Macroeconomics Theory II Francesco Franco FEUNL February 2016 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 1 / 18
Road Map Research question: we want to understand businesses cycles. DSGE methodology: produce a laboratory that fits the data and allows policy experiments. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 2 / 18
Road Map Starting point: one sector growth model with stochastic productivity (Ramsey). Kydland and Prescott (1982) argue that shocks in a competitive economy can jointly explain the cyclical and the long-run properties of the data. 1 Shocks, Uncertainty: much of what happens is unexpected, and natural way to get booms and slumps. 2 Basic intertemporal choice between Consumption and Saving and Labor-Leisure. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 3 / 18
Road Map Useful because: To understand frictions it is important to understand the dynamic effects of shocks in a competitive economy. Class by class we will augment the model with labour markets and other ingredients such as more realistic investment decisions, frictions, heterogeneity. Supply shocks can nevertheless play a role. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 4 / 18
Objects to be explained Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 5 / 18
Objects to be explained Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 6 / 18
Objects to be explained Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 7 / 18
Objects to be explained Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 8 / 18
Benchmark Competitive Model The representative agent (there is one) experiences utility according to: U = E " Â i=0 b i u(c t+i ) W t # where C t is consumption, 0 < b < 1isthesubjectivediscountfactorand W t is all the information at time t. FromnowonIwilluseE t [.] =E [. W t ] to indicate the conditional expectation: Rational Expectations, i.e. the mathematical expectations using all available information. Example: roll a dice at time t the result is dt 1 roll again a dice at time t + 1theresultis dt+1 2 Define the sum S t+1 = dt 1 + dt+1 2. E t 1 [S t+1 ]=7; E t [S t+1 ]=3.5 + dt 1. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 9 / 18
Notes Notice that we are assuming: additive time separability: consumption habit (almost every current model), history of consuption affects the MU of current consumption. time preferences: heterogeneity, calendar time, distance and time consistency. All variables are assumed to be stationary (trending variables have been filtered or de-trended). Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 10 / 18
Benchmark Competitive Model Output,Y t,isproducedaccordingtoaneoclassicalproductionfunction: Z t F (K t 1, N t ). K t 1 is capital chosen in t 1andavailableint, N t is labor and Z t is total factor productivity. Capital accumulates according to: K t (1 d)k t 1 = I t where I t is gross investment and 0 apple d apple 1isthedepreciationrate.Z t will be the source of our randomness and evolves according to G (Z ). Output can be used both for consumption and investment: C t + I t apple Y t. Each period households are endowed with one unit of labor,n t = 1, and they have K 1 before t = 0. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 11 / 18
Information and problem All variables with a subscript t (except the exogenous process Z t )haveto be chosen based on W t.the social planner solves: # " Â max t b {C t+i,k t+i } i=0e i u(c t+i ) i=0 s.t. C t + K t = Z t F (K t 1, N t )+(1 d)k t 1 N t = 1 K 1, Z 0 Z t G (Z ) Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 12 / 18
Solution To talk about existence, uniqueness, etc, etc, you need to use Stochastic Dynamic programming: V (K, Z )=max K 0 U(ZF (K, 1)+(1 d)k K 0 )+ b V (K 0, Z 0 )g(z 0 Z )dz. Here we bypass DP foundations and focus on the necessary conditions for optimality. Form the Lagrangian: L = max E t {C t+i,k t+i } i=0 and take the foc. 2 4 Â b i i=0 0 @ apple u(c t+i ) Ct+i + K l t+i Z t+i F (K t 1+i, 1) t+i (1 d)k t+i 1 13 A5 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 13 / 18
Optimality the first order conditions: L l t : C t + K t = Z t F (K t 1, 1)+(1 d)k t 1 L C t : u C (C t )=l t L K t : l t = be t [l t+1 (Z t+1 F K (K t, 1)+(1 d))] TVC : lim t b T l T K T = 0 T! Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 14 / 18
Optimality u C (C t )=l t marginal value of consumption must be equal to the marginal value of wealth. We can define the expected gross return on capital: and the second foc becomes: R t = Z t F K (K t 1, 1)+(1 d) l t = be t [l t+1 (Z t+1 F K (K t, 1)+(1 d))] marginal value of capital must be equal to the marginal value of capital tomorrow plus the expected gross return on capital as of today. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 15 / 18
CRRA Convenient to parametrize the model using a CRRA utility function for the utility: u(c )= C 1 h 1 1 h where h is the coefficient of relative risk aversion and s = 1 h elasticity of intertemporal substitution. is the apple h Ct 1 = be t R t+1. C t+1 In its stochastic version smoothing and tilting still apply and you have to augment the analysis with second moments. Link between Finance and Macro. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 16 / 18
CRRA You have seen it without uncertainty: and talked about: 1 smoothing: br S 1, 2 tilting: s! (0, ). C t+1 C t = (br t+1 ) s, Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 17 / 18
Steady State Non stochastic steady state:z = Z (all variables with an upper bar are steady state value variables). Define the discount rate r : b = 1/(1 + r). R = 1 b ZF K ( K, 1) d = r Ȳ = ZF( K, 1) C = Ȳ d K Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
Effects of a permanent increase in Z... MPK increases so that K has to increase. Output increases both for Z and K (amplification?). On C ambiguous effect: Income effect pushes consumption up, substitution effect pushes consumption down. Increases Y, S and I. If the shock is transitory C responds less, S more. Do the comovements matches the qualitative facts of the business cycle? Try a shock to b. Competitive equilibrium: A sequence {C t, N t, K t, R t, W t } t=0 so that:given K 1 and market W t and return R t the representative household solves # " Â max t b {C t+i,k t+i } i=0e i u(c t+i ) i=0 s.t. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
N s t = 1 C t + Kt s = W t Nt s + R t Kt s 1 lim t R 1 t! t Kt s = 0 (Competitive equilibrium continued )given {R t, W t } t=0 the representative firm solves: max {K d t,n d t } t=0 s.t. Z t F (Z ) Z t F K (K d t 1, 1) W t N d t (R t 1 + d)k d t 1 (Competitive equilibrium continued ) Markets clear: N s t = N d t = N t K s t = K d t = K t C t + K t = Z t F (K t 1, 1)+(1 d)k t 1 We already know by the welfare theorem that this competitive equilibrium corresponds to the optimum we found above. What we learn from studying Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
the decentralized economy is insights on the economics: reaction of households and firms to prices of factors, etc... We could use numerical SDP to try to find the global solution. This technique is not widely used for Business cycles models because not suited for models with externalities or distortionary taxation. Here log-linearize the necessary equations characterizing the equilibrium and solving for the recursive equilibrium law of motion with the method of undetermined coefficients. It basically consists in taking a first order Taylor approximation around the non-stochastic steady state to replace all the equations by approximations which are linear functions in the log deviations of the variables. (log-linear instead of linear for it allows to interpret the coefficient as elasticities being the log-deviations unit-less). Here is an example on how to proceed. Assume Y = ZK a and define Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
ˆx t C t C C : C t + K t (1 d)k t 1 = Z t Kt a 1 ln(c t + K t (1 d)k t 1 ) = ln(z t Kt a 1) C Ȳ ĉt + K Ȳ ˆk t (1 d) K Ȳ ˆk t 1 = ẑ t + a ˆk t 1. ĉ t = Ȳ C ẑt + K C R K ˆk t 1 C ˆk t. The economic interpretation is as before: if ẑ > 0, productivity is higher than in steady state, this allows higher production and the household can afford higher consumption. The coefficient Ȳ transform the percentage C change in total factor productivity in consumption units. Higher productivity increases investment as well and therefore ˆk t will increase and can offset the positive effect of ẑ t on ĉ t. Proceeding in a similar way for each necessary condition you get two more Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
conditions: ˆr t = (1 b(1 d)) ẑ t (1 a) ˆk t 1 0 = E t (h(ĉ t ĉ t+1 )+ ˆr t+1 ). Finally we have to assume a process for total factor productivity. It is convenient to assume an AR(1) process: ẑ t = rẑ t 1 + e t e t N(0, s 2 e ). You could reduce the system in two first order stochastic difference equations in ĉ t and ˆk t or in a second order stochastic difference equation in ˆk t.wewillkeepthesystemasitisandsolveitwiththemethodof undetermined coefficients. Method: given the state variables ẑ t (exogenous) and ˆk t 1 (predetermined), guess a linear recursive law (policy Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
function) for the control variables: ĉ t = n ck ˆk t 1 + n cz ẑ t ˆk t = n kk ˆk t 1 + n kz ẑ t ˆr t = n rk ˆk t 1 + n rz ẑ t. Take the first approximation for the resource constraint: ĉ t = Ȳ C ẑt + K C R K ˆk t 1 C ˆk t and replace the great ratios by their steady state values Ȳ C = rr+d r r+(1 a)d = µ 1 and K R = C (1+r r)a r r+(1 a)d = µ 2. ĉ t = µ 1 ẑ t + µ 2 ˆk t 1 +(1 (1 a)µ 1 µ 2 ) ˆk t {z } µ 3 Doing the same for the other equations: ˆr t = µ 4 ẑ t + µ 5 ˆk t 1 E t Dĉ t+1 = se t ˆr t+1 ẑ t = rẑ t 1 + e t, Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
To solve for the coefficients n for ĉ t : n ck ˆk t 1 + n cz ẑ t = µ 1 ẑ t + µ 2 ˆk t 1 + µ 3 ˆk t n ck ˆk t 1 + n cz ẑ t = µ 1 ẑ t + µ 2 ˆk t 1 + µ 3 n kk ˆk t 1 + n kz ẑ t given that this must hold for every ẑ t and ˆk t 1 we have that: n ck = µ 2 + µ 3 n kk n cz = µ 1 + µ 3 n kz. From the Euler equation we obtain that: 0 = E t (h(ĉ t ĉ t+1 )+ ˆr t+1 ) 0 = E t (h(n ck ˆk t 1 + n cz ẑ t n ck ˆk t n cz ẑ t+1 )+n rk ˆk t + n rz ẑ t+1 ) 0 = (h(n ck ˆk t 1 + n cz ẑ t n ck n kk ˆk t 1 + n kz ẑ t n cz rẑ t ) from which we obtain: +n rk n kk ˆk t 1 + n kz ẑ t + n rz rẑ t ) (n rk hn ck ) n kk + hn ck = 0 (n rk hn ck )n kz + hn cz hn cz r + n rz r = 0 Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
Finally from the ˆr t : n rz = µ 4 n rk = µ 5. Now you have 6 equations and 6 unknowns...solve for n kk you ll get a quadratic equation, choose the stable root: r µ 1 2 µ 3 + s µ 5 µ 1 2 µ 3 µ 3 + s µ 2 5 µ 3 + µ 4 2 µ 3 n kk = 2 2 the other coefficients follow. Looking at the elasticities is ugly but informative on the economics. Notice that n kk (and therefore n ck ) does not depend on the persistence of the technology shock r (but s plays a role). The laws of motion give us the time series model for the endogenous variables. Clearly the exogenous process for ẑ t ultimately determines the processes for the endogenous variables. Assume that 1 < r < 1, then: ẑ t = e t 1 rl. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18
you find that the capital stock follows an AR(2) (determined one period in advance): ˆk t = n kzẑ t (1 n kk L) = n kz e t (1 rl)(1 n kk L) output consumption and the gross return on capital follow an ARMA(2,1) process: ŷ t = (an kz n kk )L + 1 (1 rl)(1 n kk L) e t ĉ t = (n ckn kz n kk n cz )L + n cz (1 rl)(1 n kk L) ˆr t = (n rkn kz n kk n rz )L + n rz (1 rl)(1 n kk L) From the time-series model we can gain some intuition on the working of the model as well on its properties. The only intertemporal mechanism in the stochastic growth model is the consumption saving choice that we discussed in steady state so let s look at it. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18 e t e t.
n ck is an increasing function of s : if capital increases, income increases and this effect does not depend on s; contemporaneouslytherealreturn decreases which induces a positive substitution effect that is stronger the greater is s. difference: now we see the magnitudes. n cz is increasing in r if s is small and decreasing if s is large: with small s the income effect dominates and the latter is stronger the larger is r while is s is large substitution effects are stronger and a very persistent shock increases the real interest rate today and in the future encouraging saving. first model, you knew it: C S decision. still we have learned quite a bit in terms of methods. play with the file example0.m and learn for instance that: no multiplier effect on output. notice that you need a persistent technological shock for capital accumulation to kick in and have important consequences on dynamics. Francesco Franco (FEUNL) Macroeconomics Theory II February 2016 18 / 18