Closed economy macro dynamics: AD-AS model and RBC model. Ragnar Nymoen Department of Economics, UiO 22 September 2009
Lecture notes on closed economy macro dynamics AD-AS model Inflation targeting regime. With reference to an earlier lecture, we know that the money targeting regime will be qualitatively similar. We will return to that regime in the open-econony part of the lectures, because there will be qualitatuve differences between m-targeting and π-targeting in the open economy. Static and adaptive expectations compare with rational expectations in lecture 7. The RBC model IAM: ch 19 IDM: Ch 2.8.4
Dynamic analysis; inf-target & static expectations Stability of solution We start from equation (20) in part II of slide set to closed economy AD-AS model: (y t ȳ) = β(y t 1 ȳ) + β(z t z t 1 ) αβs t (1) with: α = β = α 2 h (1 + α 2 b) 1 αγ + 1 Since both γ and α are positive (as long as h > 0): 0 < 1/(1 + γα) < 1 so we conclude that the solution of (1) is (dynamically asymptotically) stable.
Deterministic solution 0.05 5 percent above equilibrium initally 0.04 0.03 0.02 0.01 Parameter values from IAM p 565 and p IAM 569 α = 0.8 7 8 γ = 0.2 Parameter values set to fit annual data See graph for references to IAM 1900 1920 1940 1960 1980 2000 2020 2040
Stochastic solution y gap (both shocks) y gap (only demand shock).025.000.025 1900 1920 1940 1960 1980 2000 2020 2040 z s α and γ as in the previous graph z t = 0.7z t 1 + ε zt 0.02 0.00 0.02 1900 1920 1940 1960 1980 2000 2020 2040 s t = 0.3s t 1 +ε st ε zt IN(0, 0.01) ε st IN(0, 0.005)
Dynamic analysis I Dynamic multipliers and impulse-responses Assume a steady-state situation initially, in period t 1, with s t 1 = v t 1 = 0. Then v t = 1 but v t+1 = v t+2..all equal to zero. Hence we have a temporary demand shock. Using (1): y t 1 z t = > 0 v t 1 + γα v t 1 y t = 1 z t = v t 1 + γα v t 1 + γα v t y t+2 1 y t+1 = v t 1 + γα v t y t+1 1 1 + γα ( 1 1 + γα 1) z t < 0 v t Set zt v t = 1 The first multiplier is positive, but less than one, since the interest rate is increased in the period of the shock.
Dynamic analysis II Dynamic multipliers and impulse-responses The second multiplier is negative, since SRAS has shifted up in the same period as SRAD shifts back to its original position. All the subsequent multipliers, or impulse responses as they are called in IAM, are also negative, but they are diminishing in magnitude. See figure 19.8 in IAM, which you can try to replicate with the aid of the Excel programs that you have from the Lecture plan page and/or seminars. Inflation responds more smoothly to a temporary demand shock. This due to two features of the model. 1 Demand shocks only affect inflation indirectly, through y t. 2 Movements in y t are smoothed by inflation expectations, which weights heavily in inflation dynamics. See for example figure 19.6 and 19.8 in IAM.
Temporary demand shock: graphical analysis π t π 1 π 2 π * AD LRAS S R A S S R A S 0 2 C y 2 y 0 A y 1 B y t Initial equilibrium in A Temporary shock in period 1 gives equilibrium B Equilibrium in period 3 is in C. Long-run is back in A
Inf-target & static expectations:permanent shocks I In the case of a permanent demand shock, all the multipliers are positive but declining: δ 0 = δ 1 = δ j = 1 1 + γα 1 1 + γα + 1 1 + γα ( 1 1 + γα 1) = ( ) 1 j+1 1 + γα 0 j ( 1 ) 2 1 + γα Besides, we know, from the interpretation of the long-run model, that the long-run multiplier has to be zero. The algebra is seen to confirm this. But what happens in the model?
Inf-target & static expectations:permanent shocks II Note first that due to stability, we know that the system reaches a new stationary equilibrium also after a permanent shock, so we only need to find out what that new steady-state is. Go back to the system of equations: y t ȳ = α 1 (g t ḡ) α 2 (r t r) + v t, (2) r t = i t π e t+1, (3) i t = r + π e t+1 + h (π t π ) + b (y t ȳ), (4) π e t+j = π t+j 1, for j = 0, 1, (5) π t = π e t + γ(y t ȳ) + s t. (6) M t P t = e m 0 Y m 1 t e m 2i t. (7) and note the these equation also hold in steady state.
Inf-target & static expectations:permanent shocks III Then, impose the long-run equilibrium conditions and Finally, define y t = y π t = π = π. v = 0 since a permanent demand shock must be due to g, and s = s, s 0 for the possibility of a permanent change in s in the AS schedule.
Inf-target & static expectations:permanent shocks IV y ȳ = α 1 (g ḡ) α 2 (r r), AD (8) y = ȳ 1 s, AS (9) γ Think of the ȳ, ḡ and r in (8) and (9) as the long-run values that apply before a permanent shock shock. Explicitly: y ȳ 0 = α 1 (g ḡ 0 ) α 2 (r r 0 ), and (10) y = ȳ 0 1 γ s (11) with r and y as the endogenous variables.
Inf-target & static expectations:permanent shocks V Permanent demand shock: r g = α 1 α 2,and y g = 0 Since there is no increase in the supply side determined natural-rate of unemployment, the real interest rate must increase to offset the permanently higher public real expenditure. This is the economics behind the mathematical result we had above: that the cumulated sum of the dynamic multipliers is zero. Permanent (positive) supply shock y s = 1 γ > 0 and r s = 1 γα 2 < 0 Since the natural (non-accelerating) level of now has increased, the real interest rates comes down in order to increase demand.
Adaptive expectations I Which properties of the model are dictated by the choice of static inflation expectations in the basic version of the closed economy AD-AS model? In order to investigate, consider adaptive expectations instead. In the short-run model, replace π e t = π t 1 by π e t π e t 1 = (1 φ)(π t 1 π e t 1), 0 φ 1, (12) or, equivalently: π e t = φπ e t 1 + (1 φ)π t 1. (13) Note first that the SRAD is unaffected since in the Taylor rule has been specified in such at way that π e t+1cancels out when we substitute to derive the SRAD function.
Adaptive expectations II With adaptive expectations, the SRAS function π t = φπ e t 1 + (1 φ)π t 1 + γ(y t ȳ) + s t (14) replaces π t = π t 1 + γ(y t ȳ) + s t. Short-run model. Since the SRAD function is unchanged, and SRAS is given by (14), we conclude that the short-run model is unaffected by changing the model of expectations. The impact multipliers are therefore unaffected. Long-run model It is also the same as with static expectations. This is typical for models with expectations: the long-run model is always unaffected by changes in the specification of how expectations are formed.
Adaptive expectations III The dynamic analysis This is affected, since π e t 1 enters into the model, not only π t 1 as in the case with static expectations. Intuitively however, dynamic stability is not endangered. This is because, from (13) wr have: π e t 1 = (1 φ) φ j π t 1 j j=0 which is a generalization of static expectations. IAM p. 578 and 579 contain the detailed analysis, with the final equations for y t ȳ in (41) and for π t in (42).
A simple RBC model I Macro production function: Y t = K γ t (A t N t ) 1 γ, 0 < γ < 1, (15) where Y t is GDP in period t, K t denotes the capital stock, and N t is population in period t. A t is the technology parameter, which is however not a constant in this model: Model for technological progress ln A t = g A t + a s,t, 0 < g A < 1, (16) where the random part a s,t is given by the autoregressive equation: a s,t = αa s,t 1 + ε a,t, 0 α < 1 (17) where ε a,t is an unpredictable technology shock.
A simple RBC model II An essential assumption is that saving is proportional to income: S t = sy t, 0 s < 1 (18) where s is the constant saving rate (this is modified in advanced RBC models). As a simplification, we assume that capital equipment lasts for only one period, so that K t = S t 1. (19) It is the infallible mark of RBC models that the labour marked is assumed to be in equilibrium in each period. Labour supply, N S t, is a function of the relative wage w t / w t : N S t = N( w t w t ) ɛ, ɛ > 0. (20)
A simple RBC model III w t is the steady state wage and ɛ is the labour supply elasticity. When the wage is equal to the steady state wage, labour supply is also equal to its steady state value. Labour demand is obtained by assuming optimizing behaviour by a macro producer, meaning that the wage rate is equal to the marginal product of labour in each time period. As shown in IDM, ch 2.8.4 (or in IAM ch 19.4) we have the following final equation for the log output gap: y t ȳ = γ(1 + ɛ) 1 + γɛ (y t 1 ȳ) + (1 γ)(1 + ɛ) a s,t. (21) 1 + γɛ Hence, we have the important result that also in this equilibrium model, there will be periods of booms (positive output-gap), and troughs (negative output-gap)
Simulated solution of the RBC model 0.03 0.02 0.01 0.00 0.01 α = 0.1, ɛ = 4, γ = 0.5 ε a,t N(0, 0.01) 0.02 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
RBC and AD-AS model I In the RBC model, the labour market is in equilibrium in each period, sometimes with positive output-gap, sometimes with negative output-gap The business cycles generated by the RBC model are therefore often called equilibrium business cycles. Unlike the cycles of the AD-AD model, where there is downward pressure on nominal price growth as well as on the real wage level when u t u > 0, and upward pressure when u t u < 0. Another difference is that there is only room for supply shocks in the RBC model, there are no aggregate demand shocks. During the last 25 years there has been a synthesis in the form of so called New-Keynesian models and the RBC model. The synthesis is called DSGE (dynamic stochastic general equilibrium) models.
RBC and AD-AS model II The main ingredient of the DSGE is a RBC core model for the real economy, and a new theory of nominal price adjustment for inflation, the so called New Keynesian Phillips curve. DSGE has been successful in shaping the thinking of inflation targeting central banks. The financial crisis has started a new controversy. For example the current Krugman debate Krugman: www.nytimes.com/2009/09/06/magazine/06economict.html?_r=1 Cochrane: http://faculty.chicagobooth.edu/john.cochrane/research /Papers/krugman_response.htm
RBC and AD-AS model III The Economist features in July. Lucas response: http://www.economist.com/businessfinance/ displaystory.cfm?story_id=14165405