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Hello,
I am studying at the Samara State University (Russia) and I have applied to study at several British Universities, including LSE, for MSc in Economics. Unfortunately, in Russia we neither study General Equilibrium nor Mathematical Theories of Economic Growth. I will have a test on these topics in two weeks time and I really need your help with two sample questions. Libraries in Samara do not have any books on these topics and so I am preparing for the test by answering sample questions. It will be invaluable if you could help me with the questions below. I know that they are easy and I could have been to answer them myself if I had some relevant literature. I would like to ask you to help me with the questions below. Thank you a lot for your help in advance.
Yours sincerely,
Pavel Erochkine
(1)
Consider an exchange economy with two commodities and two individuals, whose utility functions and endowments are, respectively
u1 (x) = x1 + f (x2), w1 = (w11, w12), f ' (.) > 0, f '' (.) < 0,
and
u2 (x) = x1 + g (x2), w2 = (w21, w22), g ' (.) > 0, g '' (.) < 0.
Show that any two Pareto optimal allocations where xh >> 0, h = 1,2, involve the same consumptions of the second good for both individuals. Again, assuming that at a competitive equilibrium the both individuals consume strictly positive quantities of both commodities, and that f (x2) = g (x2), compute the competitive equilibrium prices and allocations. Are they unique?
(2)
Consider the following economy. There is a representative family whose utility is given by:
(Integral from 0 to infinity) ò (e ^ [-(p - n) ^ t]) * log (ct) dt,
Where p is the discount rate, n >= 0 is the population growth rate, ct is per capita consumption in period t, and log is the logarithm. "^" means "to the power of". The production function is A*(kt^a), where kt is per capita capital and a (- [0, 1] is the capital share. Capital depreciates at the rate b (- [0,1] and the discount rate p is assumed to exceed the population growth rate n > 0.
(i) Write down the planner's problem.
(ii) Derive the dynamic equations that characterise the solution to the planner's problem.
(iii) Find expressions for the steady state values c* and k*. Show that in steady state the utility of the representative family is finite and the transversality condition is satisfied.
(iv) Depict the complete time path of the solution to the planner's problem on a phase diagram.
(v) Consider a specific parametrization of the economy where p = 0.05, n = 0.01, b = 0.05, A = 10, a = 0.3. By taking a linear approximation in the neighbourhood of the steady state, find the relationship between ct and kt on the saddle path. Using this expression, calculate dc (0) / dn.
(vi) Using your result from (v) and a phase diagram, explain what happens after an unexpected, permanent increase in population growth.
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General equilibrium 
