An Analysis of the Dynamics of the Vintage Capital Lijia Mo∗, Rouch Financial Group, Lincoln, NE February 7, 2016
This paper explores the route that idiosyncratic shock impacts vintage capital distribution and aggregate employment by changing the fraction of plants that choose to adjust. The neutral technological shocks increase job reallocation and aggregate employment. The increasing adjustment hazard affects the job reallocation by impacting the marginal product of capital for all vintages, therefore the capital circulates from the updated sector 0 to old sector 1. Then labor forces in the old sector with old technology are endowed with more capital, which boosts their relative wages and the labor forces in old sector increase. Aided with the endogenous adjustment ratios, the vintage model gets a new explanation. JEL Codes: E1 E2 O3 Key Words: Vintage Capital, Investment Adjustment, Technological Change, Labor Market.
∗
[email protected]
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1 Introduction
Is technological progress good or bad for workers? This question is interesting to the average person in the street as well as to professional economists. The information technology revolution induces advent of the Internet economy which has spurred our curiosity further about this matter. We can guess that some workers with outmoded skills will be losers, but that other workers will be made more productive by having better tools at their disposal. What will be the main storyline in the next century’s history books? Will the final analysis be that technology replaced flesh-and-blood workers, and result in generally higher unemployment and lower real wages? Or will it be that humans become more valuable as they could use their time more effectively? Answering the questions requires an understanding of vintage model, which embodies the different levels of technology productivity into the contemporaneous capital and worker and using the modern numerical methods to analyze the model dynamics to reveal the coexisting driving factors. Furthermore, investigating the cross-sectional plant distribution according to capital levels and over time evolution by tracking the behavior of plants with same vintage of capital stock for reason of economic inequality under uncertainty as a general equilibrium irreversibility investment analysis paves the foundation for a labor market analysis. Economists mostly believe that changes in the growth rate of productivity have profound influences on the quality of workers’ economic lives. Cadiou, Dees and Laffargue [2000] used the French data to reveal the job creation and destruction in France. Previous efforts made by Jovanovic [1998], who introduced the vintage capital model to reveal the aggregate employment fluctuation on the technology shock. Michelacci and Lopez-Salido [2005] used structural VAR models to reveal that technological progress can lead to the destruction of technologically obsolete jobs and cause unemployment. They found that neutral technology shocks increase job destruction and job reallocation, and reduce aggregate employment. The neutral technological progress
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prompts waves of Schumpeterian creative destruction. The previous empirical literature uncovered one key stylized fact that: it was an important route that aggregate shocks affected aggregate employment by changing the fraction of plants that choose to adjust. Investigation of investment dynamics in Thomas [2002] answered the question of whether the individual economic agents’ lumpy decision matters in the aggregate. The answer is sometimes. The recent literature has a general theme that the micro-level quasi-fixity in capital has important implication for aggregate investment and more generally, for aggregate economic activity. Thomas [2002] model is consistent with several empirical results. First, plant-level capital adjustment exhibits long episodes of relative inactivity punctuated by lumpy investments such as the U.S. data in the Longitudinal Research Database(LRD) used in Doms and Dunne [1998] showed that between 25 and 40 percent of the typical plant’s total investment expenditure over a 17 year sample is concentrated into a single large episode. Second, the probability of an investment spike rises in the level of capital imbalance, as Caballero, Engel and Haltiwanger[1995] demonstrated an upward sloping empirical adjustment hazard (what is it?). Lastly, this upward sloping adjustment hazard is evident in terms of time-since-adjustment. The emphasis has been made by studying how the interaction of upward sloping adjustment hazards with time-varying distributions of plant-level capital can produce large effects on aggregate investment demand. Thomas (2002) model contains both empirical evidence on determining the aggregate impact of non-convex micro-level capital adjustments, namely rising adjustment hazards, and time-varying plant distributions. The results show that when the business cycle is assumed to originate from exogenous changes in aggregate productivity, the state-dependent adjustment economy exhibits a striking negative result, that is, lumpy investment has only minor consequences for the reduced-form relations between productivity shocks and real aggregate quantities. The fact that the plants have lengthy periods of investment inactivity (i.e. irreversibility), and their ability to change the timing of their investment at relatively small cost, make the irrelevance very misleading (in
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what sense?). The explanation is that a household’s preference for smooth consumption profiles restrains shifts in investment demand. Thomas [2003b] use a similar approach to study S,s inventory accumulation and shows that these adjustment probabilities are functions of the difference between plants’ actual and target employment, therefore concludes that it is an important route that aggregate shocks affect aggregate employment by changing the fraction of plants that choose to adjust. Thomas and Khan [2003] used a similar approach to study (S,s) inventory accumulation and showed that these adjustment probabilities are functions of the difference between plants’ actual and target employment, therefore concludes that it is an important route that aggregate shocks affect aggregate employment by changing the fraction of plants that choose to adjust. Standard partial adjustment model relates current employment number to target or desired employment. But partial adjustment model is inconsistent with the behavior of individual plants. Plants exhibit nonlinear responses to shocks. It is difficult to determine equilibrium when the aggregate state involves a distribution of production units. State dependent adjustment frameworks extending to general equilibrium are limited. Generalized (S,s) model were first studied by Caballero and Engel [1999] to explain the observed lumpiness of plant level investment demand. It allows us to examine the influence of deep parameters on the adjustment process. With a large number of plants, the model is similar to the traditional partial adjustment model in that it yields a smooth market labor demand. Caballero [1993] construct a general framework for studying aggregate employment changes that can incorporate a variety of assumptions about how adjustment hazards are related to aggregate conditions. A generalized partial adjustment model in which individual production units adjust in a discrete and occasional manner is studied by Thomas [2002], but smooth adjustment is at the aggregate level. In the generalized model the adjustment rate is an endogenous function of the state of the economy, while the traditional partial adjustment model
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uses time -invariant aggregate partial adjustment. Impulse response establishes that it retains the basic features of gradual partial adjustment. By undertaking (S,s) analysis within a general equilibrium framework, the influence of aggregate shocks on equilibrium adjustment pattern can be systematically studied. Frictional labor market and the decentralized production structure assumptions necessitate the matching model in Pissarides [2000], research on the relation between embodied productivity growth and unemployment in Aghion and Howitt [1994] and Mortensen and Pissarides [1998]. They pioneered the research on the relation between embodied productivity growth and unemployment in a frictional labor market. In their models new capital adjustment is frictionless, as a result, vacancies are all consisted of the newest capital. In addition, Hornstein, Krusell and Violante [2002] modeled the existence of vacancy heterogeneity induced by vintage capital under these structures. An upgrading option is valuable, only if it is very costly for firms to meet workers, therefore their result indicates that the quantitative differences between upgrading and replacement is negligible. This section reviews several previous models of vintage capital relevant to the one discussed in this paper. Cadiou, Dees and Laffargue [2000] presents a vintage capital model assuming putty clay investment and perfect foresight. The traditional drawback of a putty-clay production function is the presence of variables with long leads and long lags. As putty-clay technology involves some stickiness in the production process, these works can investigate properly the sluggish adjustment of production factors to shocks. This framework clearly studies movements in job creation and job destruction related to economic obsolescence, replacement of productive capacity and expectations over the lifetime of the new units of production. On the contrary to recent works developing model in continuous time, their model is in discrete time. They identify the echo effect characterizing vintage capital models and the related dynamics of job creation and job destruction. This model is also proved useful to explain the medium-term movements in the distribution of income between production factors that putty-putty models lack. A
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key feature of the putty-putty specification is that it illustrates quite well the change in the wage share in value-added in France during the last three decades. A key feature of the putty-putty specification is that all the vintages of capital have the same capital intensity, which is the same case in the model discussed in this paper initiated with adjustment ratio. On the contrary, they expect the current technology to be only available for the newly created units of production. However, this is also in the similar case to this paper. This is precisely what the putty-clay specification does, where current economic conditions affect the capital intensity of the new production units (their technological choice) and the number of these units created (investment in the economy). Other production units maintain their original technology as they were endowed at their creation. Current economic conditions affect the profitable and nonprofitable production units facing the scrapping. Therefore the aggregate capitallabor ratio changes gradually with the fluctuation of investment and the scrapping of obsolete production units. Putty-clay investment provides mid-term dynamics in the distribution of income. This specification has some other advantages: the irreversibility of investment is embedded in the model and firing costs can easily be introduced, which gives a convincing foundation to the stickiness of employment. However, the putty-clay technology encounters a serious drawback: its implementation in a macroeconomic model is difficult for two reasons. Firstly, the model has a long memory, because it keeps track in working order of all the vintages of capital created in the past. Therefore the model has “variables with long lags”. Secondly, the investors’ planning horizon extends far into the future. The decision concerning the creation of new production units involves forward variables that cover the expected lifetime of these units. The model has then “variables with long leads”. The vintage capital model in Jovanovic [1998] explains the occurrence of income inequality under four similar assumptions: new technology is embodied in production tools; quantities of capital and labor are matched in fixed proportions; capital quality and labor skill are complements and assignment is frictionless. If machine quality and
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skill are complements, a worker who is with the best machine will acquire more skill therefore earn higher wages, which is different from the case in the paper, therefore inequality will persist. In this model the efficiency of each vintage of capital is endogenous and it varies when the economy is not on its balanced growth path. The study explains income inequality but is lack of the studies of inequality in consumption and wealth, one would need to carry out the full dynamics, presumably from an initial condition under which all agents start out equal. Recent discussions on growth theory emphasize the ability of vintage capital models to explain growth facts. An AK-type (Y = AK)endogenous growth model with vintage capital was studied by Boucekkine, Licandro, Puch and Rio. [2005], which was different from previous production function with the property of diminishing returns to capital. This AK model is absent of diminishing returns to capital, which is the key difference between this endogenous growth model and the one discussed in my paper. The inclusion of vintage capital leads to oscillatory dynamics governed by replacement echoes, which influences the intercept of the balanced growth path. The convergence is non-monotonic due to the existence of replacement echoes. As a consequence, investment rates do not move in a locked step with growth rates. This coincides with the conclusion of my paper. To characterize the complete resolution of the model, they developed analytical and numerical methods that should be of interest for the general resolution of endogenous growth models with vintage capital. These features, which are in sharp contrast to those from the standard AK model, contribute to explaining the short-run deviations observed between investment and growth rate time series. Their findings also indicate that there is much to learn from the explicit modeling of variable depreciation rates. An extension of this line of research is to include an endogenous decision for the scrapping time. However, in the model discussed in my paper, the extension is achieved by an endogenous state dependent investment adjustment ratio. A is a constant measuring the amount of output being produced for each unit of capital. One extra unit of capital produces A extra units of output, regardless of how much
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capital being employed. If
∆K K
= sA − δ is positive, the income grows forever, even if
without the assumption of exogenous technology progress. Therefore a simple change in the production function can alter the predictions about economic growth. This is such, because the numerical methods can be used to deal with time dependent and state dependent leads and lags. In the model discussed in their paper savings leads to growth temporarily but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. On the contrary, in the endogenous model, saving and investment lead to persistent growth. In addition, all models being discussed so far are unable to explain the the relation between vintage capital embodied productivity growth and unemployment in a frictional labor market. This paper attempts to explore the route how aggregate shocks affect aggregate employment by changing the fraction of plants they choose to adjust. Based on the analysis of the vintage model in Thomas [2002], the conclusion shows that the technology shock will lead the labor productivity increase and total labor force number decrease correspondingly. This paper also analyzes the high persistence characterizing the dynamics of firms’ neutral technology and the frequency of firms’ capital adjustment. The neutral technological shocks increase job reallocation and aggregate employment. In the vintage model, different technologies coexist. This vintage capital led inequality can be used to explain today huge economic inequality. This paper analyzes the high persistence that characterizes firm neutral technology dynamics and firm capital adjustment frequency. The paper proceeds as follows: Section 2 explains the theoretical background of the model. Section 3 introduces how does the model set? Section 4 executes the Toolkit to analyze the dynamic stochastic model easily. Section 5 explains the impulse response functions, the second moment and the policy implication. Section 6 calibrates the model using the standard parameter choices. Section 7 concludes.
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2 Explain the model
It is as one key stylized fact uncovered in the previous empirical literature that an important route through which aggregate shocks affect aggregate employment is to change the fraction of plants they choose to adjust. In the model the adjustment rate is an endogenous function of the state of the economy. The generalized model is not observationally equivalent to traditional partial adjustment model with time-invariant aggregate partial adjustment. The general equilibrium model of irreversible investment studied in Thomas (2002) answers the central question whether the intermittence and lumpiness observed at the firm level also affects investment at the aggregate level once the general equilibrium effects, such as consumption-smoothing motives leading to endogenous price adjustments, are taken into account. Plants face reversibility constraint specified in Thomas (2002) equation (7) and (8), which leads to lumpy investment at the plant level. The likelihood is that the firm will undertake a discrete adjustment in its capital stock increases as the deviation of its current capital stock (K0,t+1 ) from its desired level increases, which yield an increasing adjustment hazard. The stochastic nature of the adjustment costs specified in Thomas (2002) equation (9) leads to investment that is lumpy at the plant level, but smoother at the aggregate level. The model is based on a generalized (S,s) framework (S,s) policy is individual deterministic policy with economy state-dependent investment adjustment ratio αjt . An adjustment hazard, following econometric literature on discrete choice, is defined as the probability that an individual production unit makes a discrete change in a particular date. S,s policy is individual deterministic policy. Different from the traditional (S,s) model, this model uses the stochastic adjustment costs with a probabilistic adjustment thresholds that can capture the rising hazards to simultaneously yield lumpy plant-level investment and smooth aggregates. To model the capital difference in the process of technology change, a vintage capital
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framework where capitals and labors are costly to create units of capital and different ages, corresponding to technologies with different productivity levels is introduced. A unit measure of production units is differentiated by their stocks of capital with different vintage J. Production unit last acquires new capital j periods in the past with the subscript j. The number j of firm types means the vintage, which is endogenously determined and varies with factors such as the average inflation rate and elasticity of production demand. Depending on the current economy state and each plant’s adjustment cost, each plant chooses to invest or not. The judgement is based on the function: flow profits of plants = output - wage payments - investment - adjustment cost. If it is beneficial to invest, a current cost payment ξwt occurs for the production in date t + 1, where wt means the real wage at date t. All plants share the same production technology and same distribution of adjustment costs. The cross section distribution of the establishments over capital levels is summarized by the distribution of plants over vintage groups. Each vintage group contains a marginal plant whose investment cost is just worthwhile for it to adjust. All plants with the same or less than this critical investment cost draw have the willingness to invest. An adjustment ratio αjt in each vintage group can be derived as a function of the adjustment, a known cumulative distribution G(ξ). If it is not beneficial to invest, the plant’s capital stock at date t + 1 remains the same as after production in date t. The flow profits of the producers are returned in lump-sum fashion to households. The figure 1
1
illustrates the distribution of investment plants
evolving over time, which depends on the state of the economy.
1
Quoted from Thomas [2002]
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Figure 1: The fraction of plants over vintages
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3 The model
The competitive equilibrium is determined by solving a sequence of variables: {Ct , nj,t , Nt , ijt , αjt , Θj,t+1 , kj,t }∞ t=0 , where njt , ijt , αjt , θj,t+1 , kj,t are variables with vintage j = 1... . Competitive equilibrium allocations are determined by solving of a social planner problem indicated by Bellman equation, as in the form of function 1. Bellman equation
V (Kt , Θt ) =
max
Ct ,nt ,Nt ,it ,αt ,Θt+1 ,k0,t+1
[u(Ct , 1 − Nt ) + βEt V (Kt+1 , Θt+1 )]
(1)
subject to the budget constraints:
Ct +
1 X
θjt αjt ijt ≤
j=0
1 X
γ ν θjt zt kjt njt
(2)
j=0
Nt ≤
1 X
θjt njt +
j=0
1 X
θjt Ξ(αjt )
(3)
j=0
θ1,t+1 = θ0t (1 − α0t )
(4)
θ0,t+1 = α0t θ0t + α1t θ1t
(5)
Ct is the consumption, and current consumption is financed in forms of wages and profits from plants. k0t+1 is the next period to be adjusted target capital endowment of all the plants choosing to adjust in this period, Rj,t+j+1 is the gross rate of return on the capital, kjt are the capital endowment in this period of all the plants. The cross section distribution of the establishments across groups is summarized by two vectors indexed by vintage. Kt = {kjt }, is the vector of capital level across vintage group. The expectation Et is used as the information of date t+1 is not known at date t. Θt = {θjt }, θjt is the fraction of plants currently owning each vintage j capital level. The investment adjustment ratio αjt means in period t a fraction of vintage j firms decides to adjust. 0 < β < 1 is the discount factor parameter.
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Equation 2 specifies that household consumption cannot exceed aggregate production net of plants’ adjusting investments. Each plant’s production applies to Cobb-Douglas production function characterized by diminishing returns with respect to variable inputs in production. γ and ν are the parameters defining the exponential of capital and labor in production function, following the diminishing returns assumption they should be summed less than 1. Producers use labor and capital as variable inputs. ijt is the investment. The final good can be either consumed or invested. The adjustment rate αj responses smoothly to the aggregate state of economy. Equation 3 constrains that the household’s work hours cannot be less than weighted sum of employment in production and adjustment activities across groups. Ξ(αjt ) is the average adjustment cost paid from each group, in unit of labor and conditional on the fraction of plants investing. The figure 1 illustrates the distribution of investment plants evolution over time, which depends on the state of the economy. In period t, a fraction αjt of vintage j plants decides to invest, and at the same time, a fraction 1 − αjt of plants decide not to invest. As an assumption, all vintage J plants choose to invest, therefore αJt = 1. The total fraction of investment plants is expressed by equation 4. The fraction of non-investment plants is expressed as in equation 5.
k1,t+1 = (1 − δ)k0t
(6)
k0,t+1 = (1 − δ)k0t + i0t
(7)
k0,t+1 = (1 − δ)k1t + i1t
(8)
u(Ct , 1 − Nt ) = logCt + ζ(1 − Nt ) At = Xt zt
(9) (10)
ρ zt = zt−1 et
(11)
t ∼ N (0, σ2 )
(12)
Equation 6 defines the k1t+1 , which denotes vintage 0 capital plants at time t, k0t , become
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the one year vintage capital plants at time t + 1, if they dont adjust. δ is the parameter defining the depreciation rate of the capital. If the plant choose not to invest at the end of period t, its capital stock at time t + 1 remains the same as pervious period except the depreciation. If it is not beneficial to invest, the plant’s capital stock at date t + 1 remains the same as after production in date t. Equations 7 and 8 defining the i0t and i1t respectively, are the capital evolution equations. To detrend the time series, divide output, consumption and investment by zt , and the capital stock by zt−1 to eliminate the growth. All detrended variables are along balanced growth path at steady states. Household owns the portfolio of plants and supplies labor. Its endowment, following Hansen (1985) of indivisible labor, one unit of time per period is split to leisure Lt and market activities Nt . Household values consumption and leisure in each period with momentary utility u(ct , 1 − Nt ) in form of function 9. ζ is the parameter determining to which extent the labor produces a negative utility. Xt is the trend component and evolves with growth rate ΘA . Assuming Xt = 1, the equation becomes At = zt . The zt is a permanent aggregate technology shock which follows a mean zero AR(1) process in logs. ρ is the autocorrelation of technology shock.
Ξ(αjt ) =
Z G−1 (αjt )
G(ξ) = αjt G−1 (αjt ) = Bαjt ξ B B 2 Ξ(αjt ) = α 2 jt G(ξ) =
xdG(x)
(13)
0
(14) (15) (16) (17)
Plants may adjust labor usage frictionlessly, and a fixed labor cost occurs in adjusting capital stock. In equation 13, Ξ(αjt ) is the average adjustment cost conditional on the fraction of adjusted plants. Any individual production unit faces a random cost ξ. The discrete adjustment has the randomized fixed costs ξ, which is independently and identically distributed across time with a known cumulative distribution G(ξ) and p.d.f g(ξ). The random adjustment costs are assumed as convexity, whose distribution
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matches microeconomic data on the frequency of price adjustment. To facilitate the comparison with the time-dependent models, the distribution of the fixed costs takes the four-parameter form: G(ξ) = c1 + c2 tan(c3 ξ − c4 ). The function takes quite a flexible form and could be concave, convex, almost linear, or S-shaped depending on different assumptions of parameters. This cost is drawn from a time-invariant distribution over interval [0,B], where B is finite upper bound. The fixed cost takes the form as G(ξ) = α + a(τ − Bξ )−Ψ . The boundary assumption G(0) = 0 < G(ξ) < 1 = G(B) can be used to determine the parameters α and a, whose values are equal to τ and -1. The parameters τ and Ψ govern curvature and the latter is set to be -1. The equations 14, 15, 16, 17 can be obtained after simple algebra. ξ is a linear function of αj,t , which means the higher adjustment ratio is, the higher adjustment cost the investment plant will have.
4 Analyze the model 4.1 First order conditions In this section, I will implement this model in Toolkit which include find first order necessary conditions and other necessary constraints that characterizing the equilibrium, calculate the steady state, log-linearize the equations around the steady states, solve for the recursive law of motion, therefore obtain the impulse responses and HP-filtered moments and calibration. The model can be solved by using the techniques of dynamic programming. The first order conditions can be obtained from the lagrangian. Set lagrangian
V (kt , θt ) =
max
Ct ,Nt ,αt ,Θt+1 ,k0,t+1
[u(Ct , 1 − Nt ) + βEt V (kt+1 , θt+1 ) + λt (2) + v0t (5) + v1t (4)] (18)
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First order conditions ∂L : wt G−1 (α0t ) + i0t ∂α0t ∂L :0 ∂Ct ∂L :0 ∂(1 − Nt ) ∂L :0 ∂K0t+1
= v0t − v1t
(19)
1 Ct
(20)
= λt −
= −ζ + wt λt
(21)
= −λt + Et [βλt+1 R0t+1 + β 2 λt+2 (1 − δ)(1 − α0t+1 )R1t+2 ]
(22)
γy0t+1 + (1 − δ)α0t+1 (23) k0t+1 γy1t+2 = + (1 − δ) (24) k1t+2 βλt+1 = Et [ (πj,t+1 + αj,t+1 v0,t+1 + (1 − αj,t+1 )vj+1,t+1 − wt+1 Ξ(αj,t+1 (25) ))] λt
R0t+1 = R1t+2 ∂L : vjt ∂θjt+1
πj,t+1 = yj,t+1 − wt+1 nj,t+1 − αj,t+1 ij,t+1
The first order conditions of the plants’ maximization problem can be summarized in two equations which are the budget constraint as in 2 and the following Euler equation 22. The equation 22 is the Lucas asset pricing equation. The steady states can be solved by dropping the time indices of the first order conditions and the constraints. The investment decision of the plants should take three factors into consideration: Its real value, if it invests v0t , gross of the adjustment cost. It can be obtained by setting equation 25 j = 0. Its value if it does not invest. Its currently realized fixed adjustment costs. vjt as in the expression in equation 25 is the real value of the plant that last set its price j period ago. It is expressed in the way of dynamic program to get the optimal price satisfying the Euler equation.
λt+1 λt
is the marginal utility of consumption D1 u(Ct , 1−Nt )
future to current. It is used as the appropriate discount factor for future real profits. Combined with β, it can get the present value of next period’s expected value.
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(26)
4.2 Steady states value
The steady states of the model can be obtained by dropping the time subscripts and stochastic shocks of the equations listed above. The variables with bars are steady state values. By studying the vintage characteristics of all variables in the model, we can find the way to solve the steady states values. There are only four variables without vintage, wt , Ct , Nt , zt , representing real wage, consumption, employment and exogenous technology shock. The remaining variables are all with vintage subscript, namely called vintage variables. The key to solve steady states value of each vintage variable is finding of the relationship between each two contiguous vintages by write-off the non-vintage variables. Starting with Euler equation, all the steady states values can be expressed as the functions of the state dependent adjustment ratios and real wage rate. 1 − β 2 (1 − δ)2 (1 − α ¯ 0 ) − β(1 − δ)¯ α0 ¯ + (1 − δ)¯ α0 R0 = γ 2 β + β (1 − α ¯ 0 )(1 − δ) 1−ν ¯ 0 − (1 − δ)¯ R α0 y¯0 = γ k¯0 1 ¯0 −R β ¯1 = R β(1 − δ)(1 − α ¯0) ¯ y¯1 R1 − (1 − δ) = ¯ γ k1 w ¯ C¯ = ζ 1 θ¯0 = 2−α ¯0 θ¯1 = 1 − θ¯0 k¯0 =
θ¯0 ky¯¯00
+ θ¯1 (1 −
δ) ky¯¯11
C¯ − θ¯0 α ¯ 0 δ − θ¯1 (2δ − δ 2 )
(27) (28) (29) (30) (31) (32) (33) (34)
y¯0 y¯0 = k¯0 ¯ k0 ¯i0 = δ k¯0
(36)
¯i1 = (2δ − δ 2 )k¯0
(37)
k¯1 = (1 − δ)k¯0
(38)
y¯1 y¯1 = k¯1 ¯ k1
(39)
(35)
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w¯ y¯0 = n ¯0 ν y¯1 w¯ = n ¯1 ν y¯0 n ¯ 0 = y¯0 ( n¯ 0 ) y¯1 n ¯ 1 = y¯1 ( n¯ 1 )
(40) (41) (42) (43)
B B ¯ = θ¯0 n N ¯ 0 + θ¯1 n ¯ 1 + θ¯0 α ¯ 02 + θ¯1 2 2 β 1 v¯1 = [¯ y1 − w¯ ¯ n1 − ¯i1 + wB(¯ ¯ α0 − ) + ¯i0 ] 1−β 2 β 1 2 v¯0 = [¯ y0 − w¯ ¯ n0 − wB(¯ ¯ α0 − α ¯ ) − ¯i0 ] 1−β 2 0 B g¯0 = y¯0 − w¯ ¯ n0 − α ¯ 0¯i0 + α ¯ 0 v¯0 + (1 − α ¯ 0 )¯ v1 − w¯ α ¯ 02 2 g¯1 = y¯1 − w¯ ¯ n1 − ¯i1 + v¯0 − w¯ ∗ B/2 z¯ = 1
(44) (45) (46) (47) (48) (49)
4.3 Log-linearized first order conditions
The final step of the analysis is the log-linearization of the necessary equations defining the equilibrium. This step replaces the dynamic nonlinear equations by dynamic linear equations. This step can be fulfilled by expressing of the linear formation in percent deviation from the steady state. The expressions are in the same order as the steady state equations.
¯0 + β 2R ¯ 1 (1 − δ)(1 − α ¯ 0 (−Cˆt+1 + R ˆ 0t+1 ) + .. 0 = Et [(β R ¯ 0 ))Cˆt + β R ¯ 1 (1 − δ)(1 − α ˆ 1t+2 ) − β 2 R ¯ 1 (1 − δ)ˆ ...β 2 R ¯ 0 )(−Cˆt+2 + R α0,t+1 ](50) ¯0R ˆ 0,t+1 + γ y¯0 (ˆ 0 = −R y0,t+1 − kˆ0,t+1 ) + (1 − δ)ˆ α0,t+1(51) k¯0 ¯1R ˆ 1,t+2 + γ y¯1 (ˆ 0 = −R y1,t+2 − kˆ1,t+2 )(52) k¯1 0 = −ˆ y0t + zˆt + γ kˆ0t + ν n ˆ 0t(53) 0 = −ˆ y1t + zˆt + γ kˆ1t + ν n ˆ 1t(54) 0 = wˆt − Cˆt(55)
18
0 = k¯0 kˆ0t+1 − (1 − δ)k¯0 kˆ0t − ¯i0ˆi0t(56) 0 = (1 − δ)k¯0 kˆ0t − k¯1 kˆ1t+1(57) 0 = k¯0 kˆ0t+1 − (1 − δ)k¯1 kˆ1t − ¯i1ˆi1t(58) 0 = zˆt + γ kˆ0t + (ν − 1)ˆ n0t − wˆt(59) 0 = zˆt + γ kˆ1t + (ν − 1)ˆ n1t − wˆt(60) 0 = θ¯0 α ˆ 0t + α ¯ 0 θˆ0t + θˆ1t − θˆ0t+1(61) 0 = θˆ0t + θˆ1t(62) B B 2 ˆ ¯N ˆt + (¯ ¯ 0 )θ0t + (¯ n1 + )θˆ1t + n ¯ 0 θ¯0 n ˆ 0t + n ¯ 1 θ¯1 n ˆ 1t + B α ¯ 0 θ¯0 α ˆ 0t(63) 0 = −N n0 + α 2 2 0 = y¯0 θ¯0 yˆ0t + y¯1 θ¯1 yˆ1t + (¯ y0 − ¯i0 α ¯ 0 )θˆ0t + (¯ y1 − ¯i1 )θˆ1t −C¯ Cˆt − ¯i0 θ¯0 α ˆ 0t − ¯i0 α ¯ 0 θ¯0ˆi0t − ¯i1 θ¯1ˆi1t(64) 0 = −B w¯ α ¯ 0 wˆt − B w¯ α ˆ 0t − ¯i0ˆi0t − v¯1 vˆ1t + v¯0 vˆ0t(65) 0 = Et [β¯ g0 (Cˆt − Cˆt+1 ) − v¯0 vˆ0t + β y¯0 yˆ0,t+1 − β¯i0 α ¯ 0ˆi0,t+1 + β¯ v0 α ¯ 0 vˆ0,t+1 + β¯ v1 (1 − α ¯ 0 )ˆ v1,t+1 ... ... − β(w¯ ¯ n0 + B/2w¯ α ¯ 0 )wˆt+1 − β w¯ ¯ n0 n ˆ 0,t+1 − β(¯i0 + v¯1 − v¯0 + B w¯ α ¯ 0 )ˆ α0,t+1 ](66) 0 = Et [β y¯1 yˆ1t+1 − β¯i1ˆi1t+1 + β¯ v0 vˆ0,t+1 − β/2(2w¯ ¯ n1 + B w) ¯ wˆt+1 − β w¯ ¯ n1 n ˆ 1,t+1 .... ...β¯ g1 (Cˆt − Cˆt+1 ) − v¯1 vˆ1,t ](67) 0 = −ˆ zt + ρˆ zt−1 + t(68)
In the above equations the variables with hat are the log-deviations of the corresponding variables from their steady state, which can be regarded as the approximate percentage deviation. The calibration of the model can be achieved based on the log-linearized equations. Some of the values in the calibration are standard and others are based on guess. For example, the adjustment ratio and wage are based on the reasonable assumption.
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Figure 2: Long-term impulse response of capital, output, consumption, wage, and employment to aggregate technology shock with constant returns to scale production
5 Model result and answer 5.1 Technology shock and plant distribution It is worth to mention that all shocks I will discuss in this paper are one percent deviations of the respective variables from their steady state values. The positive technology shock has the positive effect on all variables except population density 1 in the case of constant returns to scale production technology. The impact of technology shock on wage and consumption, adjustment ratio and population density 0 are the same. On the contrary, in the case of decreasing returns to scale production function technology shock has a negative effect on capital variables.
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Figure 3: Long-term impulse response of capital, output, consumption, wage, and employment to aggregate technology shock with decreasing returns to scale production
Figure 4: Long-term impulse response of investment, labor and plant value to aggregate technology shock
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Figure 5: Long-term impulse response of investment, labor and plant value to aggregate technology shock
Figure 6: Long-term impulse response of labor and plant value to aggregate technology shock
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Figure 7: Long-term impulse response of adjustment ratio and population density to aggregate technology shock
The impulse response functions(of all important variables except the population density and investment 0 and 1, because they response too significantly) to a technology shock are shown in the figure 2 and 3. Technology shock has fraternized effect of distinguishing the value and capital among vintages, whereas distinguishes antagonistic output, investment and population density over vintages. Why does the technology shock has negative impacts on capital and investment with decreasing returns to scale but positive impacts on them with constant returns to scale production? Why does the technology shock have a similar effect on wage and consumption, adjustment ratio and population density 0? In Figure 2 to 5, why does the capital stock 0 response prior to capital stock 1 to a shock? Output and labor 0, 1 response at the same level at time 0, but output and labor 0 more than output and labor 1 at time 1. Investment 0 responses more than investment 1 in all time horizon. Variables with both vintages response to the technology shock positively. The reason is straightforward, because technology shock will impact positively more on up-to-date ones than the outdated economy. In the continuous time horizon after
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time 0, the up-to-date output and labor decline less to the technology shock than the outdated economy. New capital does not attain its full potential as soon as they are introduced, but rather their productivity can stay temporarily below the productivity of older capital that was introduced some time age with outdated technology. This feature is attributed to learning effects. When a new technology arrives, the most skilled worker labor 0 abandons his machine and switches to the best one. Thereafter the second best worker labor 1 gets the machine just abandoned by the most skilled one, and so on. This process continues until the least skilled worker scraps his machine. This process is called learning-by-doing(LBD). In addition, the technology shock is labor augmenting, i.e. the Harrod-neutral technical process, not capital augmenting, i.e. the Solow-neutral technical progress, therefore a positive shock increases labor productivity of both old and current vintages, consequently increase the aggregate labor demand with more investment being injected into the economy. The technology shock leads labor productivity increase and total labor force number increase correspondingly but decrease eventually. Therefore the wage or consumption of per efficiency of labor unit will increase. The wage increase is due to the positive effect of the increase in labor productivity, therefore the labor force decrease anyway. In absolute value, labor 0 responses positively more than labor 1 in later time after the shock. Due to the inhomogeneous production labor force, this study treats men of different vintages, distinguished by their age and training. Men of current vintage 0, e.g. those with currently training are more productive than those of previous vintage 1, therefore more easily to be out of supply and expensive on the labor market. Obviously in absolute value, the technology progress will influence more the current vintage labor force than the old vintage labor force. The reason can be revealed by organizational learning also. Organizational learning is reflected by an increase in the productivity of labor at the plant level: in a steadystate equilibrium where labor is mobile, productivity is equalized across plants, and wage inequality disappears. In absence of learning effects, the anticipation of a future technical
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shock embodied in new capital investment results in a transitional phase shown as a slowdown of economic activity. During the waiting period between the announcement and the actual availability of the new technology, the existing firm choose to wait for new investment and the new plants prefer to delay entering. This is why investment 0 responses more positively than investment 1. Therefore, their output falls temporarily until their full acceptance of the new technology, the fulfillment of the process learningby-doing. Numerically the per capita outputs of vintage 1 sector is the same as of the vintage 0 sector. From the equation 40 and 41 listed above, the per capita outputs are same over y¯j n ¯j
=
holds for adjacent
y¯j ¯j k
all vintages, i.e.
because
w ¯ C
w ¯ ν
at steady states for all j. A constant ratio equal to (1 − δ)
γ+ν−1 1−ν
for all j at steady states. It is the same for the consumption,
= ζ for all t. What makes the difference? The remaining variables with
vintage subscript determine the changes. The adjustment ratio of vintage 0 plant increases and so does the population density of plant 0. Therefore the total number of the plants with vintage 0 choosing to add their investment increases, the opposite case suits for the vintage 1 plants. In this way, the vintage 0 investment increases and the vintage 1 investment decreases. Investment leads output increase and i/y ratios are constant but not the same way for any vintage. Under the constant returns to scale, y/k ratios are the same across vintages, but under the decreasing returns to scale, y/k ratios increase with vintage. Because the marginal products of capital for all vintages are different, for example, R0t+1 = is smaller than R1t+2 =
γy1t+2 k1t+2
γy0t+1 +(1−δ)α0t+1 , k0t+1
+ (1 − δ), due to the increasing adjustment hazard in the
inflation economy and along the balanced growth path, the adjustment ratio α0t+1 is less than α1t+1 , which is equal to one. Full capital mobility induces a general equilibrium feedback that enlarges inequality: factor-price equalization requires capital to flow to the sector 0 to sector 1. Therefore labors in the sector 1 with old technology are endowed with more capital, which boosts their relative wages, the labor forces in sector 1 increases. The change of the output is significantly different for vintage 1 from vintage 0, which
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Figure 8: Long-term impulse response of outputs to capital 0 deviation
is mainly caused by the changes of the capital stocks to the technology shock are all positive and but in greater absolute values in comparison with the changes of labor. The two production factors have different responses to the technology shock, which work oppositely in determining the outputs to technology shock. The vintage 1 output is positive, because the marginal product of labor is more positive therefore offset the negative marginal product of capital, overall the output shows a positive change of vintage 1. Similarly, the vintage 0 output is negative, because the labor marginal productivity is not positive enough to offset the negative marginal product of capital. Output responses to capital deviation show that to both shocks of capital 0 and capital 1 deviation influence both output 1 and output 0 first response negatively. Output 0 and 1 oscillates around stead state as a response for capital 0 shock but dont oscillate for capital 1 shock. This means capital shock only lead to the respective vintage output increase. The vintage capital model can explain the rise of income inequality. If new machines are always better than old ones and if society can not provide everyone with a new machine
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Figure 9: Long-term impulse response of outputs to capital 1 deviation
all of the time, inequality results. If technology resides in machines and if a firm or worker must use just one technology at a time, a variety of machines will be in use, and workers’ productivity will differ, because not everyone can be given the latest vintage machine all the time. Inequality originates from the limited capacity of the capital goods sector. If machine quality and skill are complements, a worker who is with the best machine will acquire more skill, and inequality will persist. This inequality led by vintage capital can be used to explain today huge inequality in per capita outputs among countries. Income disparity is not a consequence of different initial conditions, but the result of different investment choices made by each economy. The vintage capital model has a natural nonconvexity (or lump-sum, fixed adjustment cost), when machines are indivisible. New machines are better than old ones. Under the assumption of complements between new technologies and skills, the new machines will be used by the most skilled workers. Therefore, the inequality will increase. Due to the non-convexity, small differences in skills will be transferred to larger differences in productivity. This is contrary to the convex, where the improvement of existing machines by small increment takes place instead of producing some much better machines. The revenue elasticity reduces greatest,
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B γ ν δ ζ β ρ σ α ¯ 0 w¯ .002 .42 T2 (.325 T1) .580 .060 3.6142 .954 .9225 .0134 .95 5
Table 1: Parameters choice when the adjustment cost increases under the independently and identically distributed across firms shock assumption.
5.2 Parameters choice Then to calibrate the model at annual frequency. The value of the parameters used in the model are summarized in Table 1. The adjustment cost ξ has a uniformly distributed cost, is independent draw from 0 to B, where the upper bound B 0.002 is chosen to match the observations by Domes and Dunne (1998). 1) Plants raising their real capital stocks by more than 30 percent (lumpy investors) comprise 25 percent of aggregate investment. 2) These investors constitute 8 percent of plants. The capital share of output is 0.325 and the labor share of output is 0.58, i.e. decreasing returns to scale (DRTS) or 0.42 and 0.58 in the case of increasing returns to scale (IRTS) as consistent with direct U.S estimates in King, Plosser and Rebelo (1988). The rate of depreciation matches a long-run investment-to-capital ratio of 0.076 in Cooley and prescott 1995. The ζ for the preference for leisure means that 20 percent of available time is spent in market work. The discount factor β implies an average annual interest rate of 6.5 percent, which complies with long-run per-capita output growth of 1.6 percent per year. The σ and ρ of exogenous stochastic process for productivity is estimated from Solow
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Table 2: Autocorrelation Table DRTS (HP-filtered series),corr(v(t+j),GNP(t))
σ% popdensity0 0.2145 capital0 0.0142 capital1 0.0142 value0 0.0090 output0 0.0155 output1 0.0174 labor0 0.0132 labor1 0.0154 wage 0.0044 adjratio 0.2210 investment0 0.3566 investment1 0.1730 consumption 0.0044 employment 0.0116 popdensity1 0.2145 value1 0.0083 technology 0.0086
Autocorrelation Table (HP-filtered series) corr(v(t+j),GNP(t)) t-3 t-2 t-1 t t+1 t+2 t+3 0.00 0.32 -0.48 0.82 -0.59 -0.10 0.01 -0.09 -0.29 0.38 -0.65 0.71 0.11 -0.03 -0.03 -0.09 -0.29 0.38 -0.65 0.71 0.11 -0.20 0.26 -0.40 0.78 0.06 -0.10 -0.15 -0.12 0.26 -0.52 1.00 -0.52 0.26 -0.12 -0.08 0.27 -0.34 0.47 0.19 -0.63 0.27 -0.06 0.24 -0.51 0.97 -0.71 0.32 -0.09 -0.02 0.26 -0.29 0.35 0.13 -0.70 0.35 -0.25 0.17 -0.32 0.62 0.28 -0.04 -0.15 0.00 0.32 -0.47 0.81 -0.56 -0.13 0.00 -0.04 -0.13 0.43 -0.67 0.88 -0.37 -0.09 -0.08 -0.19 0.32 -0.28 0.26 0.48 -0.46 -0.25 0.17 -0.32 0.62 0.28 -0.04 -0.15 -0.10 0.34 -0.38 0.80 -0.16 -0.22 -0.17 0.03 -0.00 -0.32 0.48 -0.82 0.59 0.10 -0.11 -0.01 -0.42 0.65 -0.61 0.70 0.04 -0.15 0.30 -0.34 0.74 0.04 -0.20 -0.20
residuals measured using Stock and Watson (1999) data on U.S. output, capital and total employment hours in 1953-1997.
5.3 Second moment Table 2 and 3 report the standard deviations and the cross-correlations with GNP for the HP-filtered series in the model, which can be used to study the fitness of the model to the real business cycle data. All moments are obtained by Toolkit using the frequency domain based calculations after filtering the series with Hodrick and Prescott (HP) filter. Details on the HP-filter can be found in Hodrick and Prescott (1997) and the frequency domain representation and Fourier transformation of the HP-filter see King and Rebelo (1993). It is shown that output with vintage 0 is totally correlated with the GNP in the real business cycle data, and most variables except capital 0 and investment 0 are positively correlated with the GNP. On the contrary, most of other capital 0 and
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Table 3: Autocorrelation Table CRTS (HP-filtered series),corr(v(t+j),GNP(t))
σ% popdensity0 0.0791 capital0 0.0064 capital1 0.0064 value0 0.0064 output0 0.0141 output1 0.0152 labor0 0.0111 labor1 0.0127 wage 0.0038 adjratio 0.0816 investment0 0.1340 investment1 0.0787 consumption 0.0038 employment 0.0116 popdensity1 0.0791 value1 0.0064 technology 0.0086
Autocorrelation Table (HP-filtered series) corr(v(t+j),GNP(t)) t-3 t-2 t-1 t t+1 t+2 t+3 0.00 0.32 -0.48 0.72 -0.74 -0.14 0.02 -0.20 -0.31 -0.11 -0.21 0.92 0.20 -0.05 -0.06 -0.20 -0.31 -0.11 -0.21 0.92 0.20 -0.31 -0.10 -0.24 0.91 0.21 -0.04 -0.16 -0.23 0.02 -0.18 1.00 -0.18 0.02 -0.23 -0.17 0.08 -0.12 0.84 -0.12 -0.45 0.09 -0.18 0.08 -0.14 0.98 -0.34 0.03 -0.23 -0.11 0.13 -0.07 0.75 -0.25 -0.54 0.16 -0.33 -0.15 -0.24 0.85 0.34 -0.01 -0.17 -0.02 0.20 -0.09 0.73 -0.72 -0.18 0.02 -0.11 -0.10 0.15 -0.09 0.88 -0.52 -0.19 -0.16 -0.18 0.05 0.05 0.70 0.27 -0.60 -0.33 -0.15 -0.24 0.85 0.34 -0.01 -0.17 -0.19 0.08 -0.06 0.94 -0.08 -0.31 -0.26 0.05 0.02 -0.19 0.10 -0.72 0.74 0.14 -0.05 -0.20 -0.30 -0.15 -0.20 0.91 0.21 -0.22 0.04 -0.09 0.94 0.03 -0.27 -0.26
investment 0 are negatively correlated with the GNP. In addition, simulation provides information about how different production setting affect business cycle properties of macroeconomic variables, such that one may understand better the effects of stochastic component shock on vintage capitals. Simulations show that the right magnitude of fluctuations is obtained when the vintage capital with up-to-date technology. It shows that the correlation structure of the production and demand components match the empirical business cycle patterns. Business cycle simulation examine quantitative properties of the equilibrium fluctuation by numerical simulations. One asks whether the different vintage capital with different productivity oscillations would add up in our model to the observed aggregate fluctuations and generate the business cycle patterns. The answer is true when the vintage capital with the latest productivity technology. The study explains the mechanism for the positive correlation of the business cycle variables and the positive correlation be-
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tween production factor components.
5.4 Policy implication to education, investment and savings A government should adopt policies that boost national savings, improve the ability to translate saving into productive investment and accelerate the demographic transition. The endogenous growth theorists, led by stanford’s Paul Romer, argue that it is a mistake to separate the determinants of the efficiency of labor from investment- that investments both raise the capital-worker ratio and increase the efficiency of labor since workers learn about the new technology installed with the purchase of new modern capital goods. Under this theory, government policies to boost national savings and investment rates is weakened. Next step is to implement this mechanism in a dynamic general equilibrium model to explore an investment driven business cycle. The investment fluctuation is characterized by the synchronized oscillation across sectors. The sectors are linked each other by derived factor demand when each sector uses other sectors’ product as intermediate inputs. Their interaction forms a positive feedback in capital adjustments in the network of input-output relations. Suppose a capital adjustment takes a form of discrete decision. Then there is a chance of a chain-reaction of investment in which an investment in one sector triggers an investment in another sector, and so on. This chain-reaction turns out to be represented by a branching process in a partial equilibrium of product markets. It is indicated that the total size of the chain-reaction can exhibit a very large variance in some parameter range.
6 Conclusion remarks This paper adapts the neoclassical business cycle model to allow for lumpy capital adjustments within each plant, and the vintage model occurs thereafter. In the vintage
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capital model, the economy can not replace all its old capital at each date, instead they renew their capital step by step, and different technologies exist at the same time. If technology resides in machines and if a firm or worker uses just one technology at a time, a variety of machines will be in use, and workers’ productivity differs. The vintage capital leads the inequality that can be used to explain nowadays huge inequality in per capita outputs among countries. Income disparity is not a consequence of different initial conditions, but the result of different investment choices made by each economy. Being the pioneer of its kind, this paper has studied a vintage model with endogenous adjustment ratio and analyzed how the aggregate technical shocks affect aggregate employment by changing the fraction of plants that choose to adjust. Conclusion based on this model and the toolkit approach specified in Uhlig [1999] is that the technology shock leads the labor productivity to increase and total labor force number to decrease correspondingly. The numerical analysis of the vintage model reveals the fact that the vintage capital does well in explaining the economic inequality. The neutral technological shocks increase job reallocation and reduce aggregate employment. The increasing adjustment hazard influences the job reallocation by affecting the marginal product of capital for all vintages, which is increasing function over vintages, for example, R1,t+2 is greater than R0,t+1 . Full capital mobility induces a general equilibrium feedback that enlarges inequality: factor-price equalization requires capital to flow from the sector 0 to sector 1. Then labors in the old sector with old technology are endowed with more capital, which boosts their relative wages, the labor forces in old sector will increase. Aided with the endogenous adjustment ratios, the previous vintage model like Jovanovic [1998], which indicates a worker with the best machine will acquire more skill therefore earn higher wages, gets a new explanation. Contrary to the real business cycle model, if the plants can not upgrade their neutral and investment-specific technology without replacing part of current workforce, the model is a standard vintage model, where technological progress is entirely embodied into new jobs. Due to the limitation of frictionless assumption of the neoclassical business cycle model,
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papers follow Pissarides [2000] random matching model to imitate the bargain between a vacant firm and a worker. Though the result of how the economy reacts to the shocks depends on difference assumptions, this paper takes more advantage and is significantly different from them by introducing a frictional capital adjustment framework. Business cycle simulations and empirical evidences show that the up-to-date variables are most reliable to report the business cycle and individual fixed investment plays a strong role in aggregate dynamics. Simulations show that the right magnitude of fluctuations is obtained when the vintage capital is with up-to-date technology, while the correlation structure of the production and demand components matches the empirical business cycle patterns.
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