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Consider the article on government debt by Elmendorf and Mankiw (1999). In their footnote 12, p. 1635, they present a formula for the long-run multiplier on capital w.r.t. the level of government debt in a closed economy. The formula, which is based on the Blanchard OLG model, indicates a negative long-run multiplier. Population growth, technical change, and retirement are ignored. a) Using our standard notation (i.e., replacing their with their with and their with ) show by phase diagram analysis that the negative sign is correct. Comment. Hint: set up the model as in Lecture Notes, Ch. 13. b) Derive Elmendorf and Mankiw's formula, using our standard notation. Their formula appears different from the corresponding formula in Lecture Notes. Is there a real difference or are the two formulas equivalent? Comment. c) In their footnote 11, p. 1635, Elmendorf and Mankiw present a formula for the long-run multiplier on national wealth w.r.t. the level of government debt in a small open economy. Is the absolute value of the long-run multiplier smaller or larger than that for the closed economy? Why?

A fiscal sustainability gap indicator with the Danish economy in mind, October 2005.2 Consider the government budget in a small open economy (SOE) with perfect mobility of financial capital, but no mobility of labor. The real rate of interest at the world capital market is a positive constant + 0 where is a constant rate of (Harrod-neutral) technical progress and is a constant rate of growth of the labor force. The aggregate production function, = ( ) has CRS. Time is continuous. Let = GDP, = government spending on goods and services including elder care and health services, = transfer payments including public pensions, = gross tax revenue, = public debt, all at time and in real terms (i.e., measured with the output good as numeraire). Assume the future is known with certainty and that budget deficits are exclusively financed by debt issue (no money financing). Initial public debt, 0 is positive. a) Ignoring business cycle fluctuations, what is the growth rate of GDP? Write down the time path of b) Write down an expression for the real primary budget surplus, at time . c) Write down a condition that the time path of must satisfy, as seen from time 0, for fiscal policy to be solvent. Translate this to a condition for the time path of the primary surplus-income ratio, It is convenient to introduce ( + ) Let , and Suppose grows at the same rate as d) What can we conclude about the time path of ? It is well-known that many industrialized countries feature an increasing elderly dependency ratio due to longer life span and lower fertility. Figure 5.1 shows this for Denmark. Figure 5.2 shows projected paths of the primary f) Is current (2005) fiscal policy, which we may call P sustainable? Why or why not? Suppose a suggested new policy design, P0 implies that the path of remains unchanged, but the path ( ) =0 is replaced by the path ( 0 0 ) =0 with time profiles 0 = 0 0 (1 ) 0 = 0 0 + (1 ) g) Write down an expression for the primary surplus-income ratio at time according to the new policy P0 . h) Find the minimum initial primary surplus-income ratio, 0 0 required for the fiscal policy P0 to be sustainable as seen from time 0. Hint: R 0 = 1 for any constant 6= 0 As a sustainability gap indicator at time 0 we choose 0 0 0 0 i) Illustrate the gap in the diagram from question e). How does 0 depend on: 1. the debt-income ratio at time 0? 2. the adjustment speed ?

3. the spending-income ratio, 4. the growth-corrected interest rate presupposing , and are independent of the growth-corrected interest rate? Hint: If no unambiguous answer as to the sign of the effect can be given, write down a criterion in the form of an inequality on which the sign depends. Comment. j) In fact an increase in the interest rate is likely to affect namely by reducing partly through the higher tax revenue from postponed taxation of labor market pensions and partly through the induced increase in wealth accumulation, which implies higher future tax revenue; there are also potential counteracting factors such as a possible increase in tax deductibility due to increased interest payments. Can this matter for the conclusion to i.3)?

Consider the government budget in a small open economy. Time is continuous, the time unit is one year, and there is no uncertainty. Let and be non-negative constants and let = 0(+) = real GDP, = real government spending on goods and services, = real net tax revenue ( = gross tax revenue transfer payments), = real public debt, = real interest rate, a constant. Assume that the budget deficit is exclusively financed by issuing debt. a) Write down an equation showing how the increase in per time unit is determined. Consider a scenario with 0 0+ 0 and = a positive constant less than one b) Find the maximum constant which is consistent with fiscal sustainability. Hint: the differential equation + = where and are constants, 6= 0 has the solution = (0 ) + where = Consider another scenario, where there is a deficit rule saying that 100 per cent of the interest expenses on public debt plus the primary budget deficit must not be above 100 per cent of nominal GDP, i.e. + ( ) (*) where 0 1 0 and = nominal public debt, = price level, = + = nominal interest rate, = the inflation rate which we assume constant and non-negative. c) Is the deficit rule of the SGP of the EMU a special case of (*)? Comment. d) Let Derive the law of movement (differential equation) for assuming the deficit rule is always binding. Suppose is such that (1 ) + + . e) Find the time path of f) Let the steady-state value of be denoted and assume 0 Will explode or converge towards over time? g) How does depend on ?