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Welfare arrangements and fiscal sustainability in the "ageing society" (from the exam Jan. 2005). Consider a small open economy (henceforth SOE) with a government sector. For simplicity, assume: 1. Perfect mobility of goods and financial capital across borders. 2. Domestic and foreign financial claims are perfect substitutes. 3. No labor mobility across borders. 4. No uncertainty. 5. Perfect competition on all markets. There is at the world market for financial capital a constant (real) rate of interest 0 The SOE has (adult) population equal to and a labor force equal to where both and are constant. Due to retirement we have The technology of the representative firm is given by = ( ) () where () is a neoclassical production function with CRS, and and are output and capital input, respectively. The whole labor force is employed. We treat time as continuous, and the time unit is one year. The symbol represents a technology factor ("" for "efficiency of labor") growing at the constant rate 0 that is, = by choosing measurement units such that 0 = 1 There are no capital adjustment costs. The rate of physical capital depreciation is 0 and is constant. Firms maximize profit.

a) Find an expression showing how the capital intensity chosen by the firm is determined. Comment. b) Show how the equilibrium real wage is determined and that it can be written = 0 Let denote government spending on goods and services. Suppose is primarily eldercare including health services. Specifically, assume = ( ) 0 where the factor of proportionality, is a constant. Let denote transfer payments including pensions. Assume = ( ) 0 1

where is the "degree of compensation", a constant. Further, let denote gross tax revenue and assume = ( + ) 0 1 where the tax rate is constant (capital income taxation, consumption taxes etc. are ignored). Finally, let denote real public debt and assume that any budget deficit (whether positive or negative) is exclusively financed by changes in (no money financing). Initial debt, 0 is positive. This level of debt as well as are assumed of "moderate" size, allowing a (0 1) to be consistent with a balanced budget. c) Write down an equation describing how the budget deficit and the increase per time unit in public debt are linked. d) Determine the primary surplus, , and its growth rate. How does depend on ? Assume . Let 0 denote the minimum size of the initial primary surplus consistent with fiscal sustainability. e) Find 0 What is the sign of 0? Comment. Hint: if is a positive constant, then R 0 (0) = 1 From now, suppose 0 = 0

f) Find Comment. g) Determine the path over time of the debt-income ratio Illustrate the time profile of in a diagram. Comment. Hint: the differential equation + = where and are constants, 6= 0 has the solution = (0 )(0) + where = Suppose that, in analogy with the Blanchard OLG model with agedependent labor supply, = + where is a constant "retirement rate" (prescribed by law), and is a constant "death rate", so that 1 is a rough indicator of "life expectancy", i.e., expected life time (as adult). As a crude representation of the much debated supposed increase in life expectancy of future generations, imagine that the government at time 0 0 becomes aware that from time 1 = 0+35 years, life expectancy for a young person just entering the labor force will be 10 instead of 1 where 0 (of course, in the real world this demographic change will not be a once for all change, but a gradual change, but for simplicity this is ignored). Population size remains equal to the constant h) In a diagram draw the time profile of ln as it would be in case there is no change in fiscal policy. Is the current fiscal policy sustainable? Hint: consider either the present discounted value of future primary surpluses as seen from time 1 or the time path of the debt-income ratio. Let 0 denote the minimum size of the (constant) tax rate required for fiscal sustainability from time 1 assuming and to be unchanged for ever and no change in taxation before time 1 (Policy I). i) Find 0 Determine the sign of 0 Comment. Now assume instead that at time 0 the government decides to incur a budget surplus (including interest payments) until time 1 such that the debt-income ratio in the time interval (0 1) gradually falls according to = where is a positive constant large enough such that at time 1 one has = 0. The plan is to accomplish this not by changing but by temporary and gradual adjustments of and/or j) Find the required value of Hint: if = a constant, then = 0 + R 0 = 0 + ( 0) Further, the plan is, for 1 to let and be back at their pre 0 level and to let take the minimum value, 00 now needed to obtain fiscal sustainability from time 1 (Policy II). k) Find 00 Determine the sign of 00 0 Comment. Suppose that at time 0 an alternative policy is proposed, namely to let and stay at their pre 0 level forever and at time 1 adjust such that fiscal sustainability is obtained (Policy III). l) Find the required