1 The social planner problem

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1 The social planner problem max C t;k t+ U t = E t X t C t () that can be written as: s.t.: Y t = A t K t (2) Y t = C t + I t (3) I t = K t+ (4) A t = A A t (5) et t i:i:d: 0; 2 max C t;k t+ U t = E t " X t C t t # s.t.: A t K t = C t + K t+ A t = A A t et

2 The Lagrangian of the problem is: L 0 = X t t Ct + t At K t C t K t+ The rst order conditions 0 = t Ct t = 0 = t + t+ ( ) A t+ = 0 0 = A t Kt C t K t+ = t 2

3 The complete (non linear) model By using the rst one into the second one, we obtain: together with C t = E t C t+ A t+ ( ) Kt+ (6) Y t = C t + K t+ Y t = A t K t A t = A A t et and write the system at the s.s. C ss = C ss A ss ( ) K ss = C ss + K ss = A ss Kss A ss = A ss A sse t 3

4 The linear model: C ss ^C t = C ss A ss ( ) Kss ^C t+ + ^A t+ ^K t+ ^Yt = C ss ^Ct + K ss ^Kt+ ^Yt = A ss Kss ^At + ( ) ^K t ^A t = ^A t + t note that A ss = by simplifying: ^C t = ^C t+ + ^A t+ ^K t+ ^Y t = C ss ^Ct + K ss ^Y t = ^A t + ( ) ^K t ^A t = ^A t + t ^Kt+ 4

5 The steady state equilibrium In order to pin down parameters ; ; ; and 2, we write down the deterministic steady state equilibrium system (remember that A ss = ): C ss = C ss A ss ( ) K ss (7) = C ss + K ss (8) = A ss Kss (9) A ss = A ss A sse t (0) By simplifying: = ( ) K ss = C ss + K ss = K ss A ss = From the rst one, we get: K ss = ( ) Kss = ( ) Kss = ( K ss ) K ss hence: K ss = ( ) or K ss = ( ) from the aggregate resource constraint: C ss = K ss = ( ) from the production function: = Kss = ( ( ) ) 5

6 by solving for, yields: = [ ( )] Now substitute in the great ratio Kss to get the level K ss : K ss = ( ) [ ( )] = [ ( )] Now use the ARC to get the level of C ss : C ss = K ss = [ ( )] [ ( )] 6

7 Calibrating the model We are assuming full depreciation in each period; as such K t+ = I t ; From EUROSTAT we compute the ratio: From the A.R.C we know that: C ss = 0:8 From the s.s relation: C ss = K ss ) K ss = 0:2 K ss = 0:2 = ( ) We have two parameters (,) and one equation. We have to choose the value of two parameters; how? From the Euler equation: = ( ) Yss K ss that says that in steady state = it is equal to the capital return, which on a quarterly basis (as the model calibration is quarterly) is equal to :0062, so: So, we have: = :0062 ) = 0:9938 K ss = 0:2 = ( ) 0:9938 = 0:798 What about parameters, and 2? can be pinned down by using microeconometric studies. and 2 can pinned down by using other studies or by estimating the Solow residual, or we can calibrate them to target the volatility and the auto-correlation of the simulated output series We have pinned down, through the calibration, all the coe cients/parameters of the log-linear dynamic equilibrium system. Put it in the computer and simulate it!! 7

8 2 The social planner problem max C t;k t+ U t = E t X t C t () s.t. : Y t = A t K t (2) Y t = C t + I t (3) K t+ = ( k ) K t + I t (4) A t = A A t (5) et t i:i:d: 0; 2 (6) that can be written as: max C t;k t+ U t = E t " X t C t t # s.t.: A t K t = C t + K t+ ( ) K t A t = A A t et 8

9 The Lagrangian of the problem is: L 0 = X t t Ct + t At K t C t K t+ + ( ) K t The rst order conditions 0 = t Ct t = 0 = t + t+ ( ) At+ + ( ) = 0 0 = A t Kt C t K t+ + ( ) K t = t The complete (non linear) model By using the rst one into the second one, we obtain: together with C t = E t C t+ ( ) A t+k t+ + (7) Y t = C t + I t Y t = A t K t K t+ = ( ) K t + I t A t = A A t et 9

10 The linear model: Css ^C t = Css ( ) Ass Kss + ^C t+ + Css ( ) A ss Kss ^At+ ^K t+ ^Yt = C ss ^Ct + I ss ^It ^Y t = ^A t + ( ) ^K t K ss ^Kt+ = ( ) K ss ^Kt + I ss ^It ^A t = ^A t + t Use the s.s relations to simplify the linear model. For example, the Euler equation in s.s. is: ( Css = E t C ss ( ) A ss Kss + = ( ) A ss Kss + ) A ss Kss = ( ) hence the linear Euler equation can be written as: ^C t = ^C t+ + [ ( )] ^At+ The law of motion of capital in s.s. is: ^K t+ K ss = I ss hence, the linear version of the law of motion of capital become: ^K t+ = ( ) ^K t + ^I t The linear system is now given by: ^C t = ^C t+ + [ ( )] ^At+ ^K t+ ^Y t = C ss ^Ct + I ss ^It ^Y t = ^A t + ( ) ^K t ^K t+ = ( ) ^K t + ^I t ^A t = ^A t + t 0

11 The Steady State Equilibrium: We have to pin down the parameters.,,,, and 2. To do it, we initially write down the complete steady state equilibrium system: = ( ) A ss K ss + = C ss + I ss = A ss Kss K ss = I ss A ss = As in the simplest model, we use the Euler equation to get K ss as function of : = ( ) + K ss K ss = ( ) ( ) ) K ss ( ) = ( ) Let s use the law of motion of capital to get the s.s. ratio Iss of deep parameters only: as a function I ss = K ss ) I ss = K ss ( ) = ( ) as a function of deep parame- By using the A.R.C. we get the s.s. ratio Css ters: = C ss + I ss ) C ss = C ss ( ) = ( ) I ss Let s use the production function and the ratio Kss to nd the level :

12 = Kss ( ) = ( ) ( ) = ( ) Now use the above equation to substitute for in the great ratio Kss K ss = K ss = ( ) ( ) ( ) ( ) ( ) ( ) = ( ) ( ) Let s use I ss = K ss to get: ( ) I ss = ( ) Now use the A.R.C. C ss = I ss to get C ss : C ss = ( ) ( ) ( ) ( ) Calibrating the model: As in the previous version of the model, we start the calibration by targeting the great ratio Iss = 0:2 I ss ( ) = 0:2 = ( ) So from the A.R.C. we get the ratio Css = 0:2 = 0:8 2

13 There are three parameters to pin down:, and. For, various sources of independent empirical evidence point to 0:025. As for, we know from the Euler equation that return, which on quarterly basis is :0062, so as: is equal to the capital = :0062 ) = 0:9938 Let s use these values to solve the great ratio Iss for I ss I ss ( ) = ( ) 0:9938 ( ) = 0:2 = 0:025 0:9938 ( 0:025) = 0:75 What about parameters, and 2? Microeconometric studies show that the value of is not far from (that is a log-linear function). and 2 can be pinned down by estimating the Solow residual, by using other studies, or we can calibrate them to target the volatility and the auto-correlation of the simulated output series. Put the linear model in the computer and simulate it ^C t = ^C t+ + [ ( )] ^At+ ^K t+ ^Y t = C ss ^Ct + I ss ^It ^Y t = ^A t + ( ) ^K t ^K t+ = ( ) ^K t + ^I t ^A t = ^A t + t 3

14 Calibration C ss Target ratio = 0:8 I ss = 0:2 Technology = 0:75 = 0:025 Preferences = = 0:9938 TFP = 0:0053 = 0:9 Download GDP, total consumption and investment from the US Department of Commerce - Bureau of Economic Analysis (BEA) database and the population aged 6 and older (P OP +6 ) are taken from the Bureau of Labour Statistics (BLS), Current population surveys. Table below summarizes data sources and the manipulations of original time series used to construct the series used for the computations shown in Table 2. Hatted variables indicate the cyclical component after the H-P ltering with smoothing parameter set at 600. Table : Data Variable Source De nition Database Table/Code Transformation Y BEA Real GDP (base year: 2005) C BEA Real Consumption (base year: 2005) I BEA Real Investment (base year: 2005) P OP +6 BLS Working age population 6 years and older NIPA table..5 NIPA table..5 NIPA table..5 LNU Yt b = ln Ct b = ln It b = ln Y t P OP +6;t C t P OP +6;t I t P OP +6;t Table 2: simulated vs. actual series Variable Model U.S. Data by :53 :57 bc = by 0:66 0:80 bi = by 3:00 4:70 Note that bi > by > bc 4

15 The markets of production inputs can be described graphically as follows: r w K S =K(t ) L S = r w r r ( > 0 ε ) K D ( r) t w ( > 0 L D ( w) ε ) t K K K L Factor inputs do not increase after the shock (" t ) since capital is predetermined and labor supply is xed. Factor inputs remunerations increase and this a ects agents behavior Households expectations of greater labor and capital income.) Consumption and savings increase ) Investment and capital increase) Y increases In the case of an autoregressive process for the TFP ( > 0), as long as A t is higher than his stationary path Y t increases Since labor supply is xed, labor remunerations is strongly pro-cyclical (in contrast with stylized facts) 5

16 3 Decentralized economy with labor choice King Plosser and Rebelo (988) Households: max fc t;l t;k t+g t=0 " X U t = E 0 t L + # t log C t + t=0 s.t.: w t L t + ( + r t ) K t = C t + K t+ ( ) K t the Lagrangian of the problem is given by: L = E 0 X t=0 t L + X t log C t +E 0 t [w t L t + ( + r t ) K t C t K t+ + ( ) K t ] + t=0 The FOCs = t C t t = 0 t = 0; ; t = t L t + t w t t+ = t + E t [ t+ ( + r t+ + )] = 0 lim 0 ( t K t ) t! = 0 the F.O.Cs. w.r.t C t and K t+ lead to the standard Euler equation: C t = E t + rt+ C t+ From the rst two F.O.Cs we obtain the labor supply function: L t = wt C t (8) 6

17 As in standard microeconomic theory labor supply increases with real wage and decreases with consumption. NOTE THAT: According to equation (8) households try to equate the marginal rate of substitution between labor and consumption to the real wage. By consider the ratio of the labor F.O.Cs. in t and t +, we obtain:: t L t t+ L t+ = tw t t+ w t+ (9) Rewrite the Euler equation as follows equation (9): t t+ = ( + r t+ ) and use it into Lt L t+ = ( + r t+) w t w t+ This equation drives the intertemporal substitution of labor supply. Labor supply in period t increases with the actual wage and with the interest rate and decreases with future (expected) wage 7

18 Firms Firms choose labor and capital services to maximize their pro ts: Max = A t Kt L t w t L t r t K t K t L t;k t The F.O.Cs are given by: w t = A t Kt L t = Y t L t r t + = ( ) A t K t L t = ( ) Y t K t The complete model + rt+ = E t Euler C t C t+ C t L t = w t labor supply w t = A t Kt L t = Y t L t r t + = ( ) A t K t L t = ( ) Y t K t labor demand capital demand Y t = C t + I t A.R.C Y t = A t K t L t Prod. func. K t+ = ( ) K t + I t P.I.M A t = A A t et T.F.P. 8

19 The linear model C ss ^Ct = + r ss C ss ^Ct+ + r ss C ss ^r t+ Note that from the Euler equation in s.s. we know that: = ( + r ss ) and r ss =. Hence, ^C t = ^C t+ ( ) ^r t+ ^C t + ^L t = ^w t ^w t = ^Y t ^Lt ^r t = ^Y t ^Kt ^Y t = ^A t + ( ) ^K t + l t ^Y t = C ss ^Ct + I ss ^It ^K t+ = ( ) ^K t + ^I t ^A t = ^A t + t 9

20 The steady state From the complete (non linear) model we know that we have to pin down the parameters,,,, and 2. To do it, we initially write down the complete steady state equilibrium system: = ( + r ss ) C ss L ss = w ss w t = L ss r ss = ( ) K ss = C ss + I ss = A ss Kss L ss K ss = I ss A ss = Let s equate the Euler equation with the demand for capital, and solve it with respect to the ratio Kss = ( + r ss ) r ss = ( ) K ss K ss ( ) = ( ) From the P.I.M. in s.s. we get: From the A.R.C, we get: I ss = K ss ( ) = ( ) C ss = I ss = ( ) ( ) 20

21 Let s equate the demand and labor supply: C ss L ss = L ss and solve it w.r.t L ss L ss = Y + ss C ss Now use the ratio Css into L ss : C ss ( ) ( ) = ( ) ) Yss C ss = L ss = Yss + + = C ss ( ) ( ) ( ) ( ) ( ) ( ) + Let s use factor inputs into the production function: = A ss Kss L ss ( ) = ( ) ( ) ( ) ( ) + Yss = ( ) ( ) ( ) + ( ) ( ) = ( ) ( ) ( ) ( ) ( ) + Now substitute in the great ratios to get the level of C ss, I ss, L ss and w ss as a function of deep parameters. 2

22 Calibrating the model As usual, we start the calibration by targeting the great ratio Iss I ss ( ) = 0:2 = ( ) from the A.R.C. we get the ratio Css = 0:2 = 0:8 There are three parameters to pin down:, and. For, various sources of independent empirical evidence point to 0:025 on quarterly basis. As for, we know from the Euler equation that return, which on quarterly basis is :0062, so as: = :0062 ) = 0:9938 is equal to the capital Let s use these values to solve the great ratio Iss I ss for 0:9938 ( ) = 0:2 = 0:025 0:9938 ( 0:025) = 0:75 What about parameters, and 2? and 2 can be pinned down by estimating the Solow residual, by using other studies, or we can calibrate them to target the volatility and the autocorrelation of the simulated output series. is a crucial parameter that we use to perform a sensitivity analysis Table 3- benchmark parametrisation C ss Target ratio = 0:8 I ss = 0:2 Technology = 0:75 = 0:025 Preferences = 0:3 = 0:9938 TFP = 0:0025 = 0:9 22

23 Model performance Relative Standard Dev. Corr ^Y ; ^Xt Variable Model ( = 0:3) Model ( = 3) U.S. Data Model ( = 0:3) U.S. Data by :57 bl = by 0:45 0:4 0:66 0:92 0:2 bwm = by 0:60 0:86 0:42 0:95 0:2 0:9 with =0:03 bc = by 0:5 0:5 0:80 0:89 0:82 bi = by 3:30 3:3 4:70 0:96 0:74 Labor market mechanism can be graphically described as follows: w S Lt low elasticity (sigma=3) w(l) S t L high elasticity (sigma=0.3) w(h) w ( > 0 L D ε ) t L L(l) L(h) L 23

24 Problems with RBC models: by construction:. It predicts that all unemployment is voluntary. 2. No place for monetary policy. 3. RBC models cannot reproduce the low correlation between output and wages. Problems with calibration:. To match labor market volatilities we need a very at labor supply curve. With steeper labor supply curves, the model does much worse. 2. Consumption volatility puzzle. Excessive consumption smoothing 3. The size of the shocks used in the RBC model to match the data are unrealistically large (lack of ampli cation mechanism). Although technology may advance at an irregular pace, there is no retreat in technological progress to explain recessions. Big recessions require big supply shock 24

25 4 Fiscal policy in RBC model Households max fc t;l t;k t+g t=0 " X U t = E 0 t L + # t log C t B 0 + t=0 s.t.: w t L t + (q t ) K t = C t + K t+ ( ) K t T R t The La- where T R t are net transfer and q t is the rental cost of capital. grangian of the problem is given by: X L = E 0 t L + t log C t B t=0 X +E 0 t [w t L t + (q t ) K t C t K t+ + ( ) K t + T R t ] t=0 The FOCs = t C t t = 0 t = 0; ; t = t B 0 L t + t w t t+ = t + E t [ t+ (q t+ + )] = 0 lim 0 ( t K t ) t! = 0 Use consumption and capital F.O.C to obtain the Euler equation: = E t C t + qt+ C t+ From the rst two F.O.Cs we obtain the labor supply function: 25

26 B 0 C t L t = w t (20) According to equation (20) households try to equate the marginal rate of substitution between labor and consumption and the real wage. Equation (20) can be written as: L t = wt B 0 C t As in standard microeconomic theory labor supply increases with real wage and decreases with consumption. 26

27 Firms: Pro t maximization: Max = ( t ) A t Kt L t w t L t q t K t L t;k t w t = ( t ) A t K t L t = ( t ) Y t L t q t = ( t ) ( ) A t K t L t = ( t ) ( ) Y t K t Government The government budget constraint is: G t = t Y t T R t where t is a (distortionary) tax rate. T R t are net government transfe (or lump sum taxes)r. G t denotes government purchases. Using the households resource constraint together with rms pro t and the government budget constraint, we obtain the aggregate resource constraint: Y t = C t + I t + G t In the case of lump-sum taxes put t = 0:This implies that public expenditure is completely nanced by negative (lump sum) transfers from the household to the government constraint (T R t = G t ). 27

28 The complete model = E t C t L t = wt B 0 C t + qt+ C t+ w t = ( t ) A t K t L t = ( t ) Y t L t q t = ( t ) ( ) A t K t L t = ( t ) ( ) Y t K t Y t = C t + I t + G t G t = t Y t T R t Y t = A t Kt L t K t+ = ( ) K t + I t A t = A ss A t et G t = G G ss G G t e G t 28

29 The steady state Let s use the Euler equation and the demand for capital to determine K ss : = + q ss q ss = ( ss ) ( ) K ss K ss = ( ss) ( ) ( ) From the P.I.M. in s.s. we get Iss : By denoting Gss C ss = I ss = K ss = ( ss) ( ) ( ) = g, from the A.R.C, we get Css : I ss G ss = g ( ss) ( ) ( ) Let s equate the demand and labor supply and solve it w.r.t L ss B 0 C ss L ss = ( ss ) L ss = Now use the ratio Css into L ss : L ss = L ss ( ss ) B 0 C ss + + ( ss ) ( ( )) B 0 ( g) ( ( )) ( ss ) ( ) Let s use the factor inputs (L ss and K ss = production function: ( ss)( ) ( ) ) into the = ( ss ) ( ) ( ) B 0 ( ss ) ( ( )) ( g) ( ( )) ( ss ) ( ) Now substitute in the great ratios to get the level of C ss,k ss, I ss, G ss as a function of deep parameters. + 29

30 Calibrating the model From data we set Gss = g = 0:2 and I ss = 0:2 From previous studies we set ss = 0:222, = 0:99 and = 0:025, from the Euler equation = + q ss, hence q ss = 0:035.Since = 0:025, this implies: 0:2 = Iss = Kss ) Kss = 8 From the s.s relations we know that: K ss = ( ss) ( ) ( ) Let s use the calibrated values of, and ss to solve the great ratio Kss w.r.t., so as to obtain = 0:64 From the A.R.C C ss = I ss g = g ( ss) ( ) = 0:6 ( ) Let s use the labor market equilibrium to x parameter B 0 in order to target the overall time devoted to work at L ss = 0:33, that can be interpreted as 2=3 of the overall available time of 24 hours. L ss = B 0 = + ( ss ) ( ( )) B 0 ( g) ( ( )) ( ss ) ( ) ( ss ) ( ( )) (L ss ) + ( g) ( ( )) ( ss ) ( ) Di erent values for, imply di erent values for B 0 30

31 The linear model ^C t = E t ^Ct+ ( ( )) E t [^q t+ ] ^L t = ^w t ^Ct ^w t = ^Y t ^Lt ss ss ^ t ^q t = ^Y t ^Kt ss ss ^ t ^Y t = C ss ^Ct + I ss ^It + g Y ^G t ss G ss ^Gt = ss ^ t + ^Y t T R sst d Rt ^Y t = ^A t + ( ) ^K t + ^L t ^K t+ = ( ) ^K t + ^I t ^A t = ^A t + t ^G t = G ^Gt + G t Put in the PC and simulate a permanent (i.e. G = ) and a transitory (i.e. G < ) government spending shock, nanced with lump sum (i.e. t = 0 and T R t = G t ) and distortionary taxation (i.e. T R t = 0 and t Y t = G t ). 3

32 Permanent increase in public expenditure - nanced by lump sum taxes Since taxes are lump-sum, all that matters is the present discounted value of taxes, not their timing; Accordingly we can assume that the government budget is always balanced on a period-by-period basis Permanent increase in G with lump sum taxes

33 A permanent increase in G t acts as a permanent negative wealth shock since government absorbs resources ) Agents are poorer ) they want to work more ) the increase in labor supply is permanent ) higher return on capital higher investment more in details: C t decreases on impact following the negative wealth shock, then a partial recovery is made possible by a larger stock of capital which increases output and partially o set the reduction in wealth at a later stage, but in the s.s. is still lower than in the old one. L t increases on impact due to the negative wealth e ect. The reaction of L t triggers the ampli cation e ect of capital since its marginal product depends on the Lt K t ratio. Then, as K t increases, the individual wealth also increases and L t starts falling back to the steady state from t +. However in the long run labor supply is still higher than in the old steady state, because the capital stock is increased. The real interest rate, is high but declining along the transition path (see the Euler equation, where C decreases). Its sharp initial increase is due to the increase in the marginal product of capital, in turn allowed by the increase in employment. In the long run the capital-labour ratio is unchanged I t increases permanently, as a higher level of capital stock has to be maintained, after the investment boom. w t decreases on impact as a consequence of the increase in labour supply. However, as capital-labour ratio go back to the steady state value, it returns to steady state. Y t increases on impact, since with predetermined capital the response of output is a function of the response of labor supply. The increase is permanent since factor inputs permanently increase..the multiplier can be greater or lower than one depending on the strength of the capital accumulation reaction. 33

34 Transitory increase in public expenditure nanced by lump sum taxes Transitory increase in G with lump sum taxes and high elasticity of L t

35 Transitory increase in G with lump sum taxes and low elasticity of L t L t increases on impact due to the increased government absorption of resources, but less than in the case of a permanent increase in G because the wealth e ect is now smaller and transitory. From t + on, the labor supply start falling back towards the old steady state, since there is no wealth e ect in the long run. C t decreases on impact following the negative wealth shock. Then it starts recovering and it moves back to the old steady state. I t decreases because labor supply increases little and consumption decreases little, and there are less resources available for private use. Then it starts recovering and it moves back to the old steady state. (since the shock is transitory, so it is the negative wealth e ect, but agents want to smooth their consumption ) they work slightly more and invest less) ^w t decreases on impact as a consequence of the increase in labour supply. Y t increases on impact following the response of labor supply but much less than in the permanent spending shock case, because the response of labor supply is smaller. After this, it declines steadily towards the old steady state. 35

36 Sensitivity analysis Check the following: - If shock persistence increases (greater g ), output and labor responses more persistent. since the wealth e ect of the shocks is expected to last more. - If steady state share of G falls (lower g), dynamics almost una ected. -If elasticity of labor supply is large (lower ), hours respond more for a given wealth e ect. -NOTE THAT: The di erence between the e ect of a permanent and a transitory shock is in the response of investment, which in turn, depends on the response of the marginal product of capital (MPC). When employment changes very little (as in the case of a transitory shock) the MPC remains almost constant and the increase in the return of investment (q t ) is not su cient to induce more savings and more investment. When the employment response is strong (as in the case of a permanent negative wealth e ect) the positive e ect in the return of investment induces households to postpone consumption and to increase savings. The less is the importance of capital in production (greater ), the larger are the labor and output responses. 36

37 Transitory increase in public expenditure nanced with distortionary taxes the increase in G, nanced by a corresponding increase in the tax rate, reduces agent s incentive to work and invest, therefore reducing the tax base. As a result, the tax rate must increase relatively more to nance the increase in G. There is a strong incentive to substitute intertemporally work e ort (i.e. to postpone labour) and to reduce investment during this period. The distortionary e ect of taxes induces a reduction of labour and output which does not occur in the case of lump-sum taxes. Investment declines during the shock both because of the absorption of resources by government spending and because of low rates of return. Hence the negative e ect here is ampli ed. 37

38 Permanent increase in public expenditure - nanced with distortionary taxes As in the case of transitory increase: Y t, C t, I t, L t, w t and q t decreases but now contractions are permanent since the distortionary e ect of taxes are permanent. 38

39 Fiscal multipliers The scal multiplier is a measure of by how much real GDP changes for an extra unit of government spending. impact multiplier: ^Y t ^G t G ss multiplier at some horizon h: ^Y t+h ^G t G ss cumulative multiplier: TX ^Y t t=0 TX ^G t t=0 G ss or TX t ^Yt t=0 TX t ^Gt t=0 G ss or TX ^Y t t=0 ^G t G ss 39

40 In our model: Permanent increase in public expenditure nanced by lump sum taxes: Impact multiplier: low elasticity of labor supply ( = 5) I.M.= 0:20 high elasticity of labor supply ( = 0:5) I.M = 0:835 Cumulative multiplier: NOTE THAT: scal multiplier the greater is the response of labor, the higher is the 40

41 Transitory increase in public expenditure nanced by lump sum taxes: Impact multiplier: low elasticity of labor supply ( = 5) I.M.= 0:05 high elasticity of labor supply ( = 0:5) I.M = 0:30 Transitory increase in public expenditure nanced with distortionary taxes: Impact multiplier: low elasticity of labor supply ( = 5) I.M.= 0:2 high elasticity of labor supply ( = 0:5) I.M = 0:79 Cumulative multiplier:

42 Some considerations about the scal multipliers: The e ect of government spending shock on key macroeconomic variables is still an open issue in theoretical and empirical literature. The methodological set-up in estimating scal multipliers determines to a signi cant extent the size of the estimated multiplier. From the introduction of: Jordi Galí & J. David López-Salido & Javier Vallés, "Understanding the E ects of Government Spending on Consumption," Journal of the European Economic Association: "Empirical evidences show that scal expansion crowds out private investment but increases consumption leading to positive scal multipliers. Real Business Cycle models have di culties in accounting for a positive scal multiplier. Both private investment and consumption are crowded out by rising interest rate. The main mechanism behind Ricardian model is the negative wealth e ect associated with government spending shocks. In the standard neoclassical model as in Baxter and King (993) an increase in government spending raises the expected net present value of taxes of households. It follows that households decrease consumption and increase labour supply. Though most macroeconomic models imply a positive scal multiplier, they often di er regarding the e ects of government spending on consumption, the largest component of aggregate demand and, hence, a key determinant of the eventual impact of the policy intervention. In that regard, the textbook IS-LM model and the standard RBC model provide a stark example of such di erential qualitative predictions. Thus, the standard RBC model generally predicts a decline in consumption in response to a rise in government spending In a nutshell, an increase in (nonproductive) government purchases ( nanced by current or future lump-sum taxes) has a negative wealth e ect which induces a rise in the quantity of labor supplied at any given wage. That e ect leads, in equilibrium, to a lower real wage, higher employment and higher output. The increase in employment leads, if su ciently persistent, to a rise in the expected return to capital, and may trigger a rise in investment. In the latter case the size of the multiplier is greater or less than one depending on parameter values. On the other hand, the basic textbook IS-LM model predicts the opposite e ect, namely, an increase in consumption (and a decline in investment) as a result of an increase in government spending. The rise in consumption is caused by the higher disposable income generated from the direct e ect of government spending on the level of economic activity, combined with the assumed dependence of consumption on current disposable income. That response has 42

43 the opposite sign to the one implied by the neoclassical model, and will tend to amplify the e ects of the expansion in government spending on output, thereby increasing the e ectiveness of scal policy as a policy tool." Gali, Lopez-Salido and Valles (2007) casted scal policy into a new-keynesian sticky-price model modi ed to allow for the presence of a fraction of households whose consumption depends on current disposable income only. This fraction of households partly insulate aggregate demand from the negative wealth e ects generated by the higher levels of current and future taxes needed to nance the scal expansion. This mechanism can generate a positive response of aggregate consumption. 43

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