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EC2210

Summer Examinations 2018/19

Mathematical Economics 1B

1. Consider a two-good, two-consumer pure exchange economy. Each consumer’s consumption

set is R and ui : R is the utility function representing the preferences of consumer

i e {1 , 2}. An allocation is a vector in R with typical element x = ((xll , xl2 ) , (x2l , x22 )) ,

where xi,j denotes consumer i’s consumption of commodity j. The aggregate endowment is

ω = (ωl , ω2 ) e R. Consider the following optimisation problem:

max ul (xll , xl2 )

zR

u2 (x2l , x22 )

s.t xll + x2l

, xl2 + x22

(a) Define a feasible allocation. (10 marks)

(b) Define a Pareto optimal allocation. (10 marks)

(c) Show that any Pareto optimal allocation solves problem (1) for some choice of u2 .

(15 marks)

(d) Show that if the utility functions are continuous and strictly increasing, then any solution to (1) is Pareto optimal. (15 marks)

2. Consider a two-good, two-consumer pure exchange economy. Let ωj  denote the aggregate endowment of commodity j e {1, 2}. Let xi,j  denote consumer i’s consumption of           commodity j. The utility function of consumer 1 is:

ul (xll , xl2 ) = xll(a)x l(l)2(_)a           for  (xll , xl2 ) e R .

Likewise, for some β e (0, 1), the utility function of consumer 2 is

u2 (x2l , x22 ) = x2l(8)xl22(_)8           for  (x2l , x22 ) e R .

(a) Show that the preferences of these consumers satisfy local non-satiation. (10 marks)

(b) Suppose prices and wealth are strictly positive. Obtain each consumer’s Walrasian demand for commodity 1 as a function of prices and wealth. Explain carefully your reasoning.

(10 marks)

(c)  Let (ωil, ωi2) e R denote consumer i’s endowments of commodities 1 and 2 for

i e {1, 2}.  Find all the Walrasian equilibrium prices for this economy as function of α, β and the individual endowments. (15 marks)

(d)  Now let’s consider the effect of a change in individual endowments on equilibrium prices.

Suppose (ωil, ωi2) e R and a new endowment distribution

(ll , l2 ) = (ωll  - ∈, ω l2  - ∈) and  (2l  + ∈, 22  + ∈)  , for some ∈ > 0 but small enough

so that the new individual endowments are still positive.  Explain the eect of this

change in the endowment distribution on equilibrium prices as a function of α and β .

(15 marks)

3. Suppose a consumer has a preference ordering on R that is complete.

(a)

Define monotonicity and strict monotonicity of . (10 marks)

(b)

Show that if preferences are strictly monotone, then for any bundle x e R with x 0,

one can nd bundles x e R and x e R such that x × x × x. (10 marks)