Neoclassical Business Cycle Model

Transcription

1 Neoclassical Business Cycle Model Prof. Eric Sims University of Notre Dame Fall / 36

2 Production Economy Last time: studied equilibrium in an endowment economy Now: study equilibrium in an economy with production Will produce operational model that can be used to compare to the actual behavior of the economy in the short run 2 / 36

3 Equilibrium Definition still the same: set of prices and quantities consistent with (i) agents optimizing, taking prices as given, and (ii) markets clearing Agents: household, firm, government Large number of each kind of agent, but identical: price-taking behavior, can study representative agent problem Time lasts for two periods: present, t, and future, t / 36

4 Firm Produce output using Y t = A t F (K t, N t ) Take real wage, w t, as given Time subscript on A t : allow it to change period-to-period Different than Solow model, assume that firms own capital stock and make capital accumulation (investment) decisions Would get same results if household owned capital stock as in Solow Model 4 / 36

5 Capital Accumulation Same equation as before, with one twist: K t+1 = qi t + (1 δ)k t q: investment shock Measure of how good we are at transforming investment into capital One way to think about financial system health Assume it is the same in t and t + 1, differently than A t Terminal condition: K t+2 = 0 I t+1 = (1 δ)k t+1 q. Intuition. 5 / 36

6 Profits and Firm Value Profit: Π t = Y t w t N t I t Firm value: present value of profit/dividend: V t = Π t r t Π t+1 Firm: picks N t, N t+1, and K t+1 to maximize V t 6 / 36

7 Firm First Order Conditions Optimality conditions: w t = A t F N (K t, N t ) w t+1 = A t+1 F N (K t+1, N t+1 ) 1 = 1 (qa t+1 F K (K t+1, N t+1 ) + (1 δ)) 1 + r t Intuition: marginal benefit = marginal cost 7 / 36

8 Labor Demand First two first order conditions imply labor demand curves Labor demand is static : depends only on current period stuff Decreasing in the real wage Labor demand shifts out if A t goes up Labor demand would shift in if K t were destroyed (natural disaster) N t = N d (w t, A t, K t ) 8 / 36

9 Labor Demand ww tt NN dd ww tt, AA tt, KK tt NN tt 9 / 36

10 Investment Demand The last first order condition implicitly defines an investment demand curve Investment a decreasing function of r t Curve shifts out if A t+1 or q go up Curve also shifts out if K t goes down exogenously (natural disaster) Investment fundamentally forward-looking I t = I d (r t, A t+1, q, K t ) 10 / 36

11 Investment Demand rr tt II dd rr tt, AA tt+1, qq, KK tt II tt 11 / 36

12 Household Problem basically the same, but now household chooses amount of labor/leisure Normalize total endowment of time to 1 each period Leisure is 1 N t, where N t is hours worked Household gets utility from leisure via v(1 N t ), with v (1 N t ) > 0 and v (1 N t ) 0 Lifetime utility: U = u(c t ) + v(1 N t ) + β (u(c t+1 ) + v(1 N t+1 )) 12 / 36

13 Budget Constraints Basically look same, but have to account for endogenous income now Household income comes from wages, dividend/profit from firm, and pays taxes to government C t + S t = w t N t T t + Π t C t+1 = w t+1 N t+1 T t+1 + Π t+1 + (1 + r t )S t Combine into one: C t + C t r t = w t N t T t + Π t + w t+1n t+1 T t+1 + Π t r t 13 / 36

14 Household First Order Conditions Household chooses C t, C t+1, N t, and N t+1 to maximize lifetime utility. Optimality conditions: u (C t ) = β(1 + r t )u (C t+1 ) v (1 N t ) = u (C t )w t v (1 N t+1 ) = u (C t+1 )w t+1 Consumption Euler equation: same as it ever was: same consumption function Two new conditions: implicitly define labor supply curves in each period 14 / 36

15 Labor Supply Condition v (1 N t ) = u (C t )w t implicity defines labor supply curve Can analyze in indifference curve-budget line diagram Changes in w t : complicated effect because offsetting income and substitution effects Assume that substitution effect dominates: N t increasing in w t Simple rational: MPC is less than 1, so C t reacts less than one-for-one to one period change in w t Easy to see with log utility over consumption Let H t denote an exogenous labor supply shifter (could be preferences or related to policy like unemployment insurance). Assume increase in H t leads to more labor supply 15 / 36

16 Labor Supply ww tt NN ss ww tt, HH tt NN tt 16 / 36

17 The Government Same as before. G t and G t+1 chosen exogenously Government s intertemporal budget constraint: G t + G t r t = T t + T t r t Ricardian Equivalence holds: household behaves as though government balances budget every period 17 / 36

18 Equilibrium Conditions Labor demand: N d = N(w t, A t, K t ) Labor supply: N s = N(w t, H t ) Consumption: C t = C (Y t G t, Y t+1 G t+1, r t ) Investment: I t = I (r t, q, A t+1, K t ) Production function: Y t = A t F (K t, N t ) Market-clearing: Y t = C t + I t + G t Six endogenous variables (N t, Y t, C t, I t, w t, and r t ) and six equations Exogenous variables: A t, A t+1, q, G t, G t+1, K t 18 / 36

19 The Y s Curve Set of (r t, Y t ) pairs consistent with production function where labor market clears Will be vertical just like before, but a little more nuanced Basic idea of derivation: Start with an initial r t. See how this affects labor market-clearing (it doesn t) Plug this value of N t into the production function to get Y t Try a different value of r t Connect the dots: vertical Y s 19 / 36

20 The Y s Curve ww tt rr tt NN ss (ww tt, HH tt ) YY ss ww tt 0 NN dd (ww tt, AA tt, KK tt ) YY tt YY tt YY tt = AA tt (KK tt, NN tt ) YY tt YY tt = YY tt YY tt 20 / 36

21 The Y d Curve Set of (r t, Y t ) pairs consistent with agent optimization and Yt d = Y t, where Yt d = C t + I t + G t Basic idea of derivation: Use the expenditure line - 45 degree line diagram. Start with an r t, determines position of expenditure line Increase r t. Causes expenditure line to shift down both because of C t and I t. Intersects 45 degree line at lower point Hence, Y d slopes down 21 / 36

22 The Y d Curve YY tt dd YY tt dd = YY tt YY tt dd = CC YY tt GG tt, YY tt+1 GG tt+1, rr tt 1 + II rr tt 1, AA tt+1, qq, KK tt + GG tt YY tt dd = CC YY tt GG tt, YY tt+1 GG tt+1, rr tt 0 + II rr tt 0, AA tt+1, qq, KK tt + GG tt YY tt dd = CC YY tt GG tt, YY tt+1 GG tt+1, rr tt 2 + II rr tt 2, AA tt+1, qq, KK tt + GG tt rr tt YY tt rr tt 2 rr tt 0 rr tt 1 YY dd YY tt 22 / 36

23 General Equilibrium General equilibrium requires that all markets clear Effectively two markets here: labor (N s = N d ) and goods (Y d = Y ) Labor market-clearing: on Y s curve Goods market-clearing: on Y d curve General equilibrium: on both curves w t and r t simultaneously determined 23 / 36

24 General Equilibrium: Graphically ww tt ww tt 0 rr tt NN ss (ww tt, HH tt ) rr tt 0 YY ss YY tt NN tt 0 NN dd (ww tt, AA tt, KK tt ) YY tt = AA tt (KK tt, NN tt ) YY tt YY tt 0 YY dd YY tt = YY tt YY tt YY tt 24 / 36

25 Curve Shifts and Equilibrium Changes Effectively five exogenous variables: A t, A t+1, q, G t, and G t+1 (changes in K t cause both Y d and Y s to shift, so won t analyze here) What shifts what: Supply shock: change in A t or H t : shifts Y s curve Demand shock: change in A t+1, q, G t, or G t+1 : shifts Y d curve With Y s vertical, demand shocks don t affect output, just real interest rate and composition of output To account for observed fluctuations, model must primarily be driven by supply shocks 25 / 36

26 Effects of Changes in Exogenous Variables Variable: A t A t+1 q G t G t+1 H t Output Hours Consumption +? Investment +? Real interest rate Real wage Changes in A t and H t have similar effects except for behavior of real wage Have to look at equations to figure out how C t and I t react 26 / 36

27 Taking Model to the Data The neoclassical model of the business cycle we have developed is sometimes called the real business cycle model It is real in the sense that there are no nominal prices Fluctuations in output must be driven by supply shocks, not demand shocks Can the model provide an empirically realistic account of observed business cycles? To extent to which it can, what are the policy implications? 27 / 36

28 Trend vs. Cycle Hodrick-Prescott Filter GDP Trend Cycle / 36

29 Business Cycle Correlations Series Correlation with GDP GDP 1.00 Consumption 0.78 Investment 0.85 Hours 0.87 Real Wage 0.14 Real Interest Rate Quantities strongly procyclical (positively correlated with output) Real interest rate mildly countercyclical Real wage mildly procyclical Probably understates true cyclicality of real wage due to composition bias 29 / 36

30 Can Model Reproduce These Correlations? Potentially, yes Has to be predominantly driven by fluctuations in productivity, A t Demand shocks don t impact output Labor supply shocks produce wrong cyclicality of real wage Doesn t mean only productivity shocks, just means these must be the predominant shock 30 / 36

31 Measuring TFP Any evidence that A t moves around a lot in data? Assume Cobb-Douglas production function Given observed Y t, N t, and K t, and a bit of theory for α, can measure in data Solow residual / Total Factor Productivity (TFP): ln A t = ln Y t α ln K t (1 α) ln N t 31 / 36

32 TFP Shocks in Data TFP and Output.04 Percent Deviation from Trend Filtered GDP Filtered TFP Correlation b/w TFP and output: 0.8 Evidence in support of model 32 / 36

33 Planner s Problem What would outcome of economy be if fictitious social planner chose allocations to maximize household well-being? This tells us what best or most efficient allocation is No prices (real or nominal) in planner s problem max U = u(c t ) + v(1 N t ) + βu(c t+1 ) + βv(1 N t+1 ) C t,c t+1,n t,n t+1,k t+1 C t + K t+1 q C t+1 (1 δ) K t+1 q s.t. (1 δ) K t q + G t = A t F (K t, N t ) + G t+1 = A t+1 F (K t+1, N t+1 ) 33 / 36

34 Planner s Solution v (1 N t ) = u (C t )A t F N (K t, N t ) v (1 N t+1 ) = u (C t+1 )A t+1 F N (K t+1, N t+1 ) u (C t ) = βu (C t+1 )(qa t+1 F K (K t+1, N t+1 ) + (1 δ)) Same outcome as competitive equilibrium 34 / 36

35 Policy Implication Competitive equilibrium is efficient. Best society can do Economy may not like reductions in A t, but given that, response is as good as can be hoped for Implication: no need for stabilization policy. Can only make things worse 35 / 36

36 Criticisms of RBC Theory RBC theory heavily criticized. Non-exhaustive list: Does not generate enough hours/employment volatility No monetary non-neutrality No role for demand shocks What does it mean for A t to decline? Is A t really well-measured? Unobserved factor utilization No heterogeneity 36 / 36