Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 5

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1 Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 5. Taxation, Consumption and Ricardian Equivalence. Consider a two-period consumption allocation problem as seen in class. For simplicity, assume that the discount rate and the interest rate r are both equal to zero. The household maximizes U (C [0])+U (C []) subject to its intertemporal budget constraint, which is now extended to include lump-sum taxes T [0] and T [] in both periods. (a) Derive the reaction of the household s consumption in the rst period with respect to an increase in taxes. Distinguish the two cases in which the households either believes that the tax increase is temporary (taxes increase only in the rst period) and the case in which the tax increase is assumed to be permanent (taxes increase in the rst period and will remain high in the future). Let U 0 = U (C (0)) and T 0 = T (0) throughout the Problem Set. = r = 0. Households maximize U 0 + U subject to + C = Lw 0 T 0 + Lw T ) max [U 0 + U (Lw 0 T 0 + Lw T )] ) U 0 0 U 0 = 0, U 0 0 = U 0 ) = C temporary tax increase in t = Lw0 T0+Lw T 2 ) = 2 That is, households react less than proportional to the tax increase. The reduction of consumption in both periods is reduced by half the value of the lost disposable income. In the absence of time preferences and interest rates consumption today is as good as consumption tomorrow and as it is optimal to smooth consumption, it is reasonable to "split" the e ect on disposable income over both (all) periods. Permanent tax + = 2 2 =

2 The reduction in consumption due to a permanent tax increase is proportional in the sense that half of "today s" and half of "tomorrow s" loss of disposable income a ect consumption "today". This may also be interpreted as the whole tax decrease "today" being paid from the reduction of "today s" consumption level. Graphically, the results can be shown as follows: Figure (b) Now extend the model by considering the government s budget constraint. Assume that the government has to cover a predetermined path of expenditures (G [0] ; G []). However, the government may decide to run a budget de cit or a surplus in the rst period (it must respect its intertemporal budget constraint). Write down the government s intertemporal budget restriction. How does the consumer now react to a rst-period tax cut if she takes into account that the government has to balance its budget intertemporally? Why is it that the consumer does not change her consumption plans in response to the tax cut? Show that the tax cut increases the households savings rate out of disposable rst-period income (Y [0] T [0]) but leaves the savings rate out of total income unchanged. Government s intertemporal budget constraint: G 0 + G = T 0 + T If government expenditure is predetermined you might also write: G G 0 + G = T 0 + T If G 0 + G is predetermined, in order to satisfy the budget constraint T 0 +T is also predetermined. So whenever taxes in one period move, taxes in the other period have to o set the change exactly. However, we have seen in (a) that the households budget constraint includes T 0 + T instead of only one of the elements. Hence, as the sum remains unchanged the optimality conditions of the households consumption decision remain unchanged which means that that given the governments budget constraint, the households reaction to a change in taxes is not to react at all. Savings rate out of disposable income: S = Y D 0 Y0 T0 C0 = Y0 D Y 0 T 0 = Y 0 T 0 2

3 C = 0 ( ) 0 (Y 0 T 0) 2 Note that this is true, because we just explained that does not react to T 0. If the tax decrease were permanent we had a negative reaction of which then would cause the derivative to be greater than 0. Savings out of total income: S = Y T C = Y T Lw0 T0+Lw T C Y T = Lw0+Lw C Lw 0+Lw Note that T 0 has to be equal to ( T ). (c) Now embed the household in a closed economy and interpret Y [0] and Y [] as GDP in the two periods. What does the tax cut in (b) do to the national savings rate? Hence, what is the relationship between budget de cits and the national savings rate in this model? (Government expenditures continue to be predetermined) Let us assume that initially T 0 = G 0 and T 0 = G 0. A tax cut like the one treated in this exercise would then coincide with a rst period budget de cit. In the context of this model the national savings rate in period 0 would increase as taxes decrease. However, we have shown that this is just a temporary e ect and given the governments budget constraint the overall (that is out of the twoperiod-income) savings rate will remain unaltered. Using the budget de cit to stimulate savings therefore does only work in the present period (short-run) if households assume that the de cit will be paid o in the future. 2. Tax cuts and macropolicy. Using the results of problem 3. how would you judge the potential of a policy of stimulating consumption through a tax cut? What assumptions need to hold for this result? How does this compare to the case of a Keynesian consumption function? Arguing along the lines of the exercises we have worked on so far the potential of stimulating consumption by a tax cut is rather low. The model tells us that additional disposable income is used entirely to save for "rainy days", that is, to pay o future tax increases with the saved money. While households expect the government to - at least try to - achieve a cleared budget they expect future tax increases. While these future tax increases enter into their budget constraint and are weighted equally to present tax decreases in the absence of time preferences, optimal consumption conditions do not change due to changes in disposable income. The Kenyesian consumption function assumes that a constant fraction of disposable income goes to savings. Hence, in this case consumption will follow 3

4 the path of income, meaning that consumption will increase due to a tax cut and will decrease in the second period, when taxes need to increase in order to even the budget. While the fact that we are only considering two periods does not crucially restrict the model in this case (de ne one low-tax and one high tax period) the fact that future consumption is equally weighted to present consumption is a strong simpli cation. One major assumption that might be violated in reality is that the budget constraint can only hold if a tax increase follows after a tax cut. In reality a lot of e ort is put instead in trying to adjust G. If households believe that the de cit in t = 0 will be paid o by less government spending in t = they are likely to to use the additional disposable income on consumption in periods 0 and (depending on the interest rate). In this case the de cit would be able to stimulate consumption even if it is only temporary. However, this exercise nicely shows how households, based on very simple (although rigid) assumptions try to smooth consumption over time as opposed to the mechanic keynesian reaction to movements in disposable income. 3. Savings and the interest rate. Consider a setup in which the household has Cobb-Douglas preferences over the two periods consumption: U = (C [0]) (C []). Savings in period bear interest r, and the household has an exogenous income described by (Y [0] ; Y []). (a) Derive the optimal allocation of consumption over the two periods. Intertemporal budget constraint: Y 0 + Y +r = + C +r maximize utility under this constraint: L = C0 C + Y 0 + Y C +r C 0 = C 0 C = = ( ) C 0 C = = Y 0 + Y +r C +r = 0 (i)&(ii)! ( ) C 0 C +r C = 0 ) = ( )(+r) C 4

5 ) C = ( ) [( + r) Y 0 + Y ] h ) = Y 0 + i Y (+r) (b) How do the planned savings depend on the market interest rate? What are the policy implications of this result? planned savings: h S P = Y 0 = Y 0 Y 0 + i Y = Y (+r) 2 0 if the interest rate increases, so will savings. I.e. Households save more if they earn more interest with their savings or if the opportunity costs of "not saving" increase. An increase of the interest rate makes future consumption (relatively) cheaper than consumption today. 5

6 Figure ii) i) Lw 0 - T 0