Macroeconomics Theory II Francesco Franco FEUNL February 2011 Francesco Franco Macroeconomics Theory II 1/34
The log-linear plain vanilla RBC and ν(σ n )= ĉ t = Y C ẑt +(1 α) Y C ˆn t + K βc ˆk t 1 + K C ˆk t ˆr t = (1 β(1 δ)) ẑ t (1 α)(ˆk t 1 ˆn t ) ˆn t = ν(σ n ) ẑ t + αˆk t 1 ĉ t 0 = E t ( ĉ t+1 +ˆr t+1 ) ẑ t = ρẑ t 1 + t 1 N N σ n 1+α 1 N N σ n measures the responsiveness of labor to shocks that change the real wage or consumption taking into account that the real wage changes with the labor supply Francesco Franco Macroeconomics Theory II 2/34
Properties of the model Three crucial parameters that control the extent of the amplification effect are: the elasticity of the labor supply to the real wage: 1 N N The persistence of the exogenous shock: ρ The elasticity of intertemporal substitution: σ σ n Francesco Franco Macroeconomics Theory II 3/34
Properties of the model Figure: Simple RBC, σ = 1,σ n = 0.1,ρ= 0.97 Francesco Franco Macroeconomics Theory II 4/34
Properties of the model Figure: Simple RBC, σ = 1,σ n = 3,ρ= 0.1 Francesco Franco Macroeconomics Theory II 5/34
Properties of the data Francesco Franco Macroeconomics Theory II 6/34
Calibrating the RBC The stochastic TFP shock can be extracted from the empirical Solow residual using ln(sr t )=ln(y t ) (1 α)ln(n t ) αln(k t ) Then assuming that ln(sr t )=ln(z t )+(1 α)ln(t t ),where ln(t t )=ln(t t 1 )+γ one can extract the estimated AR(1) process for log(z t ). Rebelo and King (1999) find ρ = 0.979 and standard deviation of 0.072 $ 0("+'1#363#7!8"%"9 (/)3+'$%" $*+,'#-'# % & 0/(1/*# ' "& "% '$()$( 4%+5$6("7"(8 "$ "# %( )& )( $& $( (& (( #& #( *&.$#/ Francesco Franco Macroeconomics Theory II 7/34 9+(&, :,;)-& )&%"+5 "/ +*%(,+ " +**$,%- <-- 7,%",=-&/,%& 5&(%&35&5 $/"3# (>&.+5%"6?"4%&/6+(( @"-(&%- Figure: Solow residual quarterly
Calibrating the RBC How to choose reasonable set of parameters: 1 microeconomic empirical studies 2 long-run properties of the economy For example choose the discount factor so that the steady state real interest rate coincides with the average return to capital in the economy1 = β R, or in a Cobb-Douglas α will be the share of capital in value added which is tipycally 1/3 Figure: Standard parameters Francesco Franco Macroeconomics Theory II 8/34
Results Figure: Standard calibration of RBC Francesco Franco Macroeconomics Theory II 9/34
Results Figure: Model results Francesco Franco Macroeconomics Theory II 10/34
Evaluation Output is not as persistent in the model as much as in the data: the model lacks a propagation mechanism The standard deviation of total hours in the model is lower than the standard deviation of output, while in the data they have similar magnitude The fluctuation in hours in data can mostly be accounted for by fluctuations in employment, the extensive margin, as opposed to the fluctuations in hours per worker, the intensive margin, (Gary Hansen, (1985), and Rogerson, (1988)) Average labor productivity is highly procyclical in the model, but is much less procyclical in the data Francesco Franco Macroeconomics Theory II 11/34
Evaluation The correlation between hours worked and productivity in the model is well above 0.90, while in the data it is very close to 0 Wages in the data fluctuate substantially less than hours and output, and earnings are not correlated with output. Wages are highly procyclical in the model The standard RBC model is unable to generate comovement, when interpreted as a two sector model with a consumption sector and an investment sector One of the maintained assumption of the standard RBC model is that the technology shock is exogenous. However, the Solow residual, which is the corresponding data statistic for the technology shock, is not exogenous in the data (Hall, 1988). It is correlated with military expenditures, monetary aggregates and government consumption Francesco Franco Macroeconomics Theory II 12/34
Labor supply A few of the above issues are due an implied Labor Supply which is not sufficiently elastic. In the model, workers are not sufficiently willing to substitute for leisure in the current period with leisure in future periods. Hence, the response of hours to changes in real wages and productivity is small: ˆn t = η nw (σ n ) ẑ t + α(ˆk t 1 ˆn t ) ĉ t. This implies that in response to a technology shock, current hours do not react as much as in the data, which determines a rise in the equilibrium real wage and a rise in average labor productivity in the model ŵ t λ t Francesco Franco Macroeconomics Theory II 13/34
Labor supply: Hansen Hansen model (1985) is based on the idea that there might be important non convexities in the production process that might make varying the number of employed more efficient than varying the number of hours Assume an agent can work a fixed number of hours h (0, 1) Agent can work or not work. They pool resources together and perfectly insure consumption In this model hours do not vary, only employment does Francesco Franco Macroeconomics Theory II 14/34
Labor supply: Hansen Let π t the probability that an agent works in period t H t = π t h is the number of per capita hours Denote C u and C e.the consumption of respectively an unemployed and an employed Francesco Franco Macroeconomics Theory II 15/34
Labor supply: Hansen The Planner solves the usual dynamic problem augmented with the following: max π t u(c et, 1 h)+(1 π t )u(c ut, 1) s.t. π t C et +(1 π t )C ut = C t Assume u(c, L) =ln C + γ n ln L. So that labor elasticity at the micro level is unity. The solution implies C u = C e = C t Francesco Franco Macroeconomics Theory II 16/34
Labor supply: Hansen The expected utility of an household can be written U(C, L) = π t ln C t +(1 π t ) ln C t + π t γ n ln(1 h)+(1 π t )γ n ln(1) U(C, L) = ln C t + π t γ n ln(1 h) U(C, L) = ln C t AH t where A = γ n ln(1 h)/ h. Then the Hansen model where labour is indivisible is equivalent to a divisible labor model with preference U(C, L) For the representative agent because instantaneous utility is linear in H. Hence, the representative agent only cares about the present discounted utility cost of labor supply and is indifferent between different sequences of labor supply over her lifetime that deliver the same present discounted utility cost The labor supply is infinitely elastic at the aggregate level Francesco Franco Macroeconomics Theory II 17/34
Labor supply: Hansen Output in the indivisible labor (IL) model is considerably more volatile than the standard model ( 1.70% for calibrated parameters) The ratio of labor volatility over average labor productivity increases substantially in the IL model However, the correlation between hours and productivity, which is lower than in the standard RBC model, is still approximately 0.70, much higher than it is in the data Francesco Franco Macroeconomics Theory II 18/34
Labor supply shocks The correlation between hours worked and productivity in the model is well above 0.90 and 0.7 in Hansen model, while in the data it is very close to 0. The standard RBC model is driven by a single shock that shifts the labor demand along the labor supply and induce a positive relationship between hours and productivity To break the relationship you need a second type of shocks that shift the Labor supply The easiest way is to introduce shocks that induce a negative wealth effect on individuals making them working more. (Christiano and Eichenbaum 1992) Francesco Franco Macroeconomics Theory II 19/34
Labor supply shocks Assume government spending is governed by ln G t+1 =(1 λ) ln Ḡ + λ ln G t + ε g t where ε g t is iid N(0,σ g ) and independent of ε z t the technology shock Government spending is financed with lump-sum (non-distortionary) taxes levied on household. The government s budget is balanced in each period The resource constraint for this economy is: C t + I t + G t = Y t In the original paper U( C, L) where C = C(C, G). The case where C = C + G is the standard RBC since increases in G can be offset by reduction in C and all other variables will not change. The case of imperfect substitution is the relevant one. Works with C = C Francesco Franco Macroeconomics Theory II 20/34
Labor supply shocks Francesco Franco Macroeconomics Theory II 21/34
Endogeneity of the Solow residual We have measured technology using a Solow residual. Suppose the production function is of the form: Y = F (K, L, Z) where Z is the index of technology level. To measure the contribution of Z to Y we differentiate the production function and rearrange it to get: dy Y = F K K Y dk K + F NN Y dn N + F Z Z Y dz Z. Francesco Franco Macroeconomics Theory II 22/34
Endogeneity of the Solow residual Assume there is perfect competition and no adjustment cost to factor of production, then firms price according to marginal cost P = W /F N = R/F K which gives us replacing it: dy Y = RK PY s K dk K + WN PY s L dx X dn N + F Z Z Y dz Z S where s K,L is the share of capital or labor costs in output (they can vary in time) and S is the Solow residual, or our measure of technology. dx X is a weighted sum of input growth Francesco Franco Macroeconomics Theory II 23/34
Endogeneity of the Solow residual Our measure of technology is simply: S = dy Y dx X If we construct the residual in this way we get a highly procyclical residual. No need for estimation, or to know anything about the production function Francesco Franco Macroeconomics Theory II 24/34
Endogeneity of the Solow residual The two series are highly correlated and actually exhibit a similar volatility which means that the shock we are feeding into the model is very similar to what we are try to explain The high volatility of TFP has been questioned on the basis that it would imply implausibly large technological shocks and shows the lack of amplification of the RBC Francesco Franco Macroeconomics Theory II 25/34
Endogeneity of the Solow residual Good for Long Run but Is it reasonable to construct it to estimate technological change from year to year, or quarter to quarter? adjustment costs to capital and labor imperfect competition in which case firms price a markup over marginal cost intensity utilization of the factor of productions that are not measured by changes in their quantities Francesco Franco Macroeconomics Theory II 26/34
Endogeneity of the Solow residual What markup pricing imply: Assume P =(1 + µ)mc for there is imperfect competition in the goods market. Then the true Solow residual is: S = dy Y (1 + µ)dx X while the measured Solow residual, call it S m, is as before. Then if µ>0 and we overestimate its procyclicality if the inputs are procyclical: S = S m µ dx X. Francesco Franco Macroeconomics Theory II 27/34
Endogeneity of the Solow residual Unobserved inputs: Assume N = BHE where B is the number of workers, H is output per worker and E is effort. S = dy Y and we do not observe E. then a K dk K + α N db B + dh H + de E S = S m α N de E. Same if K = u K where u is capacity utilization Francesco Franco Macroeconomics Theory II 28/34
Endogeneity of the Solow residual Consider the following preferences: U(C, L, E) =ln C + A ln(1 ξ E t h) where 0 E t (1 ξ)/ h is effort, h is the fixed number of hours, and ξ is a fixed cost. The technology is: Y t = Z t K α t 1 hπ t E t 1 α, where π t is the faction of number employed. Francesco Franco Macroeconomics Theory II 29/34
Endogeneity of the Solow residual Two shocks: ln G t+1 =(1 λ) ln Ḡ + λ ln G t + ε g t, that are orthogonal The resource constraint is ln Z t+1 =(1 ρ) ln Z + ρ ln Z t + ε z t C t + K t (1 δ)k t 1 + G t = Y t As in the indivisible labor model, individuals face a lottery of employment, with probability π t of becoming employed. However, they are insured against unemployment, so their consumption is independent from their employment status Francesco Franco Macroeconomics Theory II 30/34
Endogeneity of the Solow residual π t must be chosen before g t and z t are known Then, when shocks to government consumption or technology hit, firms adjust along the intensive margin, by changing effort, rather then on the extensive margin,by changing employment Here the Solow residual S t = Z t E 1 α and notice that Y t /H t = Z t E 1 α t t Kt 1 hπ t α Francesco Franco Macroeconomics Theory II 31/34
Endogeneity of the Solow residual Any variable which is correlated with E will be correlated with S, even if they are not correlated with Z In the model, E responds positively to a rise in G, dueto the presence of a negative wealth effect on the supply of effort in the preferences of the representative agent Then, both the S and Y /H will rise in response to a G shock. Standard RBC accounts falsely attributes this rise to a positive technology shock. E is correlated with innovations in G and Z. Then,S will be correlated with innovations in G Francesco Franco Macroeconomics Theory II 32/34
Endogeneity of the Solow residual If a positive technology shock occurs effort will rise. The rise in effort implies that the resulting increase in the Solow residual and in average labor productivity is greater than the technology shock Then, the endogenous response of effort provides an amplification mechanism for the technology shock Francesco Franco Macroeconomics Theory II 33/34
References Robert G. King & Sergio T. Rebelo, 2000. "Resuscitating Real Business Cycles," NBER Working Papers 7534, National Bureau of Economic Research Hansen, Gary D., 1985. "Indivisible labor and the business cycle," Journal of Monetary Economics, Elsevier, vol. 16(3), pages 309-327, November Lawrence J. Christiano, Martin Eichenbau, Current Real-Business-Cycle Theories and Aggregate Labor-Market Fluctuations. The American Economic Review, Vol. 82, No. 3 (Jun., 1992), pp. 430-450 Francesco Franco Macroeconomics Theory II 34/34