EC2210 Mathematical Economics 1B Summer Examinations 2019/20

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EC2210

Summer Examinations 2019/20

Mathematical Economics 1B

1. Consider an economy which can produce two commodities (1 and 2) using capital and labour, according to the production functions

for (K1 , L1 ) e 皿

for (K2 , L2 ) e 皿

where Ki and Li are the amounts of capital and labour used to produce commodity i e {1, 2}, respectively, while yi is the amount produced of commodity i e {1, 2}. The total endowment of this economy consists of 1 unit of capital and 1 unit of labour. There are two consumers

with a utility function u(x1 , x2 ) = x 1(2) x2(2) for all (x1 , x2 ) e 皿 .

(a) Define a feasible factor allocation and a technologically efficient factor allocation.

(10 marks)

(b) Define a feasible allocation and a Pareto-efficient allocation. (10 marks)

(c) Show the factor allocation (K1 , L1 ) =╱ , 、, (K2 , L2 ) =╱ , 、 is technologically efficient. (Hint:  What optimisation problem does a technologically efficient allocation solve?) (15 marks)

(d) Is there a Pareto-efficient allocation associated to the factor allocation in (c).  What is the economic intuition? (15 marks)

2. Consider a pure exchange economy where every consumption set Xi  is non-empty and convex and every preference relation 5i  is strictly convex (i.e., if xi(′)  5i xi  and xi(′) xi then αxi(′) + (1 - α)xi  ×i xi  for all 0 < α < 1).

(a)  Define a price equilibrium with transfers and a Pareto-optimal allocation. (10 marks) (b) Show that every consumer i can have at most one satiation point. (10 marks)

(c) Show that preferences are locally non-satiated at any consumption bundle different from the single satiation point. (15 marks)

(d)  Show that any price equilibrium with transfers of this economy is Pareto-optimal.

(15 marks)

3. Consider a two-consumer, two-commodity, Edgeworth-box economy in which the total endow- ment of the economy is (2, 2). The preference relation of consumers 1 and 2 can be represented by the following utility functions

u1 (x11 , x12 ) = (x11 )0.5 + (x12 )0.5

u2 (x21 , x22 ) = min[2x21 , x22]

(a)

Define the concepts of Pareto-optimal allocation, price quasiequilibrium with transfers

and price equilibrium with transfers. (10 marks)








2022-06-02