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Problems 1. Using the steady state equations for the (u, 0) relation (BC), the job creation condition (JC) and assuming a fixed wage rate w,compare the effects on the steady state levels of u, v and o of a smaller labour productivity (Ay 0, a sectoral shock). [Formal derivation is not required; graphical analysis with economic explanation is sufficient] 2. Using the two dynamic equations for the unemployment rate u and the degree of tightness of the labour market o, derive the effect on the steady state of the economy of an anticipated future increase in the wage rate w (firms know at time to that the wage rate will increase permanently at a future date (1). Describe the dynamic adjustment of u, v and o towards the new steady state equilibrium.A Problem Set 1 A.1 Geometric Sums 1. Calculate the following geometric sum, when x * 1, as a function of "+ in particular: [r=ltitrt.to 1=0 2. State a condition on x such that the infinite geometric sum (when 1 -+ co) has a finite value. 3. Assuming that this geometric sum is finite, calculate: I'M x=ltxtrt.txt... 4. More generally, for m a positive integer, calculate: [x=xtrott..+ i=MI 5. What is the present discounted value of an infinite stream of in- comes, which grows at rate g = 2 %, starts at yo = 90000, if the interest rate is i = 3 %?B.1 Two-period Intertemporal Optimization Consider again the 2-period consumption problem of Lecture 2. That is, assume that there are two periods t = 0, 1. Denote the interest rate by r, the discount factor by B, initial financial wealth by fo, income in period 0 by yo, income in period 1 by y1, consumption in period 0 by co, and consumption in period 1 by c1. Instead of logarithmic preferences u(c) = log c, assume that preferences are given by: u(c) = - 1 1 - 0 1. Under what condition on o is this an increasing and concave util- ity function? Why should utility be increasing and concave? 2. Show using 4 different methods that: Bu'(ci) u'(co) 3. Replacing out u'(c) by its new expression, compute c1/co- 4. What is the intertemporal budget constraint? 5. Use the intertemporal budget constraint to solve for c and co. 6. Under what condition on o do you obtain the results we had in class when u(c) = logc? Why is that? 7. Assume that o = 1/2, and fo = 0, yo = $90, 000, y1 = 0, 6 = 1. What are co and c if r = 1%? What about if r = 2%? How much does co change then? How much in percentage terms? (Advice: You may use a Google Spreadsheet to do all of these calculations much faster) 8. Same questions if o = 1. (again, use a Spreadsheet) 250 FRANCOIS GEEROLF 9. Same questions if o = 2. (again, use a Spreadsheet) 10. Compare the changes in co following an increase in the real inter- est rate r in questions 7, 8, 9. Comment. 11. Application: real interest rates have gone down substantially recently. Imagine you were starting your first job today. Assuming you are maximizing utility (!), should you save more, or less, for retirement purposes, than if interest rates were higher