ECON7030 Microeconomic Analysis

Tutorial Exercise 11 (Week 12)

Question 1

Rose (consumer 1) and Mary (consumer 2) are the only two consumers in the economy.

Each of them consumes only two goods, oregano (good x) and thyme (good y), which

they also own. Consumer 1’s utility function is given by

U 1 (x, y) = ln x + ln y.

She has 2 units of x and 8 units of y. Consumer 2’s utility function is given by

U 2 (x, y) = 2 ln x + ln y.

She has 9 units of x and 6 units of y.

For this question, let p be the price of x and normalise the price of y to 1.

(a) Draw the Edgeworth Box of this economy, marking clearly the endowment point.

For each consumer, sketch the indifference curve passing through the endowment

point.

(b) Calculate the marginal rate of substitution for each consumer at the endowment

point.

(c) Who is going to sell x and buy y? Why?

(d) Derive the demands for x and y for consumers 1 and 2. (Note: there are four

demand functions that you need.)

(e) Find the equilibrium price p.

(f) Find the equilibrium consumption bundle of each consumer.

(g) Mark your answer to part (f) in the Edgeworth Box you have drawn for part (a).

Draw the budget line under the equilibrium price. For each consumer, sketch the

indifference curve passing through the equilibrium consumption bundle.

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Question 2

This is a harder question than usual, it is merely for interest (you will not be counted

off if you do not complete it). In particular, this question involves a 3-consumer, 3-good

world, which makes it “advanced”. (However, each consumer consumes only 2 out of

the 3 goods, so you can still draw indifference curves and budget lines for each consumer

separately.)

There are three consumers: Violet, Emerald and Orange. There are three goods: Red

paint (r), Blue paint (b) and Yellow paint (y). Each consumer consumes only two out

of the three goods. Violet consumes only Red and Blue paint. Emerald consumes only

Blue and Yellow paint. Orange consumes only Red and Yellow paint. Specifically, their

utility functions are given by:

uViolet (r, b, y) = min{r, b}

uEmerald (r, b, y) = min{b, y}

uOrange (r, b, y) = min{r, y}

(In other words, each of them has a perfect complement utility function over the two

goods they consume.)

This is an endowment economy (i.e., no production). Violet is endowed with 2 units

of red paint and nothing else. Emerald has 2 units of blue paint and nothing else. Orange

has 2 units of yellow paint and nothing else.

(a) Is there any double coincidence of wants between any pair of consumers? (Note:

define a “double coincidence of wants” between consumers i and j if the following

is true: i wants to buy a good that j wants to sell, and j wants to buy a good that

i wants to sell.)

(b) Normalise the price of yellow paint to 1. Let Pr be the price of each unit of red

paint, and Pb be the price of each unit of blue paint. Derive the demand functions

for each of the goods for each consumer. (Hint: You may wish to look at Tutorial

Exercise 6 Q3, noting that the “income” is now the endowment value.)

(c) Using your answer to part (b), solve for Pr and Pb in a general equilibrium of this

economy.

(d) Find the amount of each good consumed by each consumer at the general equilib-

rium you have solved.

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