on Notes Ramsey-Optimal Monetary Policy
Lawrence J. Christiano, Roberto Motto and Massimo Rostagno January 9, 2007 Contents
1 Ramsey-Optimal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Rotemberg-Sticky Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Household . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Ramsey Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Model with Calvo-Sticky Prices and No other Frictions . . . . . . . . . . . 3.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Monetary Authority . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Equilibrium Conditions of the Model . . . . . . . . . . . . . . . . . . 3.5 Analysis of Ramsey Equilibrium . . . . . . . . . . . . . . . . . . . . . 4 Adding Money to the Model with Calvo Sticky Prices . . . . . . . . . . . . 4.1 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Numerical Analysis of the Model . . . . . . . . . . . . . . . . . . . . 5 Adding Wage Frictions to the Model . . . . . . . . . . . . . . . . . . . . . 5.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Aggregate Resource Constraint . . . . . . . . . . . . . . . . . . . . . 5.3 The Other Equilibrium Conditions . . . . . . . . . . . . . . . . . . . 5.4 Analysis of the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 6 Adding Habit Persistence and Investment Adjustment Costs to the Model . 6.1 The Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . 6.2 Analysis of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Anticipated Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Pulling All the Equilibrium Conditions Together and Adding Growth . . . 9 Introducing Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Adding Financial Frictions . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 The Addional Equilibrium Conditions . . . . . . . . . . . . . . . . . . 10.2 The Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 5 5 6 7 8 11 11 11 12 12 14 18 18 22 25 31 31 38 39 40 46 46 48 56 61 64 68 69 70 74
11 Getting the Ramsey Policy Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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These notes describe a set of monetary models which have been coded into Dynare, and which can be solved for the optimal monetary policy using code recently written for use in Levin, Lopez-Salido, (2004) and Levin, Onatski, Williams and Williams (2005) (LLSLOWW). The first section below describes the logic of the algorithm. The sequence of models goes from the least complicated to the most complicated, a version of the monetary model in Christiano, Eichenbaum and Evans (2004). In each case, the code is provided in a zip file available on the website where this document is posted.
1. Ramsey-Optimal Policy
Let xt denote a set of N endogenous variables in a dynamic economic model. Let the private sector equilibrium conditions be represented by the following N - 1 conditions: X (st+1 ) f x st , x st+1 , st , st+1 = 0, (1.1) t) (s t+1 t
s |s t t
for all t and all s . Here, s denotes a history:
st = (s0 , s1 , ..., st ) , and st denotes the time t realization of uncertainty, which can take on n possible values: st {s (1) , ..., s (n)} t = prob[st ], s
so that (st+1 ) / (st ) is the probability of history st+1 , conditional on st . This economy is not `closed' because there are fewer equations than unknowns. One way to close it would be to add an equation which characterizes policy, perhaps a Taylor rule. Instead, we consider the Ramsey optimal equilibrium. Suppose preferences over x (st ) are follows: X X t st U x st , st . (1.2)
t=0 st
where (st ) is the row vector of multipliers on the equilibrium conditions. Consider a particular history, st = (st-1 , st ) , with t > 0. The first order necessary condition for optimality of x (st ) is X (st+1 ) t t+1 f x s ,x s (1.3) U1 x st , st + st , st , st+1 t) |1 | {z } | {z } t+1 t (s {z } s |s 1N 1N -1 N-1N t-1 t-1 t -1 f2 x s 0 , x s , st-1 , st = |{z} + s | {z }| {z }
1N-1 N-1N 1N
The Ramsey problem is to maximize preference by choice of x (st ) for each st , subject to (1.1). We express the Ramsey problem in Lagrangian form as follows: t+1 X X t X (s ) t t+1 t t t U x s , st + s , f x s ,x s s , st , st+1 max | {z } t+1 t (st ) | {z } t=0 st
1N -1 s |s N -11
3
after dividing by (st ) t . In more conventional notation, U1 (xt , st ) + t Et f1 (xt , xt+1 , st , st+1 ) + -1 t-1 f2 (xt-1 , xt , st-1 , st ) = 0. The first order necessary condition for optimality at t = 0 is (1.3) with -1 0. The equations that characterize the Ramsey equilibrium are the N - 1 equations, (1.1), and the N equations (1.3). The unknowns are the N elements of x and the N -1 multipliers, . We will solve these equations by first or second order perturbation using Dynare. To apply the perturbation method, we require the nonstochastic steady state value of x. We compute this in two steps. First, fix one of the elements of x, say the inflation rate, . We then solve for the remaining N - 1 elements of x by imposing the N - 1 equations, (1.1). In the next step we compute the N - 1 vector of multipliers using the steady state version of (1.3): U1 + f1 + -1 f2 = 0,
0 = U1 0 X = f1 + -1 f2 = 0 ,
where a function without an explicit argument is understood to mean it is evaluated in steady state. Write Y
so that Y is an N 1 column vector, X is an N (N - 1) matrix and is an (N - 1) 1 column vector. Compute and u as = (X 0 X) X 0 Y u = Y - X. Note that this regression will not in general fit perfectly, because there are N -1 `explanatory variables' and N elements of Y to `explain'. We vary the value of until max |ui | = 0. This completes the discussion of the calculation of the steady state. Equations (1.1) and (1.3) form a system of dynamic equations in the endogenous variables, x (st ) and (st ) . Dynare can approximate the solution to these equations using first or second order perturbations about the nonstochastic steady state. To do this, one provides a Dynare-formated code with the equilibrium conditions, (1.1), and with the utility function and discount rate in (1.2). The code written by (LLSLOWW) takes this as input, computes the equations in (1.3) symbolically and sets up (1.1) and (1.3) as a new set of Dynareformated code. Dynare can be applied to the result.
-1
2. Rotemberg-Sticky Prices
We describe the agents in the model, and then summarize the equilibrium conditions. The example is sufficiently simple that (1.3) can be computed by hand, and the law of motion of the multipliers can be established anlytically.1
1
We are grateful to Ippei Fujiwara for suggesting this example to us.
4
2.1. Household Household i maximizes discounted utility, where the period utility function is: log (Ci,t ) - The budget constraint is: Bi,t Bi,t-1 Wt = (1 + Rt-1 ) - Ci,t + hi,t + i,t , Pt Pt Pt where i,t denotes lump-sum profits and taxes. The first order necessary conditions for household optimality are: Wt , (2.1) ht Ct = Pt and Pt Ct 1 = Et , (2.2) 1 + Rt Pt+1 Ct+1 for t = 0, 1, 2, ... . 2.2. Firms Firm j maximizes profit: Pj,t (1 + ) Cj,t - MCt Cj,t - Pt 2 2 Pj,t - 1 Ct . Pj,t-1 (2.3) 2 h . 2 i,t
The first term is firm revenues, including a tax subsidy, , received from the government (this is financed by a lump-sum tax on the household). The term after the first minus sign corresponds to the labor costs incurred in producing Cj,t . We assume that to produce 1 unit of Cj,t , exp (-Zt ) units of labor are required, where Zt denotes a shock to technology. Thus MCt = Wt ht Ct = , Pt exp (Zt ) exp (Zt )
after substituting out for the real wage using (2.1). The term after the second minus sign in (2.3) is the quantity of of the final good lost when the firm chooses to adjust its prices. The quantity of goods lost is positively related to the aggregate level of output, Ct . Firm j faces the following demand curve: - Pj,t Cj,t = Ct , 1. Pt The Lagrangian representation of the firm's problem is: 1- X Pj,t+n n Ct [(1 + ) Ct+n max Et {Pj,t+n } Ct+n Pt+n n=0 n=0 - 2 Pj,t+n Pj,t+n -MCt+n Ct+n - - 1 Ct+n ], Pt+n 2 Pj,t-1+n 5
where n Ct /Ct+n represents the state-contingent value, to households, of profits. This is taken as exogenous, by the firm. The first order necessary condition associated with the optimal choice of the price level is: Pj,t (1 - ) (1 + ) Pt - Ct + MCt Pt Pj,t Pt Pj,t Ct Ct - -1 Pt Pj,t-1 Pj,t-1 Ct Pj,t+1 Pj,t+1 +Et -1 Ct+1 = 0. 2 Ct+1 Pj,t Pj,t --1
In a symmetric equilibrium, Pj,t = Pt , for all j, so that the efficiency condition associated with firms is: 1 - (1 - ) + (MCt - 1) - ( t - 1) t + Et ( t+1 - 1) t+1 = 0. ( - 1) 2.3. Equilibrium Conditions The equilibrium conditions of the model are the household's intertemporal Euler equation, 1 Ct = Et , 1 + Rt t+1 Ct+1 the firm's efficiency condition: ht Ct 1 - 1 = ( t - 1) t - Et ( t+1 - 1) t+1 , (1 - ) + - ( - 1) exp (Zt ) and the resource constraint: 2 Ct 1 + ( t - 1) = exp (Zt ) ht . 2 (2.6) (2.4)
(2.5)
According to the latter, final goods are partly consumed by households, and partly they are used up in adjustment costs, if prices are being adjusted, i.e., if t 6= 1. The law of motion for the exogenous shock is: Zt = Zt-1 + ut , ut N (0, u ) . (2.7)
In the case where monetary policy is exogenous, we specify the following Taylor rule: Rt = - 1 + ( t - ), (2.8)
where is the target inflation rate. According to this, when inflation is above target, t > , they raise the nominal rate of interest above what the rate of interest would be expected to be at the target inflation rate and in steady state, / - 1.
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2.4. Ramsey Policy We have three private sector equilibrium conditions and four endogenous variables, Ct , t , ht and Rt . Absent a monetary policy rule, we do not have enough relations to determine all three variables. The Ramsey optimum is the value {Ct , t , ht , Rt } that maximizes household utility, subject to the three equilibrium conditions. It is perhaps obvious what the optimum is. Note that if we set = 1 , t = 1, h2 = 1, Ct = exp (Zt ) ht , t -1
then (2.5) and (2.6) are satisfied. If we let the intertemporal Euler equation define the nominal rate of interest, Rt , then (2.4) is satisfied too. This is the Ramsey equilibrium because this setting of ht and Ct solves the planning problem: maximize discounted utility subject to the version of the technology in which there are no losses to price adjustment. With these preferences and technology, you cannot do better than to set t h2 = 1. Note t that t = 1, h2 = 1 is the Ramsey equilibrium only if = 1/( - 1). If, for example, = 0, t then (2.5) indicates there must be some deviation from t = 1, h2 = 1, given firm profit t maximization. The Lagrangian representation of the Ramsey problem is: X Ct 1 2 t max E0 n {log (Ct ) - ht + 1,t - Et {Rt ,ht , t } 2 1 + Rt t+1 Ct+1 t=0 t=0 ht Ct - 1 - ( t - 1) t + Et ( t+1 - 1) t+1 +2,t (1 - ) + 1 + exp (Zt ) 2 }, +3,t exp (Zt ) ht - Ct 1 + ( t - 1) 2 where n is the planner's discount rate, which may in principle be different from . In principle, we should also add as an additional constraint, Rt 1. However, ignore that in the hope that it is non-binding. The first order necessary condition associated with Rt is: 1,t From this it is evident that 1,t = 0, for all t. We simplify the derivatives by imposing this from here on. The first order necessary condition associated with ht is, for t = 1, 2, ..., -t ht + 2,t Ct + 3,t exp (Zt ) = 0 exp (Zt ) (2.10) 1 = 0, t = 0, 1, 2, ... (1 + Rt )2 (2.9)
The first order condition associated with t is: 2,t (1 - 2 t ) + -1 2,t-1 (2 t - 1) - 3,t Ct ( t - 1) = 0. n 7 (2.11)
with the understanding, 2,-1 0. Finally, the first order condition with respect to Ct is 1 ht 2 - 3,t 1 + ( t - 1) = 0 + 2,t Ct exp (Zt ) 2
(2.12)
The equations that characterize the Ramsey equilibrium are the 7, (2.4)-(??), (2.10)(2.12). There are 7 unknowns, 2,t , 3,t and xt = (Ct , t , ht , Rt , Zt ) , for t = 0, 1, 2, ... . The required initial conditions are 2,-1 = 0 and Z-1 . We solve these equations by linearizing around steady state. This requires first computing the steady state of the Ramsey equilibrium, and then linearizing those equations about the steady state. We compute the steady state of the Ramsey problem by first fixing an arbitrary value for . Then, we solve for the remaining 4 elements of x using the four equations, (2.4)-(??). Solving (2.4) for steady state R : 1+R = . Solving (2.5) and (2.6) for h: h= 1 2 1 2 1 + ( - 1) [ (1 - ) ( - 1) + ( - 1) (1 + )] . 2 (2.13)
Given h, C can be recovered from (2.6). Also (2.7) can be solved for Z. Thus, we have solved for x as a function of : x () . We now use two of the three equations, (2.10)-(2.12), to solve for the two multipliers, 2 , 3. . We adjust until values for the two multipliers can be found which set these three equations to zero in steady state. In case the model we're working with is actually the one with exogenous monetary policy, (2.8), then the steady state inflation rate is just the target rate, . 2.5. Numerical Examples We computed some examples, to illustrate the calculations. We set = 0.99, = 5, = 100, = 0.9, = 1.5, n = 0.99, = 1 , = 1, = 1. -1
As explained above, in this example the Ramsey optimal policy is t = 1, h2 = 1, Ct = exp (Zt ) ht , t and the steady state values of ht and Ct are both unity in both the Ramsey and exogenous monetary policy equilibrium. The following figure displays the path of the actual variables
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in the Ramsey and the exogenous monetary policy (`equilibrium') equilibria.
net inflation (APR) hours worked
percent of steady state
20 0
percent
0 -0.05 -0.1 -0.15 -0.2
percent of steady state
-20 -40 -60 50 0 -50 0 10 20 30 net nominal rate of interest (APR) 40
0
10
20 consumption
30
40
1
percent
0.5
-100 1.5 1 0.5 0
0
10
20 technology
30
40
0
0 10 20 30 40 deviation in real rate of interest from ss, APR 0 -20 -40 -60
0
10
20
30
40
0 equilibrium Ramsey
10
20
30
40
The shock is a one percent jump in technology, which decays over time. Hours worked and consumption correspond to the percent deviation from steady state. In the Ramsey equilibrium, the percent deviation in hours is zero and the percent deviation in consumption corresponds exactly to what technology. The technology shock creates an expectation that current consumption is high and later consumption is lower. Other things the same, this creates an intertemporal smoothing motive, which makes people want to consume less in the current period and save for the future. The Ramsey equilibrium responds to this by reducing the interest rate by precisely the amount that is required to induce people to follow the Ramsey-optimal consumption path. In the example, the reduction is quite large, nearly 40 percent! That is, the steady state interest rate is 0.01, and the reduction in the first period is -0.10. So, the actual quarterly nominal rate of interest in the first period is -0.09. When multiplied by 400 to convert to annualized percent terms, this is the -36 percent that we see for Ramsey in the figure. (This example draws obvious attention to the fact that we ignore the non-negativity constraint on the nominal rate of interest.) The monetary policy rule evidently cuts the interest rate more than what is called for in the Ramsey equilibrium. However, expected inflation falls by a lot too. Consumption is determined by the real rate of interest. The figure indicates that in the exogenous monetary policy equilibrium, the real interest rate is cut by less than in the Ramsey. As a result, the consumption smoothing motive is not undercut by enough in the exogenous monetary policy equilibrium. In particular, they cut their consumption relative to the Ramsey optimum. This leads to a fall in demand for goods, which leads to a fall in employment and marginal costs. The fall in marginal costs induces firms to cut prices and so inflation falls relative to the Ramsey 9
optimum. Next, we set = 0 and redid the calculations. The steady state inflation rate in both equilibria is unity. Consumption and hours worked are both lower, now, at 0.8944. To see why, consider the firm efficiency condition for prices: 1 + h2 - 1 = ( - 1) - ( - 1) . The term on the right of the equality is zero. As a result, 1 1 2 h= 1- , Ct = exp (Zt ) h, in the Ramsey equilibrium. So, in terms of percent deviations from steady state, the Ramsey equilibrium is (numerically) the same with = 0 and = 1/ ( - 1) . In the exogenous monetary policy equilibrium, the drop in hours and consumption is a little bigger than what it was with the tax subsidy.
net inflation (APR) 20 0
percent
equilibrium Ramsey
percent of steady state
hours worked 0.2 0.1 0 -0.1 -0.2 1 0 10 20 consumption 30 40
-20 -40 -60 50 0 -50 0 10 20 30 net nominal rate of interest (APR) 40
percent of steady state
percent
0.5
-100 1.5 1 0.5 0
0
10
20 technology
30
40
0
0 10 20 30 40 deviation in real rate of interest from ss, APR 0 -20 -40 -60
0
10
20
30
40
0
10
20
30
40
Interestingly, not only is 1,t = 0 for all t, but in the above numerical experiments, we found 2,t = 0 also. By contrast, 3,t is non-zero. Interestingly, however, the lagged values of 3,t do not appear in the state of the system, and so therefore the optimal plan is not time-inconsistent. This is true if is set to ensure an efficient steady state, or not.
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3. Model with Calvo-Sticky Prices and No other Frictions
3.1. Firms We adopt the usual assumption that a representative final good producer manufactures final output using the following linear homogenous technology: Z 1 f 1 Yjt t dj , 1 f < , (3.1) Yt =
0
Intermediate good j is produced by a price-setting monopolist according to the following technology: 1- - zt if t Kjt (zt ljt )1- > zt t Kjt (zt ljt ) , 0 < < 1, (3.2) Yjt = 0, otherwise where zt is a fixed cost and Kjt and ljt denote the services of capital and homogeneous k labor. Capital and labor services are hired in competitive markets at nominal prices, Pt rt , and Wt , respectively. The object, zt , in (3.2), is assumed to evolve deterministically: zt = zt-1 z . (3.3)
For now, we assume that zt = 1, constant, with z = 1. In the last section below, we consider the possibility, z > 1. In (3.2), the shock to technology, t , has the time series representation in (??). We adopt a variant of Calvo sticky prices. In each period, t, a fraction of intermediate-goods firms, 1 - p , can reoptimize their price. If the ith firm in period t cannot reoptimize, then it sets price according to: Pit = t Pi,t-1 , ~ where t = 1- . ~ t-1 (3.4) Here, t denotes the gross rate of inflation, t = Pt /Pt-1 , and denotes steady state inflation. th ~ If the i firm is permitted to optimize its price at time t, it chooses Pi,t = Pt to optimize discounted profits:
X j Et p t+j [Pi,t+j Yi,t+j - Pt+j st+j (Yi,t+j + zt+j )] . j=0
(3.5)
Here, t+j is the multiplier on firm profits in the household's budget constraint. Also, ~ Pi,t+j , j > 0 denotes the price of a firm that sets Pi,t = Pt and does not reoptimize between t+1, ..., t+j. The equilibrium conditions associated with firms appear in the next subsection. 3.2. Households The household maximizes utility 1-q Pt+l ct+l 1+L X d Mt+l ht j l-t Et u(c ) - L - t+l 1 + L 1 - q l=0 11
(3.6)
subject to the constraint
d k e Pt (ct + it ) + Mt+1 - Mtd + Tt+1 Wt,j lt,j + Pt rt kt + (1 + Rt ) Tt ,
(3.7)
where Mtd denotes the household's beginning-of-period stock of money and Tt denotes nome inal bonds issued in period t - 1, which earn interest, Rt , in period t. This nominal interest rate is known at t - 1. In the interest of simplifying, we suppose that in (3.6) is positive, but so small that the distortions to consumption, labor and capital first order conditions introduced by money can be ignored. The household's problem is to maximize (3.6) subject to the standard capital accumulation technology linking investment, i, to capital. 3.3. Monetary Authority The monetary authority controls the supply of money, Mts . When policy is exogenous, it does so to implement a following Taylor rule. The target interest rate is: Yt , Rt = - 1 + [Et ( t+1 ) - ] + y log Yt+ where Yt+ is aggregate output on a nonstochastic steady state growth path. The monetary authority manipulates the money supply to ensure that the equilibrium nominal rate of interest, Rt , satisfies: Rt = i Rt-1 + (1 - i ) Rt . (3.8) 3.4. Equilibrium Conditions of the Model Real marginal cost, st , can be represented as the ratio of the real cost of capital to its marginal product and the real cost of labor to its marginal product:
k rt ht kt-1
st = st =
t
(1 - )
k ~ where rt and wt denote the real rental rate on capital and the real wage rate, respectively. The household's first order condition for labor:
wt ~
t
1-
ht kt-1
(3.9)
- ,
(3.10)
L hL ct = wt ~ t
(3.11)
The household's intertemporal first order condition for the nominal return on capital: - 1 1 k + 1 + Rt+1 = 0 ct t+1 ct+1 kt - (1 - )kt-1 = it 12 (3.12)
The capital accumulation equation:
(3.13)
The definition of the nominal rate of return on capital: k k Rt = rt + (1 - ) t - 1 ct + it Here,
t
(3.14)
The resource constraint, including possible distortions due to price dispersion: (p ) f -1 t
f
is a technology shock, with the following law of motion: log
t
1- t kt-1 ht + t ,
-
(3.15)
= log
t-1
(3.16)
k Substitute out for Rt+1 in (3.12) using (3.9) and (3.14), to obtain: " # 1- 1 ht+1 - + t+1 st+1 + (1 - ) = 0 ct ct+1 kt
where t is iid with variance 2 . Substitute out for wt from (3.11) into (3.10): ~ ht kt-1 1 L hL ct st = t 1- t
(3.17)
(3.18)
Eliminate it in the resource constraint using (3.13): ct + kt - (1 - )kt-1 = (p ) f -1 t
f
We now turn to the equations pertaining to sticky prices. The equilibrium condition pertaining to p is: t 1 - p p - 1 - p t
2 t-1 1-2 1 1- f f
1- t kt-1 ht
-
(3.19)
t
1 - p
Note that when there are no sticky prices, so that p = 0, then p = 1 and (3.19) reduces to t a more standard-looking resource constraint. We also have the following equations: 1 1- 1-f f t ~ 1 - p t Kp,t - Fp,t = 0 1 - p Et - f f -1 t+1 ~ = 0 Et f z, t Yz,t st + p Kp,t+1 - Kp,t t+1 t = 2 1-2 ~ t-1 13 (we set 1 = 0) ( z, t Yz,t + t+1 ~ t+1
1 1- f
+ p
1- f 2 1-2 t-1 pt-1 t
f
1-f
f
=0
(3.20)
p Fp,t+1 - Fp,t
)
= 0
Note that when there are no sticky prices, then Kp,t = Fp,t and st = 1 , f
so that the markup is a constant (being real marginal cost, st is the reciprocal of the markup). Sticky prices in effect make the markup fluctuate. Substitute out Kp and t , replace st using (3.17) above and replace Yz,t , zt using ~ Yz,t = (p ) f -1 t 1 zt = , ct
f
1- t kt-1 ht
-
and
so that we end up with the following two equations: ( ) 1 2 1-2 1- f 1 f f 1- t (p ) -1 t kt-1 ht - + p Fp,t+1 - Fp,t = 0, Et ct t t+1
f 1- 1 f -1 f (pt ) - st + t kt-1 ht ct 1 2 1- 1- 1-f 2 t f 1-p t+1 1 - p Fp,t+1 -Fp,t 1 - p
(3.21)
(3.22)
2 t-1 1-2 t 1 1-f
p
2 1-2 t t+1
f 1-f
1- p
1-f
= 0
To conclude, we have N = 7 unknowns, (k, s, l, c, , Fp , p ), in the N - 1 equations (3.17), (3.18), (3.19), (3.20), (3.21), (3.22). These equations have been entered into the Dynare file, newsimplemodel.mod, in subdirectory stickypricesonly. We also computed the nominal rate of interest, by including the following equation: 1 1 = 0. Et - + (1 + Rt ) ct ct+1 t+1 3.5. Analysis of Ramsey Equilibrium Suppose we start in period 0, in a steady state, when -1 = and p = 1. Consider a -1 monetary policy which results in t = for t = 0, 1, ... . Then, (3.20) implies p 0
f 1- f = 1 = 1 - p + p p-1
1-f
f
.
Substituting this into (3.20) for t = 1, 2, 3, ..., produces the result, p = 0 for t = 0, 1, ... . t This policy minimizes the distortion in the resource constraint, because (it can be shown) p 1 for all t. Thus, the resource constraint is: t ct + kt - (1 - )kt-1 = t kt-1 h1- - . t 14 (3.23)
Under the stated monetary policy, we can combine (3.21) and (3.22) to obtain:
f 1- 1 f f 1- 1 f -1 -1 (pt ) - = f (pt ) - st , t kt-1 ht t kt-1 ht ct ct
or,
st = for all t. Combining this with (3.17), we infer L hL ct = t
1 , f -
(1 - )
t
ht kt-1
f
.
(3.24)
That is, the marginal rate of substitution between consumption and leisure equals the marginal product of labor, divided by the markup. The final equation of our equilibrium is the intertemporal Euler equation, (3.18), which becomes " # 1- 1 1 ht+1 t+1 - + + (1 - ) = 0. (3.25) ct ct+1 kt f Note that if f = 1, then (3.23), (3.24) and (3.25) characterize the efficient allocations for the economy with the preferences and technology that we assume. Thus, if f is nearly unity, the constant inflation monetary policy is nearly optimal. However, if f is substantially above unity, then we cannot expect the constant monetary policy to be optimal. If there is a positive markup in steady state, then it is possible that with inflation allowed to vary in the right way, the markup can fall in response to a shock, and this would improve utility relative to the scenario in which inflation is held constant. To see this, note that in (3.24) and (3.25), there appear wedges between marginal rates of substitution in preferences and marginal rates of technical transformation that are different from unity. If instead st appeared and could be made to rise, then welfare would increase. Of course, this involves complicated tradeoffs. In this economy, fluctutations in st are induced by fluctuations in inflation and these in turn introduce an inefficiency wedge, p , in the resource constraint. In addition, if welfare rises t as st rises in response to a shock, then when the shock takes on the opposite sign, st will fall and reduce welfare. All these considerations have to be balanced off against each other to determine the optimal response of inflation to a shock. We illustrate these observations with an example. Suppose = 0.99, L = 109.8, f = 1.002, = 0.40, = 0.025, p = 0.75, 2 = 0.6, L = 1, = 0.8. With this parameterization, we computed the dynamic response of ct , ht , t , kt , t , st , Rt to a one-percent shock in 0 , 0.01. We computed a linear approximation using Dynare, and so the output of Dynare is a sequence, ct , ht , t , kt , log t , st , Rt , where denotes deviation from steady state. In the case of ct , ht , kt , st , we divided the variable by its corresponding steady state. Because the shock actually fed to Dynare was 0 = 1., we interpret the output of Dynare as being in percent terms. In the case of inflation, we computed 400 ( + t /100 - 1) , where is the steady state inflation rate. Thus, we 15
report the net inflation rate, expressed at an annual rate. Similarly, in the case of the nominal interest rate, we computed 400 (R + Rt /100) , where R is the steady state nominal rate of interest. Thus, the nominal rate of interest is reported in net terms, at an annual percent rate. The response of the state of technology, log t , is expressed in percent deviation from steady state and is not further adjusted. The results are displayed in the following figure:
consumption 0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20 hours worked 3.6608 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 0 5 10 15 20 5 10 15 20 2 20 5 10 15 20 -6 real marginal cost (% dev from ss) x 10 4 3.6608 3.6608 3.6608 net inflation (APR) actual steady state
percent deviation from ss
percent deviation from ss
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest 7.85 7.8 7.75 5 10 15 20
percent deviation from ss
percent deviation from ss
Annualized, percent
rbc model sticky price model actual steady state Markup = .2 percent (f = 1.002)
This graph displays both the response of the variables in the sticky price model, as well as in the real business cycle model characterized by (3.23)-(3.25).2 Note that the quantity allocations in the RBC model and the sticky price model are essentially the same. Also, there is essentially no response in inflation or the markup in the Ramsey equilibrium, as expected. Next, we consider a very high markup of 80 percent, or f = 1.80. The response to the
The subdirectory, stickypricesonly, also contains the mod file, rbcmodel.mod, which was used to compute the dynamic response of the variables in the RBC model.
2
16
same technology shock is as follows
consumption 0.7 0.6 0.5 0.4 5 1 0.8 0.6 0.4 0.2 5 10 15 20 nominal rate of interest 10 15 capital stock 20 hours worked 3.65 3.6 3.55 0 3.5 5 10 15 technology shock 0.8 0.6 0.4 0.2 -0.2 5 10 15 20 5 10 15 20 20 5 10 15 20 real marginal cost (% dev from ss) 0 -0.05 -0.1 -0.15 actual steady state inflation (APR)
percent deviation from ss
percent deviation from ss percent deviation from ss
0.4 0.2
Annualized, basis points
percent deviation from ss
6 4 2 5 actual steady state 10 15 20
rbc model sticky price model Markup = 80 percent (f = 1.80 )
First, note how the quantities respond very differently in the Ramsey and in the RBC economy. Also, there is now a non-negligible deviation from constant inflation and constant marginal cost. Marginal cost, st drops, implying a rise in the markup. This may seem puzzling at first, since a rise in the markup is clearly not directly welfare-increasing. However, recall that this is the response to an unexpected positive shock to technology. With equal probability, the shock would have the other sign. So, to evaluate the welfare consequence of the non-constancy of inflation, one has to trade off the welfare loss of the rise in the markup with the positive technology shock against the welfare gain of the fall in the markup with the negative technology shock. Presumably, the gain associated with a fall in technology outweighs the loss associated with the rise in technology. Now consider a more `normal' setting for the markup, f = 1.20, or 20 percent. With
17
this setting, we obtain the following impulse response function:
consumption 0.35 hours worked 3.66 0.4 3.655 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 -0.02 5 10 15 20 5 10 15 20 20 3.65 actual steady state 5 10 15 20 real marginal cost (% dev from ss) 0 net inflation (APR)
percent deviation from ss
0.3
0.25 5 0.5 0.4 0.3 0.2 5 10 15 20 net nominal rate of interest 10 15 capital stock 20
percent deviation from ss
percent deviation from ss
percent deviation from ss
-0.005 -0.01 -0.015
Annualized, percent
7.85 7.8 7.75
rbc model sticky price model actual steady state Markup = 20 percent (i.e., f = 1.20 )
5
10
15
20
These responses resemble those in the low markup economy, though the differences are not completely insignificant.
4. Adding Money to the Model with Calvo Sticky Prices
We now consider the model of the previous section, with money in the utility function. 4.1. Equilibrium Conditions The household's Lagrangian problem is:
Pt+l ct+l d Mt+l+1
1 - q l=0 k d +t Wt lt + Pt rt kt + (1 + Rt-1 ) Tt + Dt + Xt - Pt (ct + it ) + Mt+1 - Mtd + Tt+1 it +t (1 - ) kt + 1 - S it - kt+1 }, it-1 1 + L 18
Et
X j
l {u(ct+l - bct+l-1 ) - L
1+ lt L
-
1-q
where Dt denotes profits and Xt is a transfer from the government: Xt = Mt+1 - Mt , and in equilibrium it must be that Mtd = Mt . Note that we have introduced money in the utility function, and investment adjustment costs habit persistence in consumption have also been introduced. The first order conditions for consumption is: 1-q Pt - 0 0 Pt t = u (ct - bct-1 ) - bu (ct+1 - bct ) - ct q , d Mt or, z,t 1 1 = - b - ct - bct-1 ct+1 - bct t mt 1-q ct
-q
,
where mt = Mt /Pt-1 . The first order condition for lt is:
L lt L = t Wt = z,t wt ,
where wt = Wt /Pt . The first order condition for Tt+1 is: t = t+1 (1 + Rt ) , or, z,t = z,t+1 1 + Rt Pt+1 , t+1 = . t+1 Pt
The first order condition for it is: 2 it it it it+1 it+1 0 0 t Pt = t 1 - S -S + t+1 S . it-1 it-1 it-1 it it Define Pk0 ,t = so that 1 = Pk0 ,t 1 - S it it-1 -S
0
t , t Pt z,t+1 + Pk0 ,t+1 S 0 z,t it+1 it it+1 it 2
it it-1
it it-1
,
d The first order condition Mt+1 :
or, after multiplying by Pt :
d q -2 t = t+1 + (Pt+ ct+1 )1-q Mt+1 , z,t = z,t+1 1- 1- q -2 + t+1 q ct+1 q mt+1 . t+1 19
or, after dividing by t Pt :
The first order condition associated with kt+1 is: k t = t+1 Pt+1 rt+1 + t+1 (1 - ) , Pk0 ,t = z,t+1 k rt+1 + Pk0 ,t+1 (1 - ) . z,t ct + it Y+
The monetary policy rule is:
- 1 + [Et (t+1 ) - ] + y log Rt = i Rt-1 + (1 - i )
+ t .
20
Putting all the equations together, and including the equations related to sticky 1-q 1 t 1 - - b - ct q = (1) ct - bct-1 ct+1 - bct mt 2 it it it z,t+1 it+1 it+1 0 0 -S + (2)Pk0 ,t 1 - S Pk0 ,t+1 S = it-1 it-1 it-1 z,t it it z,t+1 k rt+1 + Pk0 ,t+1 (1 - ) = (3) z,t z,t+1 q -2 (4) + ( t+1 ct+1 )1-q mt+1 = t+1 z,t+1 (5) (1 + Rt ) = t+1 it it = (6) (1 - ) kt + 1 - S it-1 f 1- (7) (p ) f -1 t (Kt ) lt - = t kt (8) z,t st t (1 - ) = lt 1- lt (9) st t = kt 1-f 1 f 1- f f f t ~ 1- 1 - p t f t ~ (10) 1 - p pt-1 = + p 1 - p t (11) Et z,t Yt + " t+1 ~ t+1
1 1- f
prices: z,t 1 Pk0 ,t z,t z,t kt+1 ct + it
L lt L k rt
p t
p Fp,t+1 - Fp,t
f
1 1- f ~ 1 1 - p t Kp,t 1-f t = (13) Fp,t 1 - p ct + it + t = Rt - 1 + [Et ( t+1 ) - ] + y log (14) i Rt-1 + (1 - i ) Y+
(12) Et f z,t Yt st + p
t+1 ~ t+1
f 1-
Kp,t+1 - Kp,t = 0
#
= 0
k The 14 variables to be solved using these 14 equations are z,t , ct , mt , t , it , Pk0 ,t , rt , Rt , kt , pt , lt , st , Fp,t , Kp,t .
21
4.2. Steady State The equations associated with Calvo sticky prices imply: s = Fp 1 , f z Y = 1 - p
This is as expected. When there are no price distortions in the steady state, then the firm markup is 1/s = f in the present case of a constant elasticity demand curve. The resource constraint is c + k = k l1- - . We use the supposition that profits are zero in the steady state to determine a value for . If we think of the Cobb-Douglas part of the production function as the firm's `production function', then with fixed costs, , the firm which sells Yjt must actually produce Yjt + . Given fixed marginal costs, the firm's total cost associated with selling Yjt is st (Yjt + ), in units of the final good (i.e., scaling by Pt ). The firms' revenues are Pjt Yjt , so its profits in units of final goods are Pjt Yjt - st (Yjt + ) . Pt In steady state, Pjt = Pt , Yjt = Yt for all j because our assumptions guarantee that prices and resources are not distorted in a steady state. So, the zero profit condition in steady state is 1 Y = (Y + ) , f or, (f - 1) Y = . In the steady state, profits are 100 (f - 1) of output sold, and fixed costs must be equal to this amount if profits are to be zero. Substituting in the production function, (f - 1) k l1- - = , so that (f - 1) k l1- = f Combining this with the resource constraint, we obtain: c + k = k l1- - f - 1 1- 1 k l = k l1- . f f
22
Collecting our results, we have that the steady state equations are: Pk0 ,t = 1, z = (1 - b) u0 (c - bc) - 1-q m
c-q
L lL z
1 = rk + 1 - = 1+R 1-q c z -q c z = + m m = w
c + k =
1 1- k l f 1- k 1- 1 1 s = w r 1- rk s = l 1- k 1 s = , f
which represents 10 equations in 11 unknowns: c, s, l, z , w, rk , , R, m, k, Pk0 . Another equation is provided by the assumption that transfers are made to the household to provide them with money: Mt+1 - Mt = Xt , and dividing this by Mt : Mt+1 - Mt = xt , Mt
m - m = - 1 = x. m We will just treat as an exogenous variable, with the understanding that it is actually x that is exogenous. So, by deleting from the list of 11 unknowns, we have 10 equations in 10 unknowns. We solve these equations as follows. The variables, R, s, Pk0 , rk , l/k, c/k and w are virtually immediate. Thus, Pk0 = 1, s = 1/f , R = / - 1 and rk = 1 - (1 - ) ,
so that in steady state,
23
and
so that
1- 1 l , r = f k
k
1 k 1- l r f lk = . k The resource constraint can be written: c 1 1- ck = lk - , k f
and the wage rate can be solved using the fact that rk is known and 1 # 1- " 1 . w= 1 1- 1 f 1- (rk )
We still require k, z , m. The equations that remain available to us are the following three: 1-q 1 (ck k)-q - z = (1 - b) ck k (1 - b) m 1-q ck k z z = + (ck k)-q m m L (lk k)L = w z which now reduces to two equations in two unknowns, k and m. Multiply the first equation by ck k and use the expression for z from the household's first order condition for labor: 1-q L (lk k)L 1 - b (ck k)1-q = - ck k w (1 - b) m
SSubstitute from the labor euler equation into the second of the previous two equations: 1-q (c k)1-q L (lk k)L k . 1- = w m m or, 1-q k 1+ L + a1 = a2 a0 k m 2-q k 1+L b0 k = b1 . m for known constants, a0 , a1 , a2 , b0 , b1 and the parameter, : a0 = ck b0 L lk L 1 - b , a1 = 1-q ck 1-q , a2 = , w 1-b L lk L = 1- , b1 = 1-q ck 1-q . w 24
From the second expression
so that k
1+ L
1 2- q m b1 , = k b0 k1+L
+
1 2- q
Note that the function on the left of the equality is zero at k = 0 and strictly increasing and convex. As a result, there is a unique value of k that solves this equation. In the case, = 0, this value can be found analytically: k= w 1-b 1-b ck L lk L
1 ! 1+ L
1-q w 1-b a1 b0 1+L 2-q a2 1-b k = = > 0. a0 b1 a0 ck L lk L
.
With k in hand, k/m may be found from the previous expression. Note that when = 0 then k/m = so that m = 0. We would expect that the price level would be infinite when = 0 because in this case there is no use for money. 4.3. Numerical Analysis of the Model We considered the following parameter values. = 1.03-0.25 , b = 0.63, = 0.36, = 0.025, = 0.95, and f = 1.20, p = 0.75, = 0.84, = 1 + 0.025/4, v = 0.0005, i = 0.81, = 1.95, y = 0.18, L = 1, q = -1, a = 5.
25
With this parameterization, we have l = 0.996, y/m = 4.16, k/y = 11.11. The impulse response function to
percent deviation frompercent deviation from steady statedeviation from steady state ss percent
percent deviation from steady state percent deviation from ss
Consumption 0.7 0.6 0.5 0.4 0.3 2 4 6 8 price of capital 10
investment 2
percent deviation from steady state
Response to Technology Shocks,
= 1.95 velocity = 4.1598
hours worked 0 -0.5 -1 2 4 6 8 10 nominal rate of interest (APR) -0.6
1.5 1
0.5
2 4 6 8 10 real balances, M(t+1)/P(t) 7 6 5 4 2 4 6 8 rate of inflation (APR) 10
0.4 0.3 0.2 0.1 0 2 4 6 8 state of technology 10
percent
-0.8 -1 2 4 6 8 10
-0.6 0.9 0.8 0.7 2 4 6 8
percent, net
10
-0.8 -1 -1.2 -1.4 -1.6 2 4 6 8 10
Next, we greatly increase the importance of money, by raising v to 5. In this case, we
26
obtain the following impulse response function to a technology shock:
percent deviation frompercent deviation from steady statedeviation from steady state ss percent percent deviation from steady state percent deviation from ss percent deviation from steady state
Response to Technology Shocks, = 1.95 velocity = 0.18729
Consumption 1.2 1 0.8 0.6 0.4 2 0.4 0.3 0.2 0.1 0 2 4 6 8 state of technology 10 4 6 8 price of capital 10 investment hours worked 0.5 0 -0.5 -1 2 4 6 8 10 nominal rate of interest (APR)
1.5 1
0.5
2 4 6 8 10 real balances, M(t+1)/P(t) 8 7 6 5 2 4 6 8 10 rate of inflation (APR)
percent
-0.8 -1 -1.2 2 4 6 8 10
0.9 0.8 0.7 2 4 6 8
-0.8 -1 -1.2 -1.4 -1.6 -1.8 2 4 6 8 10
10
There is a noticeable difference in the response of consumption, with consumption rising by more here. The response in the other variables is not very different. Presumably, the rise in real balances associated with the fall in the price level is making the marginal utility of consumption rise more, as the real balance effect on consumption is more important. Next, we consider the effect of reducing the importance of money, by reducing v to
percent, net
27
0.0000005. The resulting impulse responses are as follows:
percent deviation frompercent deviation from steady statedeviation from steady state ss percent percent deviation from steady state percent deviation from ss percent deviation from steady state
Response to Technology Shocks, = 1.95 velocity = 41.6548
Consumption 0.7 0.6 0.5 0.4 0.3 2 4 6 8 price of capital 10 investment 2 hours worked 0 -0.5 -1 2 4 6 8 10 nominal rate of interest (APR) -0.6
percent
1.5 1
0.5
2 4 6 8 10 real balances, M(t+1)/P(t) 7 6 5 4 2 4 6 8 10 rate of inflation (APR)
0.4 0.3 0.2 0.1 0 2 4 6 8 state of technology 10
-0.8 -1 2 4 6 8 10
-0.6 0.9 0.8 0.7 2 4 6 8
percent, net
-0.8 -1 -1.2 -1.4 -1.6 2 4 6 8 10
10
Note that with this ten-fold increase in velocity, the impulse responses have hardly change from our initial, baseline responses. Next, we set v = 0. The steady state algorithm described above works for this case, and produces m = 0. In the dynamic equations, we set v = 0 in (1) and drop equation (4) and
28
the variable, mt . With = 1.95, the following impulse response function was obtained:
percent deviation frompercent deviation from steady statedeviation from steady state ss percent percent deviation from steady state percent deviation from steady state
Response to Technology Shocks, = 1.95 velocity = Inf
Consumption 0.7 0.6 0.5 0.4 0.3 2 4 6 8 price of capital 10 investment 2 1.5 1 0.5 2 4 6 8 hours worked 0 -0.5 -1 2 4 6 8 10 nominal rate of interest (APR) -0.6
10
0.4 0.3 0.2 0.1 0 2 4 6 8 state of technology 10 rate of inflation (APR) -0.6 0.9 0.8 0.7 2 4 6 8
percent
-0.8 -1 2 4 6 8 10
percent, net
10
-0.8 -1 -1.2 -1.4 -1.6 2 4 6 8 10
Note that the results are essentially the same. In addition, the range of determinacy is unchanged from before. From this we have to conclude that there must be a mistake somewhere because this model is essentially the standard model with no money. Next, we consider the response of the system to a monetary policy shock. Here is the response to a 0.0025 shock to monetary policy. This corresponds to a 100 basis point rise in
29
the interest rate, at an annual rate.
percent deviation from steady state percent deviation frompercent deviation from steady state ss
percent deviation from steady state percent deviation from ss
Consumption 0
investment 0
percent deviation from steady state
Response to Monetary Policy Shock, = 1.95 velocity = 4.1598
hours worked 0 -0.05 -0.1 -0.15 -0.2 2 4 6 8 10 nominal rate of interest (APR) 6
percent
-0.05 -0.1
-0.05 -0.1 2 0 -0.2 -0.4 -0.6 2 1 0.5 0 -0.5 -1 2 4 6 8 4 6 8 state of technology 4 6 8 price of capital
-0.15 -0.2
10
2 4 6 8 10 real balances, M(t+1)/P(t) 0 -1 -2 -3 -4 2 4 6 8 10 rate of inflation (APR) 2.4 2.2 2 1.8
5.8 5.6 5.4 2 4 6 8 10
10
10
percent, net
2
4
6
8
10
Note that to get the interest rate up, they need to reduce the money supply. Also, the interest rate does not rise by the full 100 basis points immediately, because the fall in prospective inflation exerts a countervailing force on the interest rate.
30
Now v is reduced, and the impulse response function becomes:
percent deviation from steady state percent deviation frompercent deviation from steady state ss
percent deviation from steady state percent deviation from ss
Consumption 0
investment 0
percent deviation from steady state
Response to Monetary Policy Shock, = 1.95 velocity = 41.6548
hours worked 0 -0.05 -0.1 -0.15 -0.2 2 4 6 8 10 nominal rate of interest (APR) 6
percent
-0.05 -0.1
-0.05 -0.1 2 0 -0.2 -0.4 -0.6 2 1 0.5 0 -0.5 -1 2 4 6 8 4 6 8 state of technology 4 6 8 price of capital
-0.15 -0.2
10
2 4 6 8 10 real balances, M(t+1)/P(t) 0 -1 -2 -3 -4
5.8 5.6 5.4
10
2 4 6 8 10 rate of inflation (APR) 2.4 2.2 2 1.8
2
4
6
8
10
10
percent, net
2
4
6
8
10
This result is the same as before. Evidently, the size of v does not matter for the responses of the model economy to monetary policy and technology shocks.
5. Adding Wage Frictions to the Model
We now add Calvo-style wage frictions to the model of the previous section, following the analysis of Erceg, Henderson and Levin. We begin by deriving the equations that pertain to household wage setting. After this, we turn to implications for the aggregate resource constraint. We then display the other equilibrium conditions. 5.1. Households We derive a law of motion for the aggregate real wage (wt ) by combining the optimality ~ of the condition of of households which reoptimize their wage, with a cross-household-wage 31
consistency condition. We suppose there is a continuum of households, j (0, 1) , each of which supplies a differentiated labor service which is aggregated into a homogeneous labor good by perfectly competitive labor contractors using the following constant returns to scale technology: Z 1 w 1 w dj lt = (ht,j ) , 1 w < . (5.1)
0
Aggregate labor is sold competitively by the representative labor contractor to intermediate goods producers for wage Wt and the j th household's wage is Wj,t . The contractor hires ht,j , j (0, 1), in order to maximize profits: Z 1 w Z 1 1 w dj max Wt (ht,j ) - Wt,j ht,j dj,
ht,j 0 0
which leads to the first order condition: Wt lt w (ht,j ) or, ht,j = lt
w -1 1-w w
= Wt,j ,
Wt,j Wt
The j th household views (5.2) as a demand curve for its specialized labor services. The rules are that if the household posts a wage, Wt,j , then it must supply the services, ht,j , implied by the demand curve. Thus, the household's problem is to choose its wage rate, Wt,j . With probability, 1 - w , it can optimize its wage rate and with the complementary probability, it cannot. In this case, we suppose that it sets its wage as follows: Wt,j = w,t-1 Wt-1,j , w,t ( t-1 )w,2 1-w,2 . ~ ~ (5.3)
w 1-
w
.
(5.2)
The 1 - w households that set their wage optimally in period t all find it optimal to set the ~ same wage, Wt . The household which can optimize its wage in period t does so to optimize the following objective: Et
X i=0
( w )i {-z(hj,t+i ) + t+i Wj,t+i hj,t+i },
where t+i is the multiplier on the household's budget constraint, (3.7). The household discounts by w because it is only interested in continuation histories in which it does not reoptimize its period t wage. Here, z indicates the household's disutility of labor: z (hj,t ) = L 1+L h . 1 + L j,t
Also, t+i represents the marginal value of a unit of currency to the household in period t +i. Given the utility function 1 . t = Pt ct 32
Substituting out for hours worked using the labor demand curve, we obtain: w w X Wt+i,j 1-w Wt+i,j 1-w i Et ( w ) {-z(lt+i ) + t+i Wj,t+i lt+i }. Wt+i Wt+i i=0 This can be written Et
X i=0 i
( w ) {-z(lt+i
Wt+i,j Wt+i
w 1-
w
w +1 Wt+i Wt+i,j 1-w ) + t+i Pt+i lt+i }. Pt+i Wt+i
~ To express the household's objective in terms of Wt , it is necessary to express Wt+i,j in terms of its period t value. We adopt the following definitions: wt = ~ Then, ~ ~ ~ ~ ~ Wt+i,j w,t+i w,t+1 Wt ~ w,t+i w,t+1 Wt ~ wt wt = = = Xt,i , Wt+i wt+i Pt+i ~ wt+i t+i t+1 Pt ~ wt+i ~ Xt,i ~ Wt Wt , wt = , z,t+i = t+i Pt+i . Pt Wt
~ w,t+j w,t+1 ~ . t+i t+1 Substituting this into the household's objective, w w 1- 1- +1 X w w ~ ~ wt wt wt wt i ( w ) {-z(lt+i Xt,i ) + z,t+i wt+i lt+i ~ Xt,i }. Et wt+i ~ wt+i ~ i=0 The variable that the household must choose is wt (note, whether the household is viewed ~ ~ as choosing wt or Wt makes no difference, since wt is Wt scaled by a variable over which the household has no control). Maximizing the household's objective with respect to wt , we obtain: w 1- -1 X w w ~ wt ~ wt wt i 0 ( w ) {-zt+i lt+i Xt,i Xt,i Et 1 - w wt+i ~ wt+i ~ i=0 w 1- w ~ wt ~ w wt wt +z,t+i wt+i lt+i ~ +1 Xt,i Xt,i } = 0 1 - w wt+i ~ wt+i ~ or, after rearranging, Et
X i=0
where
( w ) lt+i
i
wt ~ Xt,i wt+i ~
w 1-
w
{
z,t+i 0 wt wt Xt,i - zt+i } = 0 ~ w
The marginal utility of leisure can be written, after taking into account that labor must always be on the demand curve w 1- !L w ~ wt wt 0 0 Xt,i zt+i z (hj,t+i ) = L lt+i wt+i ~ 33
Then, the first order condition reduces to Et
X i=0
( w )i lt+i
wt ~ Xt,i wt+i ~
w 1-
w
w 1- !L w ~ wt wt z,t+i { wt wt Xt,i - L lt+i ~ Xt,i }=0 w wt+i ~
Since wt is not a random variable at time t, we can multiply through the expectation operator w by it or by any power of it. Multiplying by (wt )- 1-w L , w w 1- 1- !L X w w w wt ~ wt ~ z,t+i i 1- 1- L ] w Et ( w ) lt+i Xt,i { (wt )[ Xt,i }=0 wt Xt,i -L lt+i ~ wt+i ~ w wt+i ~ i=0 or, Kw,t = where Kw,t Fw,t Then,
w 1- (1+L ) w wt ~ = Et ( w ) (lt+i ) Xt,i wt+i ~ i=0 w 1- X w wt ~ z,t+i i = Et ( w ) lt+i Xt,i Xt,i . wt+i ~ w i=0 w 1 ~ (wt )(1- 1-w L ) wt Fw,t . L
X
i
1+ L
L Kw,t wt = wt Fw,t ~
The infinite sums, Kw,t and Fw,t , have a recursive representations. It is crucial to exploit this fact, for computational tractability. Thus Fw,t w 1 z,t wt 1-w z,t+1 ~ = lt + ( w ) lt+1 (Xt,1 ) 1-w w wt+1 ~ w w 1- w 1 wt wt+1 ~ ~ z,t+2 + ( w )2 lt+2 (Xt+1,1 Xt,1 ) 1-w wt+1 wt+2 ~ ~ w w 1- w 1 ~ wt wt+1 wt+2 ~ ~ z,t+3 ( w )3 lt+3 (Xt+1,2 Xt,1 ) 1-w wt+1 wt+2 wt+3 ~ ~ ~ w + w 1 wt wt+1 ~ ~ wt+i-1 1-w z,t+i ~ i (Xt+1,i Xt,1 ) 1-w + ( w ) lt+i wt+1 wt+2 ~ ~ wt+i ~ w + ,
1- 1-(1+ w) w
L
.
(5.4)
34
or, Fw,t # " w 1 z,t z,t+1 wt 1-w ~ = lt + ( w ) (Xt,1 ) 1-w {lt+1 w wt+1 ~ w w 1 wt+1 1-w z,t+2 ~ + ( w ) lt+2 (Xt+1,1 ) 1-w wt+2 ~ w w 1- w 1 wt+1 wt+2 ~ ~ z,t+3 ( w )2 lt+3 (Xt+1,2 ) 1-w wt+2 wt+3 ~ ~ w + w 1 wt+1 ~ wt+i-1 1-w z,t+i ~ i-1 (Xt+1,i ) 1-w + ( w ) lt+i wt+2 ~ wt+i ~ w + } ! w 1 z,t wt 1-w 1-w ~ Fw,t+1 . + w Xt,1 = lt w wt+1 ~ w,t Wt wt Pt ~ wt t ~ = = , Wt-1 wt-1 Pt-1 ~ wt-1 ~
w 1-
(5.5)
Note,
so that wt /wt-1 = w,t /t . Substituting and rearranging, ~ ~
1
Fw,t
z,t = lt + w w
1 w,t+1
w
w,t+1 ~ 1-w Fw,t+1 , t+1
(5.6)
which corresponds to the expression we have worked with in the past. (But, (5.5) is simpler!) Now consider Kw,t : Kw,t =
1+ lt L
+ w (lt+1 )
2
1+L
+ ( w ) (lt+2 ) +
1+ lt L
1+L
wt ~ Xt,1 wt+1 ~
w 1-
w
(1+L )
wt wt+1 ~ ~ Xt+1,2 Xt,1 wt+1 wt+2 ~ ~
w 1-
w
(1+L )
w 1- (1+L ) w wt ~ = + w Xt,1 {(lt+1 )1+L wt+1 ~ w 1- (1+L ) w ~ 1+L wt+1 + w (lt+2 ) Xt+1,2 + } wt+2 ~ w 1- (1+L ) w wt ~ 1+L + w Xt,1 Kw,t+1 = lt wt+1 ~ w (1+L ) w,t+1 1-w ~ 1+ = lt L + w Kw,t+1 . w,t+1
(5.7)
35
In order for (5.5) and (5.7) to be well-defined objects, it is necessary that they be finite. This requires that the `discount rate' in (5.5) and (5.7) be less than unity in steady state: w w,t+1 ~ t+1
1 1- w
, w
w,t+1 ~ w,t+1
w 1-
w
(1+L )
< 1,
or, because of (5.3) and that w,t = t in steady state, w 1-w,2 -1
w
, w
We have completed the derivation of the wage rate from the household's first order condition. We now identify a consistency condition that must hold across all household wages, which will allow us to express the real wage, wt , just in terms of aggregate variables. ~ ~ The object, wt = Wt /Wt , will disappear from the analyis. Substituting the demand curve for the j th specialized input, (5.2) into (5.1), we obtain w w 1- # 1 Z 1" w w Wt,j lt = dj Wt 0 Z 1 w w 1 = lt (Wt ) w -1 (Wt,j ) 1-w dj ,
0
1-w,2 w (1+L ) -1
w
< 1.
lt
or, (Wt )
w 1-w
=
so that the condition across all wages is: Wt = = Z Z
1
Z
1
(Wt,j )
1 1-w
dj
0
w
(Wt,j )
1 1-w
dj
0
1-w dj +
(Wt,j )
1 1-w
1- w
Z
(Wt,j )
1 1-w
dj
w
1-w
.
Divide both sides by Wt ,
In the limits of integration, 1 - w refers to the households that reoptimize in period t, while w refers to the households that do not. Making use of the fact that whether households are selected to optimize or not is determined randomly, we can simplify the previous expression as follows: 1-w 1 1- 1 w ~t 1-w + w (~ w,t Wt-1 ) . Wt = (1 - w ) W " w,t ~ w,t
1 1- #1-w w
1 = (1 - w ) (wt )
1 1-w
+ w 36
,
or, after rearranging,
or,
Combining the optimality condition on wt , (5.4), with the consistency condition, (5.8), 1 1-w t-1 1-w ~ 1- 1-(1+ w) w L L Kw,t 1 - w t = wt Fw,t ~ 1 - w Kw,t 1 (1-(1+w )L ) ~ t-1 1-w 1 - w t 1 = Fw,t wt ~ , L 1 - w Kw,t = wt , ~ Fw,t (5.9)
1 - w wt = 1 - w
w,t ~ w,t
1 1- 1-w w
.
(5.8)
which is an expression relates which the real wage to aggregate variables only. It is interesting to consider the case, w = 0, when there are no sticky wages. In this case, (5.9) reduces to L
or, after substituting out for Kw,t and Fw,t and rearranging, w
L lt L = wt , ~ z,t
which says that the real wage in units of the consumption good, wt , is a markup, w , above ~ the household's marginal cost, L lt L /z,t , also expressed in terms of the consumption good. This is exactly what we would expect. Sticky wages also have an impact on household utility. Cross-household dispersion in wages lead to cross household dispersion in labor effort and therefore in utility. It can be verified that the cross-sectional average of utility in period t is: L log(ct ) - 1 + L
wt + wt
+ where ht is the unweighted sum of household hours and wt satisfies the following equation: 1-w ) 1 w (1+L ) w (1+ L w (1+ L ) w,t 1-w ~ 1- 1 - w w,t w w,t + ~ + wt = (1 - w ) + w wt-1 . (5.10) 1- w,t w
w(1+L ) -1
w
h1+L , t
For purposes of computing the steady state, it is convenient to wite (5.10) as follows: 1 w (1+L ) (1+ w,t 1-w ~ w1- L ) (1+ (1+ w + w1- L ) + w1- L ) w,t ~ 1 - w w,t w w wt-1 = (1 - w ) + w . wt 1 - w w,t 37
From this expression, it is evident that existence of a steady state will require w w,t w,t ~ w(1+L ) -1
w
= w
1 wt t ~ wt-1 ( t-1 )2 1-2 ~
w(1+L ) -1
w
= w
where denotes the actual steady state inflation rate and denotes the constant in the price updating equation. A `natural' specification might be = 1. However, note that if we set w small market then the power in the above expression can be quite large. For example, if w = 1.05, then w /(w - 1) = 21. Then, if 2 is small (say, 0.13) has to be only a little above unity for the condition to be violated. For example, suppose = 1.0092 (a 3.7 percent annual inflation rate), w,2 = 0.13, w = 1.05, L = 1, then w (1-w,2 ) w(1+L ) -1
w
(1-2 ) w(1+L ) -1
w
< 1,
= 1.16.
5.2. Aggregate Resource Constraint We now develop a relationship linking aggregate homogeneous labor effort in the goods market, lt , to aggregate household employment, ht , w Z 1 Z 1 Wt,j 1-w ht = hj,t dj = lt dj (5.11) Wt 0 0
= lt (wt ) 1-w ,
w
say, where
(wt )
w 1-w
= = = =
w Wt,j 1-w dj Wt 0 w w Z Z Wt,j 1-w Wt,j 1-w dj + dj Wt Wt 1- w w w Z w w Wt-1,j 1-w w,t 1-w ~ dj (1 - w ) wt1-w + w,t Wt-1 w w 1- w w w,t ~ 1-w + wt-1 , (1 - w ) wt w,t "
w 1-w
Z 1
so that
wt = (1 - w ) wt
+
w,t ~ w w,t t-1
w 1- # 1-w w w
.
We substitute out the expression for wt using (5.8):
1 1- w w
1 - w wt = [(1 - w ) 1 - w
w,t ~ w,t
+ w
w,t ~ w w,t t-1
w 1-
w
]
1-w w
.
(5.12)
38
In order for wt to have a well-defined steady state value, we require that the coefficient on w 1- w be less than unity: wt-1 w w,t 1-w ~ <1 w w,t
or, in steady state:
where wt satisfies (5.12). Note that when there are no sticky wages, so that w = 0, then wt = 1 and no adjustment is made. Combining (5.6), (5.7), (5.9) and (5.11), we obtain:
< 1. We now can write the resource constraint in terms of aggregate household employment like this: f h i1- f -1 ww -1 h - , (5.13) ct + kt - (1 - )kt-1 = (pt ) t kt-1 (wt ) t
w
(1-w,2 ) w -1
w
1 ht (w ) Et { t + w (~ w,t+1 ) 1-w w ct
w w -1
1 w,t+1
t+1
w 1-
w
Fw,t+1 - Fw,t } =(5.14) 0
1 1-w (1+L ) w,t+1 1-w ~ w 1- (1+L ) i1+L h w w 1 1 - w w,t+1 w,t+1 ~ + w wt+1 Fw,t+1 ~ (5.15) Et { (wt ) w -1 ht w,t+1 L 1 - w
1 1- 1-w (1+L ) w ~ 1 - w w,t 1 w,t - wt Fw,t } = 0 ~ L 1 - w
5.3. The Other Equilibrium Conditions Sticky wages has replaced the labor supply curve, (3.11), with the equilibrium conditions in the previous subsection. above. In terms of the equilibrium conditions for the version of the model with just sticky prices, this means that we must replace (3.17) by (3.10). Also, if we are to write the firm efficiency conditions in terms of hours supplied by workers, ht , then we must use (5.11). Thus, equation (3.10) must be replaced by ! w (wt ) w -1 ht wt ~ (5.16) st = (1 - ) t kt-1 In addition, the intertemporal Euler equation for the household, with the rental rate of capital substituted out using the firm marginal cost condition, must be replaced by: 1- w wt+1 w -1 ht+1 1 st+1 + (1 - ) = 0 - + t+1 (5.17) ct ct+1 kt 39
Finally, the pricing equations need to be modified, given the new definition of Yz,t (see (5.13)). ( ) 1 2 1-2 1- 1- w f 1 f f t Et (p ) -1 t kt-1 (wt ) w -1 ht - + p Fp,t+1 - Fp,t = 0, ct t t+1 (5.18) and f 1- 1 f -1 ww -1 h f (pt ) - st + (5.19) t kt-1 (wt ) t ct 1 1 1-f 2 1-2 1- 1-f f f 2 1-2 1- 1 - t2 1-2 1-f 1 - p t-1t p f t+1 t Fp,t+1 - Fp,t = 0 p t+1 1 - p 1 - p Note that for Fp,t and Kp,t to be finite, it is necessary that the relevant discount rates be less than unity in steady state:
1-2 -1 f
p , p
~ We now have N = 11 unknowns, (k, s, h, c, , Fp , p , Fw , w, w , w+ ). The N -1 equilibrium conditions are (5.16), (5.17), (5.13), (3.20), (5.18), (5.19), (5.12), (5.14), (5.15) and (5.10). These equilibrium conditions have been entered into the Dynare file, newsimplemodel.mod, in subdirectory stickypriceswages. 5.4. Analysis of the Equilibrium With both sticky prices and wages, it should be clear that the RBC equilibrium allocations are not attainable. From the point of view of the wedges in the resource constraint, this would require setting t and w,t to , but this would in effect fix the real wage, and the efficient allocations require that the real wage react to shocks. Consider a case in which we have a hope that the Ramsey equilibrium coincides with the RBC equilibrium. Suppose there are only sticky wages and no sticky prices, so that p = 0, w > 0. Let's see if we can identify an equilibrium that coincides with the equilibrium in the RBC economy. Suppose we set: w,t = w,t . ~ (5.20)
By (5.12), this implies that, if w-1 = 1, then wt = 1 for all t 0. This will be necessary if we're to reproduce the RBC model allocations, because wt appears as a wedge in several places and setting it to unity eliminates those wedges. For convenience, we repeat the definitions of the objects in (5.20):
(1-2 ) f -1
f
< 1.
w,t
Wt wt Pt ~ wt t ~ = = , w,t ( t-1 )w,2 1-w,2 . ~ Wt-1 wt-1 Pt-1 ~ wt-1 ~
Evidently, achieving w,t = w,t requires that there be no indexing to inflation, i.e., w,2 = 0 ~ (obviously, wages can't be the same across households - as efficiency requires - if some 40
households link their wages to inflation, which must be time varying to ensure real wage flexibility.) Now, consider (5.14) and (5.15). Imposing (5.20): Et { w,t+1 ~ ht + w Fw,t+1 - Fw,t } = 0 w ct t+1 1 1 wt+1 Fw,t+1 - ~ wt Fw,t } = 0 ~ L L
Et {h1+L + w t or, using,
wt w,t+1 ~ = wt+1 , w,t+1 = w,t+1 ~ ~ t+1 these reduce to: ht w,t+1 ~ = -Et w Fw,t+1 - Fw,t w ct t+1 w,t+1 ~ 1 = - wt Et w ~ Fw,t+1 - Fw,t . L t+1 ht 1 wt ~ , L w ct
h1+L t
Equating these two, we obtain: h1+L = t or, wt ~ . w This is just the static efficiency condition for households in the RBC model, if w = 1. Presumably, efficiency of our proposed policy will require w = 1. Now let's pursue the implications of p = 0. By (5.18) and (5.19) (and, that p = 1 by t (3.20)), 1 1- - = Fp,t , t kt-1 ht ct and 1 1- f t kt-1 ht - st = Fp,t . ct Then, equating these two, we obtain: L hL ct = t st = 1 . f (5.21)
If f = 1, then (5.21) and (5.16) imply that the intratemporal Euler equation in the RBC model is satisfied. We conclude that if f = w = 1, w,2 = 0, then an equilibrium in which w,t = w,t = for all t has the property that all three efficiency conditions of the RBC model ~ are satisfied: (i) the resource constraint, (5.13), reduces to the RBC resource constraint, (ii) conditions (3.10), (5.21) and f = 1 imply that the RBC intratemporal efficiency condition hols and (iii) condition (5.17) with wt = st = 1 corresponds to the RBC model intertemporal 41
equation. Since the three efficiency conditions of the RBC model uniquely (together with a boundedness condition) characterize the best possible allocations given preferences and technology, it follows that under the stated conditions, w,t = w,t = is the Ramsey policy. ~ We summarize these results in the form of a proposition Proposition 5.1. If f = w = 1, p = w,2 = 0, then the Ramsey allocations coincide with those of the RBC economy, and w,t = . We now compare the Ramsey and the RBC allocations. Consider the following parameter values: = 0.99, L = 109.8, f = 1.002, = 0.40, = 0.025, p = 0.75, 2 = 0.6, w = 1.05, L = 1. We consider several special cases. We begin with the case, w ' 0, f = w = 1.002, when prices are sticky and wages are not, and monopoly power is minimized. The Ramsey and RBC economy responses to a technology shock are presented below.
percent deviation from ss percent deviation from ss
consumption 0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20
hours worked 3.661 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 -4 5 10 15 20 0 -2 20 3.6605
net inflation (APR)
actual steady state 3.66 5 10 15 20 -3 real marginal cost (% dev from ss) x 10 2
percent deviation from ss
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest 7.85 7.8 7.75 5 10 15 20 actual steady state
percent deviation from ss
5
10
15
20
Annualized, percent
rbc model sticky price model
w = 1e-006 p = 0.75 f = 1.002 w = 1.002
Note that the Ramsey and RBC allocations virtually coincide, as expected
42
Now we consider the opposite case, in which w = 0.83 and p = 0.000001 :
percent deviation from ss percent deviation from ss
consumption 0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20
hours worked 4 0.4 0.2 0 5 10 15 technology shock 0.8 0 0.6 0.4 0.2 5 10 15 20 -5 -10 20 3 2 1
net inflation (APR)
actual steady state
percent deviation from ss
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest actual steady state
percent deviation from ss
5 10 15 20 -7 real marginal cost (% dev from ss) x 10 5
5
10
15
20
Annualized, percent
rbc model sticky price model
8.2 8 7.8 5
w = 0.83 p = 1e-006 f = 1.002 w = 1.002 w,2 = 0
10 15 20
The two allocations are also very similar here, consistent with our proposition. Note now inflation drops sharply with the technology shock, to allow the real wage to rise. Marginal cost is essentially constant. We now investigate how far from the RBC model you end up when the conditions for RBC=Ramsey are not satisfied. Here is what happens when you set w,2 = 0.13. It makes
43
virtually no difference, although there is some noticeable impact on Rt .
percent deviation from ss percent deviation from ss
consumption 0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20
hours worked 4 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 10 15 20 -5 -10 20 3 2 1
net inflation (APR)
actual steady state
percent deviation from ss
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest 8.1 8 7.9 7.8 5 10 15 20 actual steady state
percent deviation from ss
5 10 15 20 -7 real marginal cost (% dev from ss) x 10 0
5
10
15
20
Annualized, percent
rbc model sticky price model
w = 0.83 p = 1e-006 f = 1.002 w = 1.002 w,2 = 0.13
44
Now consider the case, p = 0.75, w = 0.83, f = 1.20, w = 1.05
consumption 0.35 0.3 0.25 5 0.5 0.4 0.3 0.2 5 10 15 20 net nominal rate of interest 7.8 10 15 capital stock 20 hours worked 0.6 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 -0.6 5 10 15 20 5 10 15 20 -0.2 -0.4 20 3.6 3.4 3.2 actual steady state 5 10 15 20 real marginal cost (% dev from ss) 0 net inflation (APR)
percent deviation from ss
percent deviation from ss
Annualized, percent
percent deviation from ss
percent deviation from ss
rbc model sticky price model
7.6
= 0.83 = 0.75 = 1.2 = 1.05
7.4 5 actual steady state 10 15 20
w
p
f
w
w,2
= 0.13
This change has a more noticeable impact on the interest rate. And, predictably, it reduces the inflation impact of the shock. In terms of the impact on quantity allocations, the effect is rather small. A surprising result was obtained by perturbing the previous parameterization, and setting
45
f = w = 1.002 and otherwise leaving the parameters unchanged. We obtained:
consumption hours worked 1.5 1 0.5 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 10 15 20 -0.1 -0.15 5 10 15 20 20 3.65 3.6 3.55 3.5 actual steady state net inflation (APR)
percent deviation from ss
0.4 0.3 0.2 5 10 15 capital stock 20
percent deviation from ss
percent deviation from ss
0.6 0.4 0.2 5 10 15 20 net nominal rate of interest 7.8 actual steady state
percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss)
-0.05
Annualized, percent
rbc model sticky price model
7.75 7.7 7.65 5
= 0.83 = 0.75 = 1.002 = 1.002
w p f w
10 15 20
w,2
= 0.13
Note how much stronger the response of the allocations in the Ramsey equilibrium are!
6. Adding Habit Persistence and Investment Adjustment Costs to the Model
6.1. The Equilibrium Conditions The marginal utility of consumption, z,t , is now 1 1 Et z,t - = 0. + b ct - bct-1 ct+1 - bct
(6.1)
With this change, 1/ct must be replaced by z,t in (5.18), (5.19), and (5.14). Adjustment costs in investment change the household first order condition for investment, (5.17): 1- w w -1 h w 1 t+1 -z,t + z,t+1 t+1 t+1 st+1 + qt+1 (1 - ) = 0, (6.2) qt kt and add a new first order condition for investment: Et [zt qt F1,t - zt + zt+1 qt+1 F2,t+1 ] = 0. 46
We suppose that F (It , It-1 ) = [1 - S(It /It-1 )] It S 00 It 2 = 1- ( - 1) It , 2 It-1 so that F1t = 1 - F2,t+1 S 00 It It It ( - 1)2 - S 00 ( - 1) 2 It-1 It-1 It-1 2 It+1 It+1 = S 00 ( - 1) . It It
Substituting this into the first order condition for investment: S 00 It It 2 00 It - 1) - S ( - 1) Et {zt qt 1 - ( 2 It-1 It-1 It-1 2 It+1 00 It+1 -zt + zt+1 qt+1 S ( - 1) } = 0. It It With the investment adjustment costs, the resource constraint is changed f h i1- f -1 (w ) ww ht -1 - ct + It = (pt ) t kt-1 t where kt - (1 - )kt-1 1 t+1 S 00 It 2 = 1- ( - 1) It 2 It-1
(6.3)
(6.4)
(6.5)
The equation defining the nominal rate of interest is: Et {
z,t+1 (1 + Rt ) - z,t } = 0
(6.6)
The variables to be determined in equilibrium are the following N = 15: Rt , z,t , It , t , qt , ht , ct , kt+1 , st , Fp , p , Fw , w, w , w+ . The N - 1 equations are (5.18), (5.19), (5.14) with ~ 1/ct replaced with z,t , (5.15), (5.16), (3.20), (5.12), (5.10), (6.1), (6.2), (6.3), (6.6), (6.4) and (6.5). With some algebra, it can be verified that the average, across all households, of period utility is: L log(ct - bct-1 ) - 1 + L
wt + wt
We use this utility function to define the objective in the Ramsey problem. The equilibrium conditions of this version of the model have been entered into the Dynare file, newsimplemodel.mod, in subdirectory sticypriceswageshabitadjust.
w(1+L ) -1
w
h1+L . t
47
6.2. Analysis of Equilibrium Consider the following parameter values: = 0.99, L = 109.8, f = 1.2, w = 1.05, = 0.40, = 0.025, 2 = 0.6, S 00 = 5.1, L = 1, b = 0.63, w,2 = 0.13, w = 0.83, p = 0.75 (6.7)
There are special cases of this model worth considering. Consider, for example, the case, S 00 = 0.000001, w = 0.83, f = w = 1.002, b = 0, p = 0.000001, w,2 = 0. This corresponds to a case that we considered before, and it's worth verifying that we reproduce it now:
consumption 0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20 hours worked 4 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 10 15 20 -5 -10 5 10 15 20 20 3 2 1 5 10 15 20 -7 real marginal cost (% dev from ss) x 10 0 actual steady state net inflation (APR)
percent deviation from ss
percent deviation from ss
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest
Annualized, percent
percent deviation from ss
percent deviation from ss
8.2 8 7.8 5
actual steady state
rbc model sticky price model
= 0.83 = 1e-006 = 1.002 = 1.002
w p f w
w,2
=0
10
15
20
Consider now the `opposite' from the above case, the one in which p = 0.75, w =
48
0.000001 :
= 1e-006 = 0.75 = 1.002 = 1.002
w p f w
percent deviation from ss percent deviation from ss
w,2
=0
0.26 0.24 0.22 0.2 0.18 5 10 15 capital stock 20
percent deviation from ss percent deviation from ss
consumption
hours worked 3.661 0.4 0.2 0 5 10 15 technology shock 0.8 0.6 0.4 0.2 -4 5 1 10 15 price of capital 20 0 -2 20 3.6605
net inflation (APR)
actual steady state 3.66 5 10 15 20 -3 real marginal cost (% dev from ss) x 10 2
0.35 0.3 0.25 0.2 0.15 5 10 15 20 net nominal rate of interest 7.85 7.8 7.75 5 10 15
5
10
15
20
percent deviation from ss
Annualized, percent
actual 0.5 steady state 0 -0.5 -1 5 10 15 20
rbc model sticky price model
20
This reproduces what we had before. To establish a suitable benchmark for this model with habit persistence, we solve the version of the RBC model with habit persistence and adjustment costs (see rbcmodel.mod). The first order condition for investment is (6.3), which, in steady state implies qt = 1. The intertemporal Euler equation for this model is (5.17) with wt = st = 1. In steady state, that equation is: 1- 1 h + 1 - , = k or, "1 # 1 - (1 - ) 1- h = . k The intratemporal Euler equation is, in steady state: k = L hL . z (1 - ) h The resource constraint is (6.4). This, combined with the capital accumulation equation, (6.5), is, in steady state: # " 1- h - k. c= k 49
The marginal utility of consumption is (6.1), which, in steady state is: z = Combining the previous three equations: (1 - b) (1 - ) 1-b or, h= # - " 1- h h = - kL hL , k k
1 h 1- 1+L (1 - ) k h i h 1- - L k
1 - b . c (1 - b)
Given this RBC model benchmark, we simulated the model with w = f = 1.002, w = 0.000001, p = 0.75, S 00 = 5.1, b = 0.63. We found the following
w = 1e-006 p = 0.75 f = 1.002 w = 1.002 w,2 = 0
percent deviation from ss
1-b 1-b
0.5 0.4 0.3 0.2 5 0.2 10 15 capital stock
percent deviation from ss
consumption
hours worked 0 -0.2 -0.4 -0.6 3.659 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 2 1.5 1 0.5 0 5 10 15 20 10 15 price of capital 20 0 -5 -10 20 3.661 3.66
net inflation (APR)
actual steady state
20
percent deviation from ss
percent deviation from ss percent deviation from ss
5 10 15 20 -3 real marginal cost (% dev from ss) x 10 5
0.15 0.1 0.05 5 10 15 20 net nominal rate of interest
Annualized, percent
5
10
15
20
7 6 5 4 5 10 15 20 actual steady state
rbc model sticky price model
Again, it looks like the Ramsey policy duplicates the RBC model. Next, we considered the case, w = 0.83, p = 0.000001. The results are consistent with
50
expectations:
percent deviation from ss percent deviation from ss
consumption 0.5 0.4 0.3 0.2 5 0.2
w = 0.83 p = 1e-006 f = 1.002 w = 1.002 w,2 = 0
hours worked 0 -0.2 -0.4 -0.6 -0.8 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 2 1.5 1 0.5 0 5 10 15 20 10 15 price of capital 20 0 -1 -2 5 20 2 0 4
net inflation (APR)
actual steady state
percent deviation from ss
0.15 0.1 0.05 5 10 15 20 net nominal rate of interest 7.5 7 6.5 6 5 10 15 20 actual steady state
percent deviation from ss
10 15 capital stock
20
5 10 15 20 -6 real marginal cost (% dev from ss) x 10 1
10
15
20
percent deviation from ss
Now let's see how far off things get when there are both sticky prices and sticky wages.
Annualized, percent
rbc model sticky price model
51
First, we simply set w = 0.83, p = 0.75 and kept f = w = 1.002, w,2 = 0. Then,
w = 0.83 p = 0.75 f = 1.002 w = 1.002 w,2 = 0
percent deviation from ss percent deviation from ss
1.5 1 0.5 5 0.4 0.3 0.2 0.1 10 15 capital stock
percent deviation from ss
consumption
hours worked 1.5 1 0.5 0 -0.5 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 10 15 price of capital 20 -0.4 -0.2 20 3.65 3.6 3.55 3.5 3.45
net inflation (APR)
actual steady state
20
percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss) 0
percent deviation from ss
5 10 15 20 net nominal rate of interest 5 0 -5 5 actual steady state 10 15 20
5
10
15
20
Annualized, percent
6 4 2 0 5 10 15 20
rbc model sticky price model
Note - starred line for price of capital is sticky price model and solid line is rbc Note that the response of the allocations is now dramatically higher than it is in the RBC model! This is the same as the striking result we observed when we were studying sticky
52
prices and wages alone. When f was changed to 1.0093 :
= 0.83 = 0.75 = 1.009 = 1.002
w p f w
percent deviation from ss percent deviation from ss percent deviation from ss
w,2
=0
consumption 1 0.8 0.6 0.4 0.2 5 0.3 0.2 0.1 10 15 capital stock
hours worked 3.7 0.5 0 -0.5 5 10 15 technology shock 0.8 0.6 0.4 0.2 -0.8 5 4 3 2 1 0 5 10 15 20 10 15 price of capital 20 20 3.6 3.5 3.4 3.3
net inflation (APR)
actual steady state
20
percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss) 0 -0.2 -0.4 -0.6
percent deviation from ss
5 10 15 20 net nominal rate of interest
5
10
15
20
Annualized, percent
6 4 2 5 10 15 20 actual steady state
rbc model sticky price model
Note - starred line for price of capital is sticky price model and solid line is rbc
3 In all cases, we did the calculations using initval in Dynare, so that our initial guess of the steady state was given to Dynare as an initial guess. Typically, since we compute the steady state exactly ourselves in ssnew.m, Dynare simply accepts our steady state. However, in the cases f = 1.009, and f = 1.02 Dynare got lost when it started with our initial conditions. In these cases, we manually verified that our steady state is correct, and then forced Dynare to accept our steady state. The results in this figure are based on this type of run. We bypassed Dynare's steady state calculation by replacing line 59 in dynare_solve.m with `if 5 > 2'.
53
When we tried f = 1.02, we obtained the following result:
= 0.83 = 0.75 = 1.02 = 1.002
w p f w
percent deviation from ss percent deviation from ss
w,2
=0
net inflation (APR)
consumption 0.8 0.6 0.4 0.2 5 10 15 capital stock
hours worked 0.5 3.6 0 3.4 -0.5 3.2 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 3 2 1 0 5 10 15 20 10 15 price of capital 20 -1 -0.5 20
actual steady state 5 10 15 20 real marginal cost (% dev from ss) 0
20
percent deviation from ss
0.25 0.2 0.15 0.1 0.05 5 10 15 20 net nominal rate of interest
Annualized, percent
percent deviation from ss
5
10
15
20
percent deviation from ss
7 6 5 4 3 5 10 15 20 actual steady state
rbc model sticky price model
Note - starred line for price of capital is sticky price model and solid line is rbc Note how the results appear to be continuous in the parameter f , and how the Ramsey allocations are converging to the RBC model calculations. When we tried f = 1.20, the
54
two almost coincide:
= 0.83 = 0.75 = 1.2 = 1.002
w p f w
percent deviation from ss percent deviation from ss
w,2
=0
net inflation (APR)
consumption 0.6 0.4 0.2 5 10 15 capital stock
hours worked 0 -0.2 -0.4 -0.6 -0.8 5 10 15 technology shock 0.8 0.6 0.4 0.2 5 2 1.5 1 0.5 0 5 10 15 20 10 15 price of capital 20 -1 -0.5 20 3.8 3.6 3.4 3.2
actual steady state
20
percent deviation from ss
percent deviation from ss percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss) 0
0.25 0.2 0.15 0.1 0.05 5 10 15 20 net nominal rate of interest
Annualized, percent
5
10
15
20
7 6 5 4 5 actual steady state 10 15 20
rbc model sticky price model
Note - starred line for price of capital is sticky price model and solid line is rbc We then set all parameters to their benchmark values, w = 1.05, w,2 = 0.13, when we
55
obtained
= 0.83 = 0.75 = 1.2 = 1.05
w p f w
percent deviation from ss percent deviation from ss
w,2
= 0.13
net inflation (APR)
consumption 0.6 0.4 0.2 5 10 15 capital stock
hours worked 3.8 0 -0.2 -0.4 -0.6 -0.8 5 10 15 technology shock 1 0.8 0.6 0.4 0.2 5 2 1.5 1 0.5 0 5 10 15 20 10 15 price of capital 20 -1 -0.5 20 3.6 3.4 3.2
actual steady state
20
percent deviation from ss
percent deviation from ss percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss) 0
0.25 0.2 0.15 0.1 0.05 5 10 15 20 net nominal rate of interest
Annualized, percent
5
10
15
20
7 6 5 4 5 actual steady state
rbc model sticky price model
10
15
20
Note - starred line for price of capital is sticky price model and solid line is rbc Interestingly, the Ramsey and RBC model allocations almost coincide now.
7. Anticipated Shocks
We now consider an alternative representation of the technology shock. We replace the representation (3.16) with log
t
= log
t-1
+ t-12 + t .
The code pertaining to this case appears in the subdirectory, `mirage'. We began by simulating this model under the benchmark parameterization, (6.7), modified so that f = w = 1.002, p = 0.000001, w = 0.83, w,2 = 0. This is the case of sticky wages and flexible prices. The figure below shows that the Ramsey allocations exactly reproduce the RBC allocations in this case. The experiment is one in which there is an expectation that techology will rise by 1 percent in period 13, an expectation which turns out not to be fulfilled. Note how sharply the real rate of interest rises ultimately with the shock. Note, too, that the real wage falls. This reflects that the real wage must be equal to the marginal product of labor, and the latter must fall. The increase in employment, therefore, reflects a positive labor supply
56
effect.
rbc model sticky price model consumption
= 0.83 = 1e-006 = 1.002 = 1.002
w p f w
percent deviation from ss percent deviation from ss
w,2
=0
percent deviation from ss
hours worked 4 0.4 0.2 0 5 10 15 20 -12 x 10 technology shock 0 1 -1 -2 -3 5 10 15 20 price of capital 0 -1 -2 3.8 3.6 3.4 3.2
net inflation (APR)
0.1 0.05 0 5 0.8 0.6 0.4 0.2 0 5 10 15 20 net nominal rate of interest 11 10 9 8 5 10 15 20 actual steady state 10 15 investment 20
actual steady state
percent deviation from ss
5 10 15 20 -6 real marginal cost (% dev from ss) x 10
5
percent deviation from ss
percent deviation from ss
10 15 real wage
20
Annualized, percent
-0.2 -0.4 -0.6 -0.8 5 10 15 20
0 -0.05 -0.1 -0.15 -0.2 5 10 15 20
Next, we made prices sticky and wages flexible, p = 0.75, w = 0.000001. We obtained
57
the following results:
rbc model sticky price model consumption 0.1 0.05 0 5 0.8 0.6 0.4 0.2 0 5 10 15 20 net nominal rate of interest 16 14 12 10 8 5 10 15 20 actual steady state 10 15 investment 20
w = 1e-006 p = 0.75 f = 1.002 w = 1.002 w,2 = 0
percent deviation from ss percent deviation from ss
percent deviation from ss
hours worked 3.662 3.6615 0.2 0 5 10 15 20 -12 x 10 technology shock 3.661 3.6605
net inflation (APR)
0.4
percent deviation from ss
actual steady state 5 10 15 20 real marginal cost (% dev from ss)
15 10 5 0
0.01 0 -0.01 5 10 15 20 price of capital 5
percent deviation from ss
percent deviation from ss
10 15 real wage
20
Annualized, percent
-0.2 -0.4 -0.6 -0.8 5 10 15 20
0 -0.05 -0.1 -0.15 -0.2 5 10 15 20
Note how the inflation rate is essentially constant now, so that the fall in the real wage is accomplished by a fall in the nominal wage rate.
58
Next, we raised w to 0.83, and obtained the following results:
rbc model sticky price model consumption
w = 0.83 p = 0.75 f = 1.002 w = 1.002 w,2 = 0
percent deviation from ss
percent deviation from ss percent deviation from ss
hours worked 0.5 3.7 0 3.65 3.6 5 10 15 20 -11 x 10 technology shock
net inflation (APR)
0 -0.2 -0.4 5 0.8 0.6 0.4 0.2 0 -0.2 5 10 15 20 net nominal rate of interest 40 30 20 10 5 10 15 20 actual steady state 10 15 investment 20
actual steady state
-0.5
percent deviation from ss percent deviation from ss
5 10 15 20 real marginal cost (% dev from ss) 0.2 0 -0.2 -0.4
20 15 10 5 0
5 0 -1 -2 -3 5
percent deviation from ss
10 15 20 price of capital
5 0.2 0.1 0 -0.1 -0.2 5
10 15 real wage
20
Annualized, percent
10
15
20
10
15
20
We have seen this sort of result several times now. Apparently, when there are sticky wages and prices, then low markups cause the Ramsey allocations to look very different from the RBC allocations. Note, for example, that the real wage now increases.
59
Finally, we consider our benchmark parameter values, (6.7):
= 0.83 = 0.75 = 1.2 = 1.05
percent deviation from ss percent deviation from Annualized, percent ss percent deviation from ss percent deviation from ss deviation from ss percent percent deviation from ss
w
p
f
w
w,2
= 0.13
net inflation (APR)
consumption
hours worked 3.8 0.4 0.2 0 5 10 15 20 -15 x 10 technology shock 0 -5
percent deviation from ss
0.15 0.1 0.05 0 5 1 0.5 0 10 15 20 net 5 nominal rate of interest 18 16 14 12 10 8 5 -9 x 10 10 15 investment 20
3.6 actual steady state 5 real marginal10 (% dev from ss) cost 15 20 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 5 real wage 20 10 15 0.2 0 -0.2 5 10 15 20 3.4
-10 5 price of capital 20 10 15
actual steady state
-0.2 -0.4 -0.6 -0.8 -1 5 10 15 20
10 15 output gap 20
2.5 2 1.5 1 0.5
rbc model sticky price model 5 10 15 20
These results are very interesting in several ways. First, the Ramsey quantity allocations are very similar to the RBC allocations. Second, note that the Ramsey real wage is now rising rather than falling. This is an important finding, and needs further explanation. We have also reported the output gap gapt = (p ) f -1 t
f
wt w -1
w
(1-)
.
This quantity is extraordinarily small, its maximum is 2.510-9 . This is in percent deviation from steady state. We investigated what happens when wages and prices are virtually completely sticky, so that the real wage cannot change, by setting p = w = 0.99 and keeping all other parameters
60
at their benchmark values. We obtained essentially the same results:
percent deviation from ss deviation from ss percent Annualized, percent percent deviation from ss percent deviation from ss deviation from ss percent percent deviation from ss
consumption
hours worked 0.4 0.2 0 0 5 10 15 20 -13 x 10 technology shock 3.66 3.658 3.656 3.654 3.652
net inflation (APR)
0.2 0.1 0 5 1 10 15 investment 20
actual steady state 5 20 real marginal10 (% dev from ss) cost 15 0
0.5 0 10 15 20 net 5 nominal rate of interest
-0.5 -0.5 -1 5 price of capital 20 10 15
percent deviation from ss
-1 5 0 -0.1 -0.2 5 10 15 20 10 15 real wage 20
18 16 14 12 10 8 5 -8 x 10 6 4 2 5
actual steady state 10 15 output gap 20
-0.2 -0.4 -0.6 -0.8 -1 5 10 15 20 rbc model sticky price model
w = 0.99 p = 0.99 f = 1.2 w = 1.05 w,2 = 0.13
10 15 20
8. Pulling All the Equilibrium Conditions Together and Adding Growth
We now consider the case, z > 0, in (3.3). With this change, the economy follows a deterministic growth path in steady state. All variables should be interpreted as scaled by zt . This causes the dynamic equilibrium conditions in the model to acquire growth rate adjustments. We repeat all the equilibrium conditions of the previous section, except the growth adjustments have been incorportated. The equations pertaining to prices are:
1 1- f f
and
1 - p p - 1 - p t
t
2 t-1 1-2
t
1 - p
+ p
2 1-2 t-1 t
p t-1
f 1-
f
1-f
f
=0
(8.1)
Et
(
z,t (p ) t
f f -1
kt-1 z
) 1 2 1-2 1- 1- w f t - + p Fp,t+1 - Fp,t = 0, (wt ) w -1 ht t+1 (8.2) 61
and z,t f (p ) f -1 t 2 t t+1
f 1- 1-2 f
p
f
In the price equations, the only required adjustment is in the production function, where kt-1 needs to be adjusted. Now, consider the wage equations. These are (5.14), (5.15), (5.12), and (5.10). In adjusting these equations, it is to be born in mind that the wage updating term, w,t+1 , does ~ not need to be adjusted, since in t he model with technological growth we suppose that the wages of non-optimizing households must be augmented by the growth rate of technology as follows: Wj,t = w,t z Wj,t-1 , w,t ( t-1 )w,2 1-w,2 . ~ ~ At the same time the definit...