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Fiscal Policies October 28, 2009 Static Endowment Model • Representative agent has preferences ( ) = (1 − ) ln() + ln() where denotes government purchases and 0 ≤ ≤ 1 is a preference parameter measuring how much is valued relative to • For these preferences, = µ 1− ¶Ã ! • Assume that people have an exogenous productivity and one unit of time supplied inelastically; so that GDP is = (1) • Resource constraint is given by + = • How would a “planner” allocate resources in this economy? max {(1 − ) ln() + ln() : + = } • Optimal allocation satisfies µ 1− ¶Ã ! = 1 + = • Solve for ∗ = (1 − ) ∗ = • Notice: and 1 − can be interpreted as expenditure shares • An increase in (productivity) implies an increase in both ∗ and ∗ (pure wealth eﬀect) • An increase in (war episode?) implies an increase in ∗ and a decrease in ∗ (100% crowding out of private consumption) Static Production Model • Can an increase in the demand for government purchases () lead to an increase in GDP? • Representative agent has preferences ( ) = (1 − ) [ln() + ln()] + ln() where 0 ≤ ≤ 1 and 0 are preference parameters • For these preferences, µ = 1− ¶Ã ! µ ¶ and = • Assume that people have exogenous productivity and one unit of time • Time can be divided between employment () and leisure (); + = 1 • Resource constraint is given by + = (note: GDP now given by = ) • How would a planner allocate resources in this economy? max {(1 − ) [ln() + ln(1 − )] + ln() : + = } • Optimal allocation satisfies µ 1− µ ¶Ã ! = 1 ¶ = 1− + = • Solution is Ã ! 1 1 + (1 − ) Ã ! 1− ∗ = 1 + (1 − ) Ã ! ∗ = 1 + (1 − ) ∗ = • with GDP equal to ∗ = ∗ = (1 + (1 − )) • So yes: an increase in can (or at least, should) lead to an increase in employment and output • Whether it will or not depends on how is financed • Analysis to this point is consistent with benevolent government that has ability to lump-sum tax • To see this, consider the following... • Government levies lump-sum tax on household sector (take as exogenous for the moment) • Household choice problem is max {(1 − ) [ln( − ) + ln(1 − )] + ln()} • Solution satisfies µ − 1− ¶ = • Now, assume that government sets = (GBC) and chooses to maximize household utility; max {(1 − ) [ln( − ) + ln(1 − )] + ln()} • Solution satisfies µ 1− ¶Ã − ! =1 • The equilibrium allocation is identical to the optimal allocation (verify) Distortionary Taxes • What if the government does not have the ability (or willingness) to lumpsum tax? • Assume that GBC is now = (a flat income tax) • Optimal fiscal policy becomes more complicated (in an interesting and realistic way) • Start with household’s problem, viewing as a parameter • Household’s budget constraint is = (1 − )(1 − ); i.e., slope is −(1 − ) • So household optimization implies µ 1− ¶ = (1 − ) = (1 − ) • Combining these two conditions... Ã which simplifies to... (1 − ) 1− µ ! = (1 − ) ¶ =1 1− • Now, this is rather interesting...we can solve for equilibrium level of employment ̂ = 1(1 + ) • Does not depend on ! (or for that matter) • What is going on here? Substitution and wealth eﬀects on employment choice exactly cancel • How does this compare to the optimal level of employment ∗? ̂ = µ ¶ 1 1+ Ã 1 1 + (1 − ) ! = ∗ • There will be “underemployment” in the competitive equilibrium • Private consumption will satisfy = (1 − )̂ • Now, impose the GBC = ̂ Note: since ̂ does not depend on is an increasing function of (no Laﬀer curve here) • So private consumption will satisfy = ̂ − • Government chooses to solve max {(1 − ) [ln(̂ − ) + ln(1 − ̂)] + ln()} • Solution must satisfy µ 1− ¶Ã ̂ − • Solving, we have... ̂ = ̂ ! =1 • It follows that ̂ = ; ̂ = (1 − )̂ and ̂ = ̂ • If government is constrained to finance with distortionary income tax, then optimal tax rate is (share of expenditure desired by households) • An exogenous increase in (an increase in the demand for government purchases) implies a corresponding increase in the tax rate • In this model, the increase in tax rate has no employment or output eﬀects (SE = WE) • Therefore, the desired increase in must come entirely from a corresponding decline in (100% crowding out again) Summary • If must be supplied by a government (rather than a market) and if must be financed with distortionary tax, then equilibrium less than socially desirable • Sometimes, this is called a “second-best” solution (the socially desirable outcome is called “first-best”) • In either case, an exogenous increase in implies an increase in and a decrease in (crowding out eﬀect) • Under first-best, less than 100% crowding out because GDP expands (under second-best, 100% crowding out because GDP unaﬀected) Business Cycle Implications • Imagine that cycle is generated by exogenous changes in • In both cases, optimal (first or second best) policies suggest that should be procyclical • Conventional wisdom suggests that should be countercyclical • Not clear why this should be the case • E.g., imagine “irrational” decline in — this may be corrected with distortionary subsidy on leisure (financed by lump-sum tax) and holding constant Dynamic Endowment Model • Preferences are given by (1 1 2 2) = (1−1) ln(1)+1 ln(1)+ [(1 − 2) ln(2) + 2 ln(2)] • For simplicity, assume = and assume that government purchases (1 2) are exogenous (it is better, but more complicated, to think of as endogenous, responding to exogenous changes in ) • For these preferences, 21 = 1 2 1 • Households have an exogenous endowment (1 2) • Exogenous interest rate (small open economy) • Assume that government finances expenditure programs with lump-sum taxes ( 1 2) • Hence, household lifetime budget constraint is ( − 2) 1 + 2 ≤ (1 − 1) + 2 • Define ≡ 1 + −12 and ≡ 1 + −1 2; and rewrite LBC as 1 + 2 ≤ − • Note: − equals lifetime disposable wealth (after-tax wealth) • This should look familiar: we’ve already studied this model for the case in which = 0 • Solution must satisfy 1 2 = 1 1 + 2 = − • This implies solution (1 2 ) is invariant to timing of taxes (important) • Now, GBC is 2 2 1 + = 1 + • Define ≡ 1 + −12; so that GBC is = • Substitute = into household optimality conditions to derive equilibrium Ã ! 1 ( − ) 1+ Ã ! 2 = ( − ) 1+ 1 = • Implication: (1 2 ) depends only on (= ) and not on timing of spending/taxes NIPA • Private sector saving is 1 = 1 − 1 − 1 • Government sector saving is 1 = 1 − 1 • So, domestic sector saving is 1 + 1 = 1 − 1 − 1 A Deficit-Financed Tax Cut • A policy frequently advocated during recession • Basic idea is hold government spending fixed and cut household taxes; increasing household disposable income, thereby increasing consumer demand • What are the eﬀects of such a policy according to our model economy? • Deficit-financed tax cut means hold (1 2) fixed and cut current taxes; i.e., ∆ 1 0 • Since 1 = 1 − 1; this means ∆1 = ∆ 1 0 (i.e., a decrease in the surplus, or an increase in the deficit) • GBC implies ∆1 + ∆2 ∆ 2 = ∆ 1 + • Or, as (1 2) is being held fixed, this implies ∆ 1 + ∆ 2 =0 • And since ∆ 1 0 this implies ∆ 2 0 (current deficit implies future taxes) • Note that this policy implies ∆ = 0 (since ∆ = 0) • Implication: deficit-financed tax cut has no eﬀect on desired consumption (1 2 ) • In fact, the timing of taxes is irrelevant (budget deficits do not matter) • Put diﬀerently, taxes and deficits (future taxes) are equivalent ways to finance a given spending program (1 2) • This result is known as the Ricardian Equivalence Theorem (RET) • Note: RET does not imply government spending does not matter (get this straight in your head) • Note: timing of taxes does matter for composition of domestic saving (between private and government sectors) • We already know that ∆1 = ∆ 1 0 (government issues bonds to pay for tax cut) • Since 1 = 1 − 1 − 1 , we have ∆1 = −∆ 1 0 (private sector saves the entire tax cut by purchasing these government bonds) • Government bonds are then used by households to pay for anticipated higher future taxes • Note that while the timing of taxes may aﬀect the composition of domestic saving, it does not aﬀect its level (therefore, this policy will not aﬀect the trade balance either) • In a closed economy version of this model (trade balance zero, endogenous interest rate), such a policy has no influence on the equilibrium interest rate • Interesting theoretical result, but what assumptions does it depend on and does it have empirical support? • Assumptions: [1] Perfect financial markets; [2] forward-looking, long-lived (or altruistic) agents; [3] lump-sum taxation • Assumption [3], in particular, appears to be grossly violated in reality; [1] and [2] surely do not hold exactly either • And yet, perhaps surprisingly, it appears very diﬃcult to reject RET in available data (perhaps assumptions are reasonably good approximations?) • Cautious lesson: deficit-financed tax-cut not likely to be as “stimulative” as one might initially have imagined (say, based on Keynesian consumption function) Keynesian: 1 = + (1 − 1) Our Theory: = 1(1 + ) and = (2 − 2) Dynamic Production Model • Preferences are now (1 1 1 2 2 2) = (1 − 1) (ln(1) + ln(1)) + 1 ln(1) + [(1 − 2) (ln(2) + ln(2)) + 2 ln(2)] • You get the idea...we can generalize more and more, building on what we have learned earlier • We can simplify, without loss of generality, depending on the questions we want to explore • Question: how should a transitory increase in desired government purchases be financed? • The real world event I have in mind is the U.S. entry into WW2 (late 1941) • I like to think of this “exogenous” event as a temporary increase in the demand for government output (in particular, military goods and services) • Ideally, we can model this as ∆1 0 ∆2 = 0 • But it will be simpler to assume = 0 and treat (1 2) as exogenous; so experiment is now ∆1 0 ∆2 = 0 • Note: this simplification will not aﬀect the answer to the question at hand (but may not be suitable for other questions) • And so, write preferences like this (1 1 2 2) = ln(1) + ln(1) + [ln(2) + ln(2)] • Note: you’ve seen these before (Re: discussion on Intertemporal Substitution of Labor Hypothesis) • If government can lump-sum tax, then Ricardian Equivalence Theorem holds (believe me) • Implication is that it does not matter how the government chooses to finance ∆1 0; it may choose ∆ 1 = ∆1 or ∆ 2 = ∆1 • In either case, the eﬀect is to decrease after-tax household wealth • Therefore, a decrease in the demand for all normal goods (1 1 2 2); which implies an increase in (1 2) and (1 2) — Note: a transitory increase in leads to a persistent increase in • What if lump-sum taxes are not available? Then Ricardian Equivalence Theorem fails: timing of taxes will matter • It turns out to be optimal to “smooth” taxes across time (smooths out distortions) • Implication: at least some of ∆1 0 should be financed with debt • Indeed, this is exactly what we saw in the U.S. during WW2