EC2210 Mathematical Economics 1B Summer Examinations 2018/19
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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 ) z∈R |
. 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 effect 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. |
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(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 find bundles x e R and x e R such that x × x × x. (10 marks) |
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2022-06-02