Accolley, Delali (2009) Investment Specific Technological Change And Indivisible Labor

Investment Specific Technology Change and Indivisible Labor Economics Department, Université Laval 2009 April 21

Delali Accolley [email protected] http://www.pdfcoke.com/accolleyd Keywords Investment specific shock, q shock, indivisible labor Acknowledgements The author of this paper has benefited from the extraordinary help of Dr. Alain Gabler, Université Laval, Quebec in programming the simulations.

Abstract The contribution of indivisible labor to an economy with investment specific technology change has been assessed. Within an indivisible labor economy, an investment specific technology shock positively and strongly impacts on hours worked. Consequently, the impact of this shock on consumption and output is stronger than within a divisible labor economy. Introducing indivisible labor has also amplified the impact of the q shock on investment in structures. As far as investment in equipment is concerned, indivisible labor has not added much to its response to q shock.

1. Introduction Jeremy Greenwood et al (1997, 2000), analyzing the post-war US data, found a negative correlation between the amount of new equipment investment and its relative price. The reported correlation coefficient between both variables computed using detrended data is -.46. They then attributed this phenomenon to a technological change specific to the equipment industry having rendered new equipment less costly and more affordable. They next amended the standard optimal stochastic growth model by distinguishing between two types of physical capital: equipment and structures, and introducing therein, alongside the usual sector-neutral technology parameter affecting the aggregate production technology, an investment-specific technology shock hitting the relative price of new equipment. Simulating the designed model reveals that investment specific technology shocks explain 30% of the variability observed in the US GNP whereas investment in new equipment makes up only 7% of this aggregate. The authors point out that this result is likely to be underestimated. According to them, including indivisible labor in the model

would amplify the responsiveness of the economy to investment specific technological shocks. Their model is sketched and simulated herein in Section 2. It has then been amended in Section 3 by introducing in indivisible labor. The purpose of the investigations carried out in Section 3 is to find out how and through which mechanism the introduction of indivisible in the model will increase the economy’s response to the shocks under consideration. A priori, It is known that an indivisible labor economy exhibits more fluctuations in hours worked but the implications of such a modification in the model on the other variables is not obvious. Section 4 concludes this project.

2. The Model The economy consists of infinitely-lived households, business firms, and government whose economic behaviour is described below.

2.1 Firms Firms produce aggregate output out of three (3) inputs: equipment, , structures, , and labor, . The production technology is CobbDouglas,

Delali Accolley _____________________________________________________________________________________ Where , , and are respectively equipment and structures depreciation rate, and real interest rate.

[2.1] Where represent respectively the total factor productivity and the equipment utilization rate. evolves over time according to the law specify below:

Whereas the depreciation rate of structures is constant over time that of equipment depends on its utilization rate and takes the following functional form:

[2.2] [2.8] where is the average gross growth rate of in [2.2] is a random variable following a first-order Markov process with transition density:

The laws of motion of equipment and structures are: [2.9] [2.10]

Firms maximize their profit defined as: Where and denote respectively the investment in equipment and structures. The stochastic variable is the inverse of relative price of new equipment and is indicative of investment-specific technological change affecting equipment only. It grows on average at the rate and follows the process:

[2.3] in [2.3] denote respectively real rental price of equipment and structure, and real wage. The first-order conditions (FOCs) from this optimization programs are:

[2.11] [2.4] The random variable in [2.11] follows the first-order ergodic Markov process:

[2.5] [2.6]

with

These conditions all state that the marginal productivity of the inputs should equate their real rental prices. From households’ point of view, lending money or investing their savings into acquiring equipment or structures being substitutable ventures, in order to avoid specialization, equilibrium in capital market requires:

[2.12]

The economy faces the following resource constraints: [2.13]

[2.7]

meaning aggregate output less adjustment costs is affected to consumption and 2

Investment Specific Technological Change and Indivisible Labor _____________________________________________________________________________________ investments in both equipment and structures. Adjustment costs are made up of costs specific to equipment and structures

[2.14]

[2.15]

2.2 Government Government proportionally levies taxes on both capital and labor incomes at the flat rates Greenwood, et al. (2000 & 1997) posit the tax revenue raised each time period by the government is rebated back entirely to agents as lump-sum transfers.1 [2.16] Where , and denote respectively equipment and structure real rental price, real wage, the lump-sum transfer, and government spendings on public goods.

2.3 Households The representative household’s maximizes his expected lifetime utility derived from consumption and leisure subject to his resource constraints. His instantaneous utility function is:

[2.17] The Euler equations from the household’s optimization problem are derived in Appendix 1 – relations [A.1.13] through [A.1.17]. Relation [A.1.13] regards the optimal choice of the utilization rate of equipment. [A.1.15] governs the intra-temporal substitution between consumption and leisure. Conditions [A.1.16] and [A.1.17] explain the representative household’s inter-temporal pattern of consumption. Finally, [A.1.14] consists of the various constraints the agent is facing.

2.4 General Equilibrium and Balanced Growth Path, and Calibration The dynamics of the whole economy is described by relations [A.1.18] through [A.1.22] in Appendix 1. As stated earlier in Subsection 2.1, total factor productivity, , and the inverse of the relative price of new equipment, , grow respectively at the rate and - relations [2.2] and [2.11]. Labor and equipment utilization rate are stationary. Adjustment costs, and , grow over time at the rate but are null along the balanced growth path. The resource constraint – relation [2.13] – and the law of motion of structure investments – relation [2.10] – suggest that and will grow at the same rate along the balanced growth path. It follows that equipment grows at a faster rate - relation [2.9]. From the aggregate production function – relation [2.1], it transpires that , which means:

1

Here, one coud follow Greenwood, et al (1995), among others, and introduce public spending in the government finance in order not to exaggerate the wealth effect occasioned by this redistribution policy.

[2.18]

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Delali Accolley _____________________________________________________________________________________ computed using US time series over the period 1954-19990.2

[2.19]

Table 2.1: Calibrated Parameters

Since all variables are not stationary, there is a need to normalize the model. Relations [A.1.23] through [A.1.27] in Appendix 1 describe the above mentioned dynamic equations as function of the normalized variables. As a result, the deterministic balanced growth path of the economy is summarized by:

Parameter Value .18 .12 .97 1.032 2.32 2.4709 .53 .4

[2.20]

[2.21]

2.5 Simulation and Findings The response of the economy to an exogenous and stochastic shock to is sketched in Figure 2.1 below. 3

[2.22] [2.23]

It appears in the fourth panel of Figure 2.1 that investment specific shock has an immediate and positive impact on the inverse of the relative price of equipment. As a consequence, this lowers the price of equipment, which occasions a slight rise in households’ investment in equipment and an important rise in the utilization rate of existing equipment stock. Investment in new equipment has not crowded out consumption and investment in structures – which also has slightly increased – for three reasons. First, the existing equipment is not fully used, . Before acquiring new equipment, firms must first increase the utilization rate of the already existing equipment stock.

[2.24] [2.25] [2.26] For the adjustment costs to be null along the balanced growth path, one needs to impose some restrictions:

[2.27] A similar reasoning shows that:

Parameter Value .4 1.59 .2365 v .056 .64 .35

.

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The value of the parameter is not from the authors but has been computed assuming a utilization rate of equipment of .9 and knowing that

The calibrated parameters used in simulating the model are reported from Greenwood, et al. (2000, p. 101) in Table 2.1 below. They were

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The authors calibrated their model to a sample of 37 annual observations on US time series. In paper, the model has been simulated over 150 periods.

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Figure 2.1: Impulse Responses to a q Shock This explains the important rise of 2.6% in observed in the fourth panel of Figure 2.1. Second, the q shock, by favoring new investment in equipment, raises output as it appears in the last panel of Figure 2.1 and creates a wealth effect. Consumption rises due to the increase in output. Thirdly, the rise in the utilisation rate of equipment increases the marginal product of equipment capital stock, , which could have promote considerable additional investment in equipment but this expected substitution effect is cancelled out by the rise in the depreciation rate of equipment which is an increasing function of the utilization rate.

3. Introducing Indivisible Labor in the Model The indivisible labor model as popularized by Gary Hansen (1985) is briefly presented. The implications of introducing this model into Greenwood, et al.,( 2000)’s contribution is then investigated.

3.1 Indivisible Labor The indivisible labor real business cycle (RBC) model, as put forth by Gary Hansen (1985), is motivated by the fact that: 

 

50% of the observed variability in total hours worked is attributed to variations in the number of people at work whereas only 20% of this variability is accounted for by the average hours worked. Most people either work full-time or not at all. Previous RBC equilibrium theories, by hypothesizing that labor market is always cleared, fail to account for unemployment and fluctuations in the rate of unemployment.

To take into account these realities, the one sector stochastic growth model has been revised so as to exclude part-time work possibility from the model, i.e. indivisibility of labor. Besides, working, in this model, does not result from individuals’ will. It is the outcome of

Delali Accolley _____________________________________________________________________________________ a lottery . Individuals are fully insured and get paid whether they work or not.

The utility function in [3.3] is linear in hours worked (and leisure) and is characterized by an infinite elasticity of substitution of leisure.

In the indivisible labor economy, each household’s instantaneous utility function is the same as the one in [2.17]. But since labor is indivisible – people work either full-time, hours, or not at all – and working is the outcome of a lottery , the representative household’s expected utility becomes:

3.2 Indivisible Labor and Investment Specific Technological Change A main feature of the indivisible labor RBC model is that it increases the volatility of the stochastic growth model for a given stochastic process for technology shock. In this subsection, the characteristic of this model With respect to investment specific shock is checked.

[3.1]

The production side and the government finance remain unchanged. Only the utility function in [2.17] has been replaced by that in [3.3]. The household’s optimization problem is presented in Appendix 2. The FOC, except that of hours worked, are the same as in the case divisible labor. The value of matching Greenwood et al (2000)’s data is .79.

Given that the proportion of households working in the economy is , the per capita hours worked is: [3.2] Substituting the relation

from [3.2] into

[3.1], one gets: [3.3] where

The response of the economy to a q shock when labor is indivisible is sketched in Figure 3.1 below.

.

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Figure 3.1 : Responses of the Divisible/Indivisible Labor Economy to a q Shock It transpires from Panel 5 of Figure 3.1 that the q shock has a stronger impact on hours worked and consequently on output (Panel 6). By raising output more than in the divisible labor economy, the q shock has induced more wealth effect, which has contributed to a significant rise in consumption. The impact of the shock on investment in structures is more important than in the case divisible labor but its impact on investment in equipment has not much changed because the impact of the shock on the relative price of equipment is the same in the two economies.

4. Conclusion This paper has investigated the transmission mechanism through which a q shock propagates

within an economy. The two main determinants of the strength of such a shock are: the utilization rate of equipment and the intertemporal elasticity of substitution of leisure. When a q shock lowers the price of equipment, before making new acquisitions, the already existing equipment is used more intensely. This explains the positive impact of the shock on . Within an indivisible labor economy, a q shock has a positive a stronger impact on hours worked. As a consequence, the impacts of this shock on consumption, investment in structures, and output are stronger than within a divisible labor economy.

5. Appendices Appendix 1: Divisible Labor 1. Household’s Optimization Problem

[A.1.1] Bellman’s equation

[A.1.2]

FOC and Euler equations [A.1.3] [A.1.4] [A.1.5] -

Envelope condition

Investment Specific Technological Change and Indivisible Labor _____________________________________________________________________________________

[A.1.6]

[A.1.7]

[A.1.8]

[A.1.9] -

Envelope condition [A.1.10]

[A.1.11]

[A.1.12]

[A.1.13]

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Delali Accolley _____________________________________________________________________________________

[A.1.14]

[A.1.15]

[A.1.16]

[A.1.17]

2. General Equilibrium Replacing the rental prices of the inputs in the above Euler equations by the marginal products and calling all the constraints and laws of motion agents face gives the equations making up the economy’s dynamic stochastic general equilibrium. [A.1.18]

[A.1.19]

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Investment Specific Technological Change and Indivisible Labor _____________________________________________________________________________________

[A.1.20]

[A.1.21]

[A.1.22]

Normalizing the model To normalize the model, new variables are defined by dividing all the non-stationary variables by their rate of growth. Define:

Defined also [A.1.23]

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Delali Accolley _____________________________________________________________________________________

[A.1.24]

[A.1.25]

[A.1.26]

[A.1.27]

3. Steady State

[A.1.28]

[A.1.29] [A.1.30]

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Investment Specific Technological Change and Indivisible Labor _____________________________________________________________________________________ Some ratios expressed as a function of the baseline parameters [A.1.31]

[A.1.32]

[A.1.33]

[A.1.34]

c

[A.1.35]

Appendix 2: Indivisible Labor

[A.2.1]

FOC [A.1.3] [A.2.2] [A.1.3] and [A.2.2] [A.2.3]

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Delali Accolley _____________________________________________________________________________________ Substituting [2.6] into [A.2.3] yields: [A.2.4]

[A.2.5]

6. Works Cited Greenwood Jeremy, Hercovitz Zvi and Krusell Per The Role of Investment-Specific Technological Change in the Business Cycle [Journal] // European Economic Review. 2000. - Vol. 44. - pp. 91-115.

Greenwood Jeremy, Rogerson Richard and Wright Randall Household Production in Real Business Cycle Theory [Book Section] // Frontiers of Business Cycle Research / book auth. Cooley Thomas F. - Princeton : Princeton University Press, 1995.

Greenwood Jeremy, Hercowitz Zvi and Krusell Per Long-Run Implications of InvestmentSpecific Technological Change [Journal]. - [s.l.] : The American Economic Review, June 1997. - 3 : Vol. 87. - pp. 342-62.

Hansen Gary D Indivisible Labor and the Business Cycle [Journal] // Journal of Monetary Economics. - [s.l.] : Elsevier Science Publishers B.V., 1985. - pp. 309-27. - 16.

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