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Question 46 (1 point) A) Using the AS-AD model developed in chapter 10, discuss the effects of an increase in taxes on the position of the IS, LM, AD and AS curves in the medium run. Then discuss the effect on output, the interest rate and the price level also in the medium run. Assume before the change, the economy was at the natural rate of output. [5 points] B) Explain how expectations augmented Phillips curve differ from the Original Phillips curve. Using the following equation, Pt = Pte (1 + m) (1 - Out + z), discuss the steps to derive the equation in inflation form, Itt =It . + (m +z) - Out [5 points] P+ can be written as Pte andIt . as Itte C) Suppose a liquidity trap exists. Explain the effects of an increase in money supply using the IS-LM model. [5 points ]Question 2 (25 marks) In a one-period model economy, there are many consumers, many firms and a government. All consumers are identical. The representative consumer is en- dowed with h units of hours, and derives utility from both consumption c and leisure 1. The utility function is w(c,!) = Inc + all, where a > 0. Let w denotes real wage. He receives labor income w(h = 1), of which T proportion is paid as tax (proportional income tax). All firms are also identical to each other and the market is perfect competitive. The representative firm hires labor and produces consump- tion good with production technology given by Y = AN, where Y is output, A is productivity and N is labor demand. All the firms are owned by the consumers. The government can not issue bonds and runs a balanced budget. That is, the tax collected by the government is equal to its spending G. a) Define the competitive equilibrium for this economy. b) Solve for the competitive equilibrium - the equilibrium wage, consumption and leisure. c) Set up a social planner's problem and derive the efficient allocations. d) Is the competitive equilibrium socially optimal? Why or why not. Question 3 (25 marks) In a two-period model economy, there are a total N number of consumers, of which N are of type A and N are of type B consumers. Consumers derive utility from current period consumption c and future period consumption c'. For all consumers, current and future consumptions are perfect substitutes. Type A consumers have MRS. = a, and type B consumers have MRScp = b, where a 0, Ny' - G' > 0 and a > 2 Ny-G a) Solve for the competitive equilibrium - the equilibrium real interest rate, and the current and future consumption allocations for each individual consumer. b) Does the Ricardian Equivalence Theorem hold in the equilibrium? Explain why or why not.146 Chapter 3 Continuity f) Prove that lim, _,tan 1 x = x/2 and lim,,tan x = -1/2. 3.50. In Exercise 3.19 you proved that, for a > 1, the function a" exists and is continuous on R. When a has the special value a = e, we obtain the natural exponential function f (x) = ex, also denoted exp (x). We know that et is continuous on R. a) Prove that f (x) = e* is strictly monotone increasing on R and maps (-co, co) one-to-one onto (0, oo). b) Prove that f-exists and maps (0, oo) one-to-one onto R. c) For simplicity, denote f- by g. Prove the following properties of g: (1) g (1) = 0; (ii) g(xy) = g(x) + g(y) for all x and y in (0, co); (iii) g(x) = rg(x) for all x in (0, oo) and all r in R; (iv) g(x/y) = g(x) - g(y) for all x and y in (0, co). The inverse function g = f is a version of the logarithm discovered by Napier in about 1594 to assist with the calculations required in the renascent science of astronomy. The properties in part (c) made such calculations, if not easy, at least less difficult. Only when its analytic properties became known did this function assume its present important status. It is now denoted g (x) = In x and is called the natural logarithm. Little could Napier have known the profoundly important role this function would come to play in widely divergent areas of mathematics and the sciences. d) Prove that In x is strictly monotone increasing on its domain (0, co).( e) Prove that In x is continuous on (0, co). f) Find lim,+ In x and lim,_ Inx. 3.51. Define a function f at points x = (x,, X2) in R2 by f (x) = tan '(x, + x?). a) Find the domain and the range of f. b) Find all points x in R where f is continuous. c) Prove that lim, x 1 0 tan (x7 + x2) = 1/2. 3.52. Define a function f at points x = (x], X2) in R by f (x) = tan '[1/(x, + x2)]. a) Find the domain and the range of f. b) Find all points x in R where f is continuous. c) Does lim, f(x) exist? Is it possible to define f(0) in such a way that fi continuous at 0?Consider two ways to obtain the change in consumer's surplu. Suppose utihty is given by Way) = (a _ 37h" + y and the budget constraint is m = pa: + y. Thus, the marginal rate of substitution is MRS = a 211:. (a) Derive the Marshallian demand for goods 3 and y. [1 mark] (b) Suppose that m = 24, a = 10. Plot the inverse demand for good 11:. For what values of 10 would you say that 117* = D? [1 mark]