Microeconomics

Project description

QN 1 (Consumer optimization: 30 marks)

A consumer has preferences for water W and food F with the following form: U (W, F) =

8W + F with marginal utilities Uw = 4Wand UF = 1. The price of W and F are identical

at 1 per unit. The consumer has a budget of 4.

(a) (7 marks) Draw the budget constraint with F on the horizontal axis and W on the

vertical axis.

(b) (7 marks) Define the marginal rate of subsitution of food for water. Calculate the

marginal rate of substitution when W = 1 and F = 3.

(c) (8 marks) What is the utility level when W = 1 and F = 3? Work out an indifference

curve going through this point. Draw this on the graph, showing where it intersects the

budget constraint and what it looks like close to the axes.

(d) (8 marks) What is the consumer optimum at the given budget constraint? Illustrate the

consumer optimum on the graph. Is the marginal rate of substitution the same as the

ratio of prices at this point? Please explain why or why not.

QN 2 (Market demand curves and equilbrium: 30 marks)

A market has two consumers. The first has preferences given by U (x, y) = xy, the second

has preferences given by U (x, y) = x

2

3 y

1

3 . The first consumer has income M1, the second has

income M2. The prices of the goods are px and py.

(a) (7 marks) Show that Marshallian demands for the first consumer are given by

x1 =M12px

y1 =

M1

2py

(b) (7 marks) Calculate Marshallian demands for the second consumer.

(c) (8 marks) Explain briefly how we calculate a market demand curve for x? Calculate this

market demand curve in terms of incomes and prices.

(d) (8 marks) The supply for good x is given by the curve xs = px. Calculate (partial)

equilbrium prices in the market for good x. Does it depend on the market for good y?

Explain briefly why or why not. How much does each consumer consume if incomes are

constant across the two consumers (i.e. M1 = M2 = M).

QN 3 (Production: 40 marks; all parts give 8 marks)

A production process has the following form

Q = F (K, L)

= (K + L)

2

such that marginal products are FK = FL = 2 (K + L). Answer the following questions:

(a) Does the production function display increasing, decreasing or constant returns to scale?

(b) Draw the isoquant associated with producing 100 units of the good. Put capital (K) on

the vertical axis and labour (L) on the horizontal axis.

(c) Find the marginal rate of technical substitution at two different points on this isoquant.

What is the elasticity of subsitution for this process?

Another firm has a similar production technology but can only operate along two discrete

processes:

F1 (K, L) =

K +

L

2

if L = 2K

F2 (K, L) = (K + L) if K = 2L

In more detail, the firm can use the first technology when it has twice as much labour as

capital. When using the second technology, the firm employs twice as much capital as labour.

Both discrete processes are perfectly divisible.

(d) If the firm has K units of capital and L units of labour such that K < 2L and L < 2K,

derive expressions for how much capital will be devoted to the first production process

an

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