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 |
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u1 (x11 , x12 ) = (x11 )0.5 + (x12 )0.5 u2 (x21 , x22 ) = min[2x21 , x22] |
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(a) |
Define the concepts of Pareto-optimal allocation, price quasiequilibrium with transfers and price equilibrium with transfers. (10 marks) |
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2022-06-02