebate the tax they pay, 

they'll be as well off as they were before the tax but will use less gasoline." Critics of this 

proposal argue that this tax and rebate plan will have no effect on gasoline demand since 

consumers will just use the rebated money to purchase more gasoline. What does economic 

analysis say about this? 

d. The Pigovians claim that their rebate plan will make everyone as well off as they were before the 

tax. Modify the diagram above to show the flaw in the Pigovian’s rebate logic and explain that 

flaw here. Specifically, add a new, post-rebate, budget constraint that illustrates the flaw in their 

rebate plan. (Hint: Think about how the rebate shifts the budget constraint.)



18. Madeline spends all of her income on pansies and petunias, which she regards as perfect substitutes. 

Pansies cost $4 per unit, and petunias cost $5 per unit. 

a. If the price of pansies decreases to $3 per unit, will Madeline buy more of them? _________. 

What part of this change in consumption is due to the income effect and what part is due to the 

substitution effect? ___________________________________________________________. 

b. If the price of pansies was $4, the price of petunias was $5, and Madeline's income for flowers is 

$120, draw her budget line in blue ink on the diagram below. Draw the highest indifference curve 

that she can attain in red ink, and label the point that she chooses as A. 

c. Now suppose that the price of petunias falls to $3 per unit while the price of pansies does not 

change. Draw her new budget line in black ink. Draw the highest indifference curve that she can 

now reach with red ink, and label her optimum choice point as B. 

d. How much would Madeline's income have to be after the price of petunias fell, so that she could 

just exactly afford her old commodity bundle at A? _____________________. 

e. When the price of petunias fell to $3, what part of the change in Madeline's demand was due to the 

income effect and what part was due to the substitution effect? ___________________________. 

Pansies 

40 

30 

20 

10 

0 10 20 30 40 

Petunias



19. Heidi receives utility from two goods, goat’s milk (M) and strudel (S), according to the utility function 

U(M,S) = M*S. 

a. Show that increases in the price of goat’s milk will not affect the quantity of strudel that Heidi 

buys – that is, show that ⁄ . 

b. Develop an intuitive argument that explains why the substitution and income effects of a change in 

P M are exactly offsetting in their effect on S in this utility function. 

20. Preferences and an extension: 

a. Show that Cobb-Douglas preferences are homothetic. 

b. Now consider the following data on consumer expenditure shares by income level and expenditure 

category and comment upon whether you think homothetic preferences are a reasonable 

assumption for modeling U.S. consumer preferences. 

Income Range: Less than $100,000 $100,000 to $149,000 More than $150,000 

Total Expenditures $ 51,607 $ 81,026 $ 118,674 

Food $ 6,669 12.9% $ 9,202 11.4% $ 11,435 9.6% 

Housing $ 15,795 30.6% $ 24,962 30.8% $ 36,971 31.2% 

Transportation $ 10,324 20.0% $ 14,982 18.5% $ 16,799 14.2% 

Health care $ 2,745 5.3% $ 3,472 4.3% $ 4,447 3.7% 

Other $ 16,074 31.1% $ 28,410 35.1% $ 49,022 41.3% 

Source: Statistical Abstract of the U.S., 2006, Table 666.

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