Диссертация (1137741), страница 11

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Agénor, 2012). Therelative weights of these two inputs,  and  , are assumed to satisfy the constant return toscale condition  +  = 1 .2.2.3. Monetary and fiscal policyThe central bank follows the Taylor rule with a non-negativity restriction on thenominal interest rate:1 + it = max{1, (1 + rt n )(1 +  t ) };   1,(2.12)where rt n is the natural interest rate and  t is an inflation rate.The government chooses expenditures Gt = GtP + GtU and sets taxes for both types ofconsumers. While Eggertsson and Krugman (2012) consider only utility-enhancinggovernment spending, this model incorporates two types of government expenditure: utilityenhancing spending GtU and productive government expenditure GtP . Public spending isfinanced by taxes Tt s and Tt b , paid by savers and borrowers, respectively, and by issuinggovernment bonds Btg , which are bought by savers.

The no-arbitrage condition implies that40For more details see Appendix 1.49the interest on public bonds it equals the interest on private borrowings. The governmentbudget constraint in real terms is:Gt + btg−1 (1 + rt −1 ) = btg + Tt ,where btg =(2.13)Btg, Tt = sTt s + (1 − s ) Tt b .PtTotal government expenditure Gt has the same composition as private consumption.

Itis represented by the Dixit-Stiglitz aggregator with the corresponding demand function for agood j : / −11Gt =   g jt−1/ dj 0− p jt , g jt = Gt  . Pt (2.14)The sequence of government expenditure, taxes and debt chosen by the governmenteach period should satisfy the intertemporal budget constraint. The correspondingtransversality condition is:btg+ Nlim=0N → (1 + r ) N(2.15)2.2.4. Steady stateIn the steady state, given the set of exogenous variables G, T , b g , D that satisfy thegovernment budget constraint G − rb g =  sT s +  bT b , the following relations hold forW , r,Y , C , C , h , h , h, b  :bsbssW=W= hs (ht s )U cs (Cts ) hb (ht s )U cb (Ctb ),(2.16),(2.17)1 + r =  −1 ,(2.18)Y =  s C s + bC b + G,(2.19)C b = −rD+ Wh b − T b ,1+ r(2.20)D+ b g = −  sb s ,1+ r(2.21)h =  s h s + b h b ,(2.22)b50Y = (G P ) ( h ) ,(2.23)W= 1, − 1 [ (G ) (h ( j )) −1 ](2.24)P Bswhere C =  s C +  b C , W = W ( s ) W (b) , G =  G , b =,  = 1 − b .P sssbbPsIn this steady state constrained agents borrow up to the debt limit, while savers do not,inflation is at zero (  t = 0 ) and Yt = Y , Dt = D , h s = h b = h .The model was log-linearized around the steady state with zero inflation.

The steadystate values are denoted by a bar and the log-deviations from the steady state by a hat. Thesteady state level of the real debt limit equals the low level of debt D = Dlow : the economycomes to the steady state after the debt limit falls from the high value to the low one. Themodel is summarized in Table 2.1.41 When considering the model in log-deviations from thesteady state, it is convenient to represent two types of expenditure as shares of the totalgovernment spending Gt = gtP + gtU . If   (0,1) is a share of productive governmentspending in total government spending, then gtP =  Gt and gtU = (1 − )Gt .Table 2.1.

Summary of the Log-linear ModelDescriptionAnalytical representationBudget constraint, borrowerCtb = Yt b +  D Dt −  D Dt −1 +  D t −  D ( it − Et t +1 − r )Euler equation, saverCts = Et Cts+1 −  ( it − Et t +1 − r )Aggregate consumptionCt =  s Cts + (1 −  s )CtbLabor supply, saverWt =  s hts (i) +  s −1 CtsLabor supply, borrowerWt =  b htb (i) +  b−1 CtbLabor market clearinght =  s hts + (1 −  s )htbProduction functionYt =  hˆt +  gtPPhillips curve t =  (1 +  )Yt −  (1 +  ) gtP −  Gt + Et −1 tResource constraintYt = Ct + Gt = Ct + gtU + gtPit = max(0, rt n +   t )Taylor ruleGovernment budgetconstraint−1Note: r = log  ,  t = log41Gt + btg−1 (1 + r ) = btg + b g ( it − Et t +1 − r ) + TtPtDbg ( + 1 −  ), it = log (1 + it ) ,  D = , b g = log ,  =,Pt −1 (1 −  )YYFor more details see Appendix 1.51=Cihii hUi −1, Cti = log t , hti = log t ,  i = hhi =   0,  i = − i c =   0, i = s, b . + 1 −YYhU ccYThe introduction of productive government expenditure leads to the Phillips curve,which differs from Eggertsson and Krugman (2012).

Inflation level is now affected not onlyby total expenditure, but also by the productive part of government spending. The parameters and  also differ from those introduced in Eggertsson and Krugman (2012): they dependon the labor elasticity of the output. 42 Productive expenditure is included separately in thePhillips curve due to its effect on the aggregate supply. The second term appears in thePhillips curve because increasing productive government expenditure raises the marginalproduct of labor, decreasing marginal costs and, thus, prices.2.3.

Fiscal multiplier and comparative statics2.3.1. Deleveraging shockFollowing Eggertsson and Krugman (2012), we assume that an unexpecteddeleveraging shock occurs in the short-term. In the long-term the economy returns to thesteady state with a low value for the debt limit.

In order to analyze the effect of the shock, themodel is split into short-run and long-run. All variables in the current period t are assumed tobe short-term variables, while the next period variables t + 1 are at their long-term levels.In the long-run the output gap equals zero, YL = 0 , with flexible price equilibrium L = 0 , and the interest rate is equal to the natural interest rate, iL = rLn = r .In the short-term the budget constraint, (2.25), and the labor income of a borrower,(2.26), are:CSb = YSb − Dˆ +  D S −  D  ( it −  L − r ) − TSb ,(2.25)YSb = YS +  −1 ( −1b −1 s − 1)GS − (1 + ) −1 gSP +  −1b −1s −1 (CLs −  (iS − Et L − r )),(2.26)YSb = WS + hSb ,where Dˆ = D high − Dand  = (1 +  −1 )( −1 +  −1 ) −  −1 −1  b−1  0 .43Y42Eggertsson and Krugman (2012) considered the one input production function, linear in labor.For  to be positive the following inequality should hold b   −1 −1 / ((1 +  −1 )( −1 +  −1 )) , for the calibratedparameters b  0.24 .4352The log-deviation of borrower’s consumption, CSb , depends positively on the deviationof the disposable income and current inflation, which reduces the real value of debt.

The realinterest rate paid for borrowings and the drop of the debt limit have a negative impact on theborrower’s consumption. Equation (2.25) coincides with the equation for the log-deviation ofthe borrower’s consumption in Eggertsson and Krugman (2012). The difference comes fromthe equation (2.26), which defines the log-deviation of labor income.

The log-deviation oflabor income, YSb , is defined by the wage and the supplied units of labor. The deviation of thewage from its steady state value is derived from the combination of the equations for the logdeviations of labor demand and supply. 44 As opposed to Eggertsson and Krugman (2012)borrowers' consumption in equation (2.26) is also negatively affected by productivegovernment expenditure due to the substitution effect between productive governmentexpenditure and labor.

This negative effect decreases with  .Savers’ consumption, defined by the Euler equation, in terms of short-term and longterm variables is:CˆSs = Cˆ Ls −  (iS −  L − r ).(2.27)As opposed to the borrowers' consumption, savers' consumption depends on theexpected future consumption and does not depend on disposable income.The equation (2.28) for aggregate demand is obtained by combining the resourceconstraint and the equations (2.25) and (2.27), which specify the consumption of two types ofagents in the short-term, taking into account that CLs = 0 and  L = 0 : s ( +  −1 ) + b D b ˆbˆYS = −(iS − r ) −D−TSb +1 − b1 − b1 − b1 +  −1 −1 s − b −1(1 +  ) −1b P+ b D S +GS −gS1 − b1 − b1 − b(2.28)Government spending affects the demand through the last two elements on the righthand side of equation (2.28).

While in Eggertsson and Krugman (2012) the governmentexpenditure increases the output gap, in this model there is an additional negative impact ofproductive government spending on the aggregate demand. The negative impact ofgovernment expenditure disappears when there are no borrowers in the economy, as it comes44For more details see Appendix 1 (Subsection 1.2.3).53from the impact of this part of spending on borrowers' labor income. This effect decreaseswith the higher labor elasticity of the output,  , and increases with  .Applying the definition of the natural interest rate to the equation (2.28), the IS curvecan be written as: s ( +  −1 ) + b D ˆYS = −(iS − rSn ).1 − b(2.29)In order to ensure that aggregate output falls with an increase in the nominal interestrate, the inequality 1 −  b  0 should hold.45 With a fall in the nominal interest rate, saversare encouraged to consume more.

The higher consumption of savers results in higher incomefor both savers and borrowers. Liquidity-constrained borrowers increase their demand as theyconsume all their additional income, which induces an increase in the output. The interest ratein the flexible price equilibrium is the natural rate, rSn :rSn = r −b s ( +  ) +  b D −1Dˆ + D b − s ( +  −1 ) +  b D  Sb1 +  −1 −1 s − b −1(1 +  ) −1b−T+G−g SP .SS−1−1−1 s ( +  ) + b D  s ( +  ) + b D  s ( +  ) + b D (2.30)The natural rate depends negatively on the value of the deleveraging shock. The fall inborrowers' consumption due to the lower debt limit decreases the natural interest rate, whichstimulates the consumption of savers. In the long-term, the natural rate is equal to the steadystate value r .

In the short-term it increases with inflation and government expenditure, whilethe deleveraging shock and taxes diminish the natural interest rate.Two possible regimes emerge from the Taylor rule. In the case of a relatively smalldeleveraging shock, the natural interest rate will remain positive and the nominal interest rateadjusts to the new level of the debt limit, stimulating the demand and offsetting the negativeimpact of the shock. If the shock of the debt limit is high enough, the natural interest rate canbecome negative and ZLB would be binding. In this case, the output gap will be negative.Eggertsson and Krugman (2012) show that in this case, the AD curve becomes upwardsloping, as inflation now increases output due to its negative effect on the real value of debt. 46The AD curve remains upward-sloping in the model with productive expenditure, although itFor this condition to be satisfied b  0.48 , based on the calibration presented in Table 2.2.The slope of the upward-sloping AD curve should be higher, than the slope of the AS curve, to ensure theequilibrium stability.

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