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Exercise 3 Consider the following two distribution functions: 0, if x 100. a) Prove that neither F(r) dominates G(x) nor G(x) dominates F(x) according to first- order stochastic dominance. b) Since both distributions have different expected values, they cannot be compared ac- cording to second-order stochastic dominance. Nevertheless, can you prove that for all risk averse individuals F(x) is better than G(x)?6. Write down consumption per worker. on as a function of It. and the savings rate. Determine the Golden Rule level of the savings rate in this economy. {E points] Solution. e={1s)k'."- At the steady state. we have i 1 3 I: c { 3} (541m) Taking log: . o: o lnc =1_ln{d+5w}+1n(ls}+1_lns. Differentiating it w.r.t. 3 gives the following rst order condition, 1 o 1 Ill__ls+lor3' Sos=o. T. Using a Solow diagram. show how the accumulation of capital per worker would change if there were an increase in the population growth rate, gm. Provide the intuition. In another diagram, draw the evolution of consumption per worker over time after this change. [3 points) Solution. The increase in 9N 1Irreould shift up the required investment curve, resulting in a lower level of k; and y; in the steady state. Intuitivcly? the original level of investment is no longer sucicnt to keep up with the growing population so capital per worker would fall. Since :3; = {1 3) ya, it sufces to study the evolution of pl. 3;. = if was at its steady state level and will change to its new steady state level which is lower. Hence, the evolution of consumption per worker decrease; from ['1 3] fr)" to {1 s] {k**}\". Make sure that the rate of change becomes slower and slower in your graph. Question Four Order statistics are very useful tools for analyzing the properties of samples. 1. Write down the general formula of the pdf and cdf for the kth order statistic of a sample of size n of a random variable X with CDF Fx(x). Question Five (Bain/Engelhardt p. 229) Let X] and X2 be a random sample of size n = 2 from a continuous distribution with pdf of the form f(x) = 2x if 0