Microeconomics...

1. Express the following in closed form (i.e., reduce to a simplified expression that does not use the summation notation). In the case of part (ii), also find the limit as n -> too S_2k 2. Find 2 Hi(i+2) 3. (a) The geometric series _1/2' converges to one. How many terms of the series are required for the partial sum _ _, 1/2' to be within 0.00005 of 1? (b) The infinite sum _ _,ip(1- p) . 01 for 0 1 . Hint: let x, =n " -1. The binomial formula then gives 2 osmn-2 x 25 (+x ) =n from which it follows that osX, S 2/(n-1) (for all n2 2 ). 7. Show that sin(1) 2-0. 8. Use the fact that lim 1+ 1 ) = e to show that lim ( 1+ ! ) = e" , and lim A(1+ " )" = Ae". 9. Examine the convergence of the sequences (i) and (ii) 3n 2n2 -1 10. Let X, = 3-n n-+ 2n -1 Find 2n -1 - and y,, 3n- -2 (i) lim , (ii) lim y, (ii) lim xy, (iv) lim x,, /y,,35. An economy satisfies all the conditions for Pareto optimality except for one producer who is a monopolist in the market for her output and a monopolist in the market for the single input that she uses to produce her output. Her production function is q = 0.5x, the demand function for her output is p = 100 -4q, and the supply function for her input is r = 2 + 2x. Find the values of q, x, p. and r that maximize the producer's profit. Find the values for these variables that would prevail if she satisfied the appropriate Pareto conditions.38. Determine taxes and subsidies that will lead the firms described in Exercise 36 to their Pareto- optimal output levels but leave their profits unchanged. What is the size of the social dividend secured by this change in allocation?2. (5 pts] If the Japanese interest rate is 0.5%, the U.S. interest rate is 2.5%% and the expected dollar-yen exchange rate is Ey = 0.01, what theory could you use to find the spot dollar-yen exchange rate? What is the value of the spot exchange rate according to the theory?44. An entrepreneur plans for a one-machine horizon. He purchases the machine for 500 dollars. Its scrap value at time T is S(T) = 500 - 40T. The rate of interest is 0.05. The machine yields a quasi-rent flow at the rate Z = 85 - 4t dollars per year. When should the entrepreneur retire this machine?(11) The domain of a function f is [a, d). Below is a sketch of the graph of the derivative of f. (a) The largest interval on which f is decreasing is (_ (b) f has a local maximum at r = (c) The largest intervals on which f is concave up are ( ) and ( (d) f has points of inflection at: r = and r = 10. MONOTONICITY AND CONCAVITY (12) The domain of a function f is [a, e]. Below is a sketch of the graph of the derivative of f. (a) The largest interval on which f is increasing is ( (b) f has local minimum at r = (c) The largest intervals on which f is concave up are ( - -) and( (d) f has points of inflection at: I = -, and r = (13) The domain of a function f is [a, k]. Below is a sketch of the graph of the derivative of f. (a) The largest intervals on which f is decreasing are ( _ _) and ( (b) f has local minima at r = , and r = (c) The largest intervals on which f is concave up are ( ) and ( (d) f has points of inflection at: r = -, and r = (14) The domain of a function f is [a, j]. Below is a sketch of the graph of the derivative of f. A (a) The largest intervals on which f is increasing are ( and ( (b) f has local maxima at r = _, and r = (c) The largest intervals on which f is concave up are ( and ( (d) f has points of inflection at: I= r= , and r = (15) Consider the function f: I re-. (a) The interval on which f is increasing is (. (b) f has a local minimum at r = (c) The interval on which f is concave down is (a - va, a + va) where a =