Transitional dynamics of fiscal policy in a general
equilibrium life-cycle model
∗
Boele Bonthuis
†University of Groningen
February 2, 2011
Abstract
In this thesis we study the effects of fiscal changes in an overlapping generations model. The model features imperfect annuities, realistic demographics and a realistic productivity profile. We find that fiscal shocks have pronounced redistributional effects on generations alive during the shock. A redistribution from retired to working agents under a consump-tion tax shock and from working to retired under a labour tax shock. These redistribuconsump-tional effects are larger than the eventual steady state effects. Announcing the shocks beforehand dampens these effects.
JEL codes: C61, D91, E62, H30, J10.
Keywords: dynamic general equilibrium model, fiscal policy, overlapping generation, transitional dynamics.
∗The author is grateful to comments and suggestions by Ben Heijdra and Jochen Mierau. Any remaining
errors are the responsibility of the author.
1
Introduction
In 1789 Benjamin Franklin wrote to Jean-Baptiste Leroy1
”In this world nothing can be said to be certain, except death and taxes.” However, although death is certain we never know when we pass away and although in most countries taxes are inevitable we may not know what taxes will be like in the future.
In this thesis we investigate the effect of fiscal policy changes in an overlapping gen-erations (OLG) model with life-time uncertainty. We study the transitional dynamics of the key variables through time. We are particularly interested in the difference in shock impact on different generations alive at the time of the shock and during transition.
In essence, the type of OLG model used in this thesis goes back to Yaari (1965) and Blanchard (1985) who first incorporated the concept of life time uncertainty in economic models. Although it was quite an improvement on the infinitely lived agent models, the mathematical implementation used by Blanchard (1985) does not resemble realistic
demo-graphics.2
We use real death rates to construct our demographics. Furthermore, instead of assuming constant productivity throughout life we incorporate a hump-shaped produc-tivity profile. First, this assures that agents face a labour market entry and retirement decision; second, it results in a hump-shaped labour supply curve.
The agent’s asset management in an OLG model with lifetime uncertainty is of partic-ular interest. Only allowing for regpartic-ular capital market investment results in the exclusion of the option to run a deficit, since agents have to meet a solvency condition upon death. As it is uncertain when they will pass away they will have to meet the solvency condition at all time. The existence of an annuity market allows agents to borrow; the annuity interest rate corrects for the probability of death during a period in life. Agents pay or receive a higher interest rate depending on their age. This interest rate can be calculated because the distribution of the probability of death for all cohorts is known. As shown by Yaari (1965) and Davidoff et al. (2003) it is optimal for agents to annuitise their wealth.
1
Franklin and Bigelow (1887).
2
However, in reality the annuity market does not function perfectly and annuitisation of wealth does not necessarily bring about the largest welfare as is shown by Heijdra et al. (2010).
The improvements in our model compared to the old OLG models with lifetime uncer-tainty, inclusion of realistic demographics and a realistic productivity profile, in combina-tion with an imperfect annuity market, result in a model that reflects the core character-istics of an economy quite accurately.
In this thesis we simulate a tax increase for both consumption tax and labour income tax. We also investigate the effect of announcing the shock ten years before the shock actually occurs. Furthermore, we simulate a tax swap, in which labour income tax is lowered but tax revenues are kept equal through a consumption tax increase; and we simulate a labour tax increase with a redistribution scheme (skewed towards the young) aimed at keeping the capital level at pre-shock value.
A similar model is used in Heijdra and Mierau (2010). They investigate various fiscal policies and their effect on the key elements in an economy in a continuous time model with endogenous growth. They show that a consumption tax can increase growth through redistribution. In effect resources are taken away from the dis-saving elderly and redis-tributed to the saving young generations. They also show that although labour taxation is detrimental to growth and consumption, growth can be restored if the right redistribution policy is implemented (i.e. a redistribution skewed towards the young).
Although the steady states in their paper can be determined accurately, the transitional dynamics from one steady state to the other cannot be determined analytically. A solution to this problem is to rewrite the model in discrete time and simulate shocks which induce transition to the other steady state. Our steady state results resemble Heijdra and Mierau (2010) quite closely. The biggest difference is that they describe a steady state growth path, while we describe a steady state with a steady state capital stock because of our neo-classical production function. Another difference is caused by the fact that we use real mortality data, which adds ten years to the model, inducing very old agents to borrow again at the end of their lives.
Bouzahzah et al. (2002) investigate the transitional dynamics in a discrete OLG model with 6 cohorts and no lifetime uncertainty. Their model features endogenous growth through a human capital accumulation function much like Lucas (1988) although they also investigate an exogenous growth scenario. Our model differs in a couple of aspects from Bouzahzah et al. (2002). First, our model includes lifetime uncertainty. Second, our model features a neoclassical production function. Third, our model features endogenous labour supply for all periods. Fourth, we investigate an OLG model with 80 cohorts.
jump up straight away and then slowly declines. Moreover, our transition path is three times as long as theirs. This effect has two main reasons. The first one is labour supply; ours being endogenous, theirs being exogenous. The second reason is the level of detail; we have 80 cohorts they have six. Our model therefore does not allow as much to smooth out within a cohort.
Heijdra and Ligthart (2010) study the transitional dynamics of a small open economy in an OLG setting. In their finite horizon scenarios the adjustment of aggregate variables is oscillating, whereas the infinite horizon produces a monotonic adjustment, much like our results. According to Heijdra and Ligthart (2010) three elements produce the cycles;
endogenous labour supply, finite horizons and the Ethier-productivity effect.3
Therefore, the difference between their finite horizon model results and the results of our model and the remarkable similarity between their infinite horizon results and our results can probably be explained by the fact that we use a different production function and study a closed economy.
The remainder of this thesis is organised as follows: in section 2, we define the model, in section 3 we calibrate the steady states of the model, in section 4 we present the simulation results of the policy shocks and finally in section 5 we draw conclusions.
2
Model
In this section we define the model to be simulated. The structure of Heijdra and Mierau (2010) is taken as a basis for the model described below. However, some very important changes are made. First of all this model is written in discrete time instead of continuous
time.4
Second, we do not describe the knife-edge case in which endogenous growth occurs. Instead we define the model exhibiting exogenous growth.
3
The Ethier-productivity effect allows for external economies of scale through input diversity.
4
2.1
Firms
The productive side of the economy is described by perfectly competitive firms with the following individual production function:
Yi,t = ZtKi,tǫKN 1−ǫK
i,t , (1)
Zt is the technology at time t, Ki,t is the amount of capital and Ni,t is the amount of
efficiency units of labour used. The first-order conditions of profit maximisation can be written as: wt = (1 − ǫK)Zt Ki,t Ni,t ǫK , (2) rt+ δ = ǫKZt Ki,t Ni,t ǫK−1 , (3)
wt is the real wage rate wage, rt is the interest rate and δ is the depreciation rate. The
capital intensity is the same for every firm, therefore, Ki,t/Ni,t ≡Kt/Nt, where Kt and Nt
are the cumulative capital stock and cumulative efficiency units of labour in the economy respectively. We define the technology function as follows:
Zt= Z0
Kt
Nt
η
. (4)
We can therefore rewrite (1)-(3) as:
Yt/Nt = Z0 Kt Nt ǫK+η , (5) wt = (1 − ǫK)Z0 Kt Nt ǫK+η , (6) rt = ǫKZ0 Kt Nt ǫK+η−1 −δ. (7)
If η ≡ 1 − ǫK the model exhibits endogenous growth a la Romer (1989), the steady state
is defined as a steady state growth path in which the capital stock rises at a constant rate
and rt≡r is independent of time. In this thesis we focus on the case in which η =< 1−ǫK.
2.2
Consumers
As mentioned before, three important features distinguish this model from other OLG models; first we introduce a realistic demographic structure, second we use a realistic labour efficiency function, and third the model features imperfect annuities.
2.2.1 Demographics
We describe a population on a steady state growth path. Which means that every genera-tion is a fixed proporgenera-tion of the total populagenera-tion. The size of the generagenera-tion born at time
t is:
Pt,t= (1 + π)tP0 = β ¯P , (8)
in which π is the population growth rate, P0 is the size of the initial generation at t=0, β
is the crude birth rate and ¯P is the population size. The size, at time t, of the cohort born
at time v is: Pv,t= P0(1 + π)v t−v Y i=0 [1 − µi], (9)
where µi is the mortality rate between age i-1 and i, we use Dutch mortality data for
2009.5 Qt−v
i=0[1 − µi] is the surviving fraction of cohort aged t-v=u. We can write Pv,t and
the relative cohort size pv,t as:
Pv,t = β ¯P (1 + π)−u u Y i=0 [1 − µi], (10) pv,t = β(1 + π)−u u Y i=0 [1 − µi]. (11)
We can see that the relative cohort size only depends on age, birth rate and population
growth rate. Aggregating (11) from the newborn cohort until the oldest cohort ( ¯D) gives:
1 = β ¯ D X u=0 (1 + π)−u u Y i=0 [1 − µi]. (12) 5
The maximum biological age at which yearly mortality data is available is 98. Throughout this thesis
age means economic age starting at 18, therefore the maximum age ( ¯D) we use is 80, hence we have 80
2.2.2 Efficiency
Although quite convenient for calculation purposes, the regular imposition of constant effi-ciency through agents’ lives is quite unrealistic. Instead we use a hump shaped productivity function: Eu = α0 1 1 + ξ0 u −α1 1 1 + ξ1 u . (13)
Using data of Hansen (1993) we can estimate the parameters for this equation using
non-linear least squares. The parameters are (standard errors between brackets); α0 = 19.805
(fixed), α1 = 19.248 (0.064), ξ0 = 0.033 (0.002) and ξ1 = 0.038 (0.002) with an adjusted
R-squared of 0.992. Maximum productivity is obtained at age 24.
A visual representation of efficiency and demographics is given in section 6.1.
2.2.3 Annuity market
In our model it is possible for agents to insure against longevity risk through annuities. Annuity firms invest the assets of agents in the capital market receiving the competitive
interest rate rt. Because some of the agents in a certain cohort die at the end of the period
the annuity firms are able to pay a higher interest rate to the surviving agents since all assets of the deceased accrue to the annuity firm. On the other hand, all debt of agents who die indebted is canceled.
Agents therefore fully annuitise their financial wealth for two reasons. First, because annuities pay a higher interest rate than the market. Second, agents would not be allowed to die indebted if they took a regular capital market loan using the market interest rates. Because of lifetime uncertainty it is therefore impossible to run a personal deficit. However, taking an annuitised loan is possible, because it already corrects for the chance of death through a higher interest rate.
More specifically, (1 + rt)Av,t−1 has been earned by the annuity firm on the assets of
cohort born at time v. A part of that cohort, µt−v, passes away; this occurs after interest is
received. The assets of the deceased are equally distributed among the surviving fraction
member therefore receives λµt−v/(1 − µt−v) of the assets of the deceased plus interest on
their own assets, giving the annuity interest rate as6
: (1 + rA v,t)Av,t−1 = (1 + rt)Av,t−1λµt−v 1 − µt−v + (1 + rt)Av,t−1, 1 + rA v,t = (1 + rt) 1 − (1 − λ)µt−v 1 − µt−v . (14)
The profit made through imperfect annuities by annuity firms is taxed away by the gov-ernment and redistributed to the agents.
2.2.4 Individual behaviour
Agents maximise their expected remaining lifetime utility. At time t the remaining lifetime utility of an agent born at time v is:
EΛv,t= ¯ D+v X τ =t Uv,τ (1 + ρ)τ −t τ −v Y i=0 [1 − µi], (15)
in which ρ is the time preference parameter and utility is defined as:
Uv,τ = lnCv,τǫc [1 − Lv,τ]
1−ǫc
. (16)
The budget identity at time t is:
Av,t = (1 + rAv,t)Av,t−1+ wv,t(1 − θL) + T Rv,t−Xv,t, (17)
where total consumption Xv,t is defined as:
Xv,t = (1 + θC)Cv,t+ wv,t(1 − θL)[1 − Lv,t], (18)
in which θL and θC are taxes on labour income and consumption respectively and T Rv,t
are government transfers. Agents are born without assets and upon certain death assets
are zero too (Av,v = 0 and Av,v+ ¯D = 0). Individual wages evolve according to:
wv,t= Et−vwt (19)
6
We use two stage maximisation to determine the consumption path and the labour supply path. The first stage, the intratemporal maximisation stage, optimally chooses between consumption and leisure, it maximises equation (16) subject to equation (18) and the non-negativity constraint on labour supply:
M axCv,t,1−Lv,t Uv,t = lnC ǫc v,t[1 − Lv,t]1−ǫc s.t. Xv,t = (1 + θC)Cv,t+ wv,t(1 − θL)[1 − Lv,t] Lv,t ≥ 0
The first order conditions are: ǫC Cv,t = (1 + θC)λ1, (20) 1 − ǫC 1 − Lv,t = wv,t(1 − θL)λ1+ λ2, (21) Xv,t = (1 + θC)Cv,t+ wv,t(1 − θL)[1 − Lv,t], (22) Lv,t ≥ 0, λ2 ≥0, λ2Lv,t= 0. (23)
For λ2 = 0 the marginal rate of substitution between consumption and leisure is:
1 − Lv,t Cv,t = 1 − ǫC ǫC 1 + θC (1 − θL)wv,t . (24)
The case in which Lv,t = 0 describes the corner solutions brought about by the hump
shaped productivity profile. This induces a clear entry and retirement decision. Using (18) and (24) we can express expenditure on leisure in terms of total consumption:
wv,t(1 − Lv,t) = Xv,t
1 − ǫC
(1 − θL)
(25) and we can express expenditure on consumption in terms of total consumption:
Cv,t = Xv,t
ǫC
1 + θC
. (26)
The second stage of maximisation, the intertemporal stage, optimally smoothes total con-sumption over lifetime. Using (25) and (26) in (16) we create the indirect utility function:
in which: Qv,t = ǫC 1 + θC ǫC 1 − ǫC wv,t(1 − θL) 1−ǫC , (28)
is the price index at time t of total consumption indicating the maximum attainable utility given wages, prices for goods (unity) and the budget constraint on total consumption.
The maximisation problem becomes:
M axXv,τ EΛv,t = ¯ D+v X τ =t ln [Xv,τQv,τ] (1 + ρ)τ −t τ −v Y i=0 [1 − µi], s.t. Av,τ = (1 + rv,τA )Av,τ −1+ wv,τ(1 − θL) + T Rv,τ −Xv,τ.
The total consumption Euler equation becomes: Xv,t+1 Xv,t = 1 + r A v,t+1 1 + ρ (1 − µt−v+1). (29) Using (14) in (29) we get: Xv,t+1 Xv,t = 1 + rt+1 1 + ρ [1 − (1 − λ)µt−v+1]. (30)
The consumption Euler equation can be constructed using (26) in (30): Cv,t+1
Cv,t
= 1 + rt+1
1 + ρ [1 − (1 − λ)µt−v+1] . (31)
2.2.5 Aggregate behaviour
We aggregate over all agents alive at time t, the aggregate per capita variables (ct, nt, at) are
defined, using (11), as:
ct = ¯ D X t−v=0 pv,tCv,t, (32) nt = ¯ D X t−v=0 pv,tNv,t, (33) at = ¯ D X t−v=0 pv,tAv,t, (34)
in which Nv,t= Et−vLv,t is efficient labour supply.
2.3
Government
The government receives taxes from consumers and the profit of the annuity firms: taxt = (1 − λ)(1 + rt) ¯ D X t−v=0 µt−v 1 − µt−vpv,tAv,t−1 +θCct+ θLwtnt. (35)
The first term on the right-hand side is the profit of the annuity firms. The government redis-tributes according to:
taxt= ¯ D X t−v=0 pv,tT Rv,t. (36)
As you can see we abstain from the possibility of wasteful government spending. The government transfers are defined as:
T Rv,t= z 1 1 + φ t−v . (37)
If φ > 0 the transfers are biased towards the young and if φ < 0 the transfers are biased towards the elderly. Using (37) we can calculate net transfers for agents:
N T Rv,t = T Rv,t− wtNv,tθL− Cv,tθC
−Av,t
(1 + rt)(1 − λ)µv−t
1 − µt−v (39)
In this expression we have incorporated the loss agents make because of imperfect annuities, which could be interpreted as an indirect capital tax.
In table 1 the entire model is summarised.
3
Steady state
In order to examine the transitional dynamics in this model we need to define a benchmark steady state. Once we have defined this steady state we can also calculate the steady states of the different scenarios after the shocks. We calibrate the model for 80 cohorts, which quite conveniently allows us to use annualised parameters. First, we calibrate the population share since this is exogenous to the rest of the system. Setting π = 0.01 and using equations (12) and the mortality data mentioned before we find that β = 0.0221.
In our benchmark calibration for the rest of the model we set ρ = 0.035, δ = 0.1, λ = 0.7, θC = 0.2, θL = 0.4 and η = 0.6.7 The tax rates are chosen to be close to OECD average tax
rates.8
Furthermore, we let ǫK adjust to ensure r = 0.04 (ǫK = 0.2743), we set Z0 to provide a
capital effective-labour ratio of 0.5 (Z0 = 0.4678) and we set ǫC to induce a retirement age of 45,
again the OECD average (ǫC = 0.2268).9 Using the benchmark parameters we can calculate the
steady states of the different scenarios we want to investigate.
7
The size of η has an effect on the steady state capital labour ratio. For values close to the knife-edge
case (η = 1 − ǫK) the higher η the lower the capital labour ratio.
8
OECD average value added tax in 2009 was 17.6, OECD average tax wedge on labour in 2009 was 36.4%, these averages have been fairly stable throughout the last decade, source: www.oecd.org .
9
Besides the benchmark model (model d), we estimate 5 more steady states:10
(a) No taxes
(b) Consumption tax (c) Labour tax
(d) Consumption tax and labour tax
(e) Same tax income as (d), lower labour tax, higher consumption tax (f) Same capital level as (d), higher labour tax
In table 2 the core characteristics of all scenarios are summarised and in the appendix (section 6.2) the graphs corresponding to situation (a) and (d) are presented.
Contrary to Heijdra and Mierau (2010) we see that a consumption tax is welfare increasing, (b) compared to (a). The positive effect of the drop in labour supply plus the redistribution effects outweigh the loss of consumption. Ceteris paribus, situation (b) with a consumption tax of 16.11% would bring about the highest welfare. However, as Altig et al. (2001) show if diversity of earnings within a generation is taken into account, an increase in consumption tax can deteriorate the welfare of the lowest paid workers. A labour tax on the other hand is always welfare deteriorating, see situation (c). Logically, a tax swap, a lower labour tax and a higher consumption tax, increases welfare, see situation (e) compared to (d). But even a labour tax increase can improve welfare if tax income is redistributed towards the young, situation (f).11
Agents younger than 42 receive in situation (f) larger lump sum payments than under situation (d), agents aged 42 and older receive less than under situation (d). Another interesting result is that if taxes exist and are reasonably high, agents start borrowing again at the end of their lives. We see that adding the realistic tail to the demographic structure (ages 70 to 80) has this effect on the asset profile. Transfers in the last few years of life are high enough and consumption low enough for agents to be able to meet the solvency condition upon certain death.
10
For the files used to calculate the steady states see: m-files at www.boelebonthuis.com, username: guest; password: welcome1.
11
4
Simulation
In this section we simulate the model that is defined in the previous sections. For each scenario steady state (d) is chosen as starting point. We simulate four scenarios:
1. Consumption tax increase 2. Labour income tax increase
3. Tax swap; lower labour tax, higher consumption tax 4. Labour tax increase with redistribution towards the young
For scenario 1 and 2 we also investigate the effect of announcing the shock beforehand. Even though, quite unrealistically, the tax shocks in each scenario are very large (10%), the transitional dynamics are shown clearly in these cases, which is what we are particularly interested in. Smaller more realistic tax increases have logically similar but smaller effects.
There are a couple of technical issues we need to address during the simulation. Due to the imperfect annuities the model predicts that, if consumption at ages close to certain death becomes low enough, agents re-enter the labour market. Furthermore, certain anticipated shocks induce young agents to enter the labour market in the last period before the shock, then exit the labour market at shock impact and re-enter later in life. In our model we exclude the possibility of re-entering the labour market altogether; the working life of an agent should be a continuous time. We do not allow agents to re-enter at high ages and do not allow young agents to ’pre’-enter the labour market before the shock lest they decide to quit at impact.
Below we will discuss the most interesting simulations results.12
4.1
Scenario 1: consumption tax increase
This scenario examines the effect of a consumption tax increase from 20% to 30%.
12
For the files used to simulate see: mod-files at www.boelebonthuis.com, for the results see:
4.1.1 Unanticipated shock
At time t = 1 agents face a consumption tax increase of 10%.
Individual labour supply drops directly in period 1 for all working ages. Moreover, the agent who was about to enter the labour market in the period of the shock stays out of the labour market for another period only to enter at age 7. Interestingly, this same agent supplies more labour at every age throughout his/her working life compared to agents with the same age at another point in time, creating a ridge displayed in figure 6.
Individual assets show a ditch at this same person (see figure 7), this is because, having anticipated to enter the labour market, this agent is heavily indebted and eventually corrects this by supplying slightly more labour.
For consumption there are two noticeable effects. One is the direct impact of consumption becoming more expensive and the second is the adjustment of labour, creating a discrepancy between those who can adjust labour supply and those who are already retired. This is expressed by the hump for agents who are retired at shock impact (see figure 8).
These effects are also visible in the change in net transfers; the net transfers for working agents increase because the labour reduction effect plus higher redistribution outweighs the effect of more expensive consumption whereas the net transfers for the retired decrease because of more expensive consumption (see figure 9).
For agents alive at the time of the shock, expected welfare is lowest for those who are in the final phase of their working life, these agents were not able to reduce their labour supply much, but at the same time they had to reduce consumption significantly for the remainder of their lives making them worse off compared to other agents. The agents mentioned before, the first ones to delay labour market entry, are the ones with the highest welfare. They can significantly reduce labour supply while wages are high because of the slow adjustment of assets and they receive relatively large sums in redistribution keeping consumption up (see figure 4).13
13
Finally, we did a quick investigation of the effect of the size of η on the adjustment path. We find; the closer η is to the knife-edge case the longer the adjustment path. Which makes sense since we get closer to the situation in which there is no steady state capital stock. For illustrational purposes we have simulated the same scenario for η = 0.3 as well (see figure 5 for the effect on technology).
4.1.2 Anticipated shock
At time t = 1 a rise in consumption tax at time t = 11 is announced.
Individuals who will be in the labour force at the time of the shock seem to postpone their labour supply, supplying more labour in the last few periods prior to the shock. The person with the highest labour supply during life is the person who is retired at shock impact. This person also consumes more compared to agents of similar ages at different times. Apart from the anticipation effects described above, the anticipated shock seems to produce similar labour supply changes as an unanticipated shock (see figure 12).
This also holds for individual assets. However, prior to the shock we see some interesting results. Agents who postpone their labour market entry at shock impact, create a ditch after the shock, as was the case under the unanticipated scenario. But before the shock their assets create a local maximum. Which can be explained by the fact that they significantly cut back their consumption prior to the shock. As might be expected we see a large increase of assets for individuals at the end of their career in the periods prior to the shock because of the large increase in labour supply for these agents (see figure 13).
The anticipation effect for individual consumption can be seen in the fact that the hump shape for retired agents, which appeared under the unanticipated scenario, has vanished (see figure 14). An interesting result is that cumulative welfare increases because of the announcement. How-ever, there are no redistributional effects of the announcement. The worst off and the best off are the same as under the unanticipated scenario (see figure 16).
4.2
Scenario 2: labour tax increase
4.2.1 Unanticipated shock
At time t = 1 labour tax is increased from 40% to 50%.
At shock impact labour supply is reduced drastically. All working agents cut back their labour supply and the retirement age falls to 40. The reason for this massive shift in retirement age is that working becomes relatively unrewarding (although wages rise) and the government transfers are very high directly after the shock. Therefore, labour supply can be cut back and consumption does not have to fall dramatically. After the shock a counter reaction can be seen. Labour supply bounces back, which even creates a hump in labour supply for agents alive at shock impact. Those at the top of the hump are the agents who were most indebted at shock impact. The hump ends at the agent born just before the shock. After him/her labour supply is fairly stable and retirement age is at its new steady state level; 42 (see figures 18 and 19).
For individual assets we see again a ditch for agents who create the ridge in labour supply. Furthermore, we see a shift of the peak of assets to earlier ages since the retirement age decreases. For the agents who decide to retire earlier than planned in the face of the shock the assets decline rapidly. Agents who are already retired at the time of the shock exhibit a similar pattern to the situation prior to the shock. The most noticeable effects of the shock die out when the newborn agent at the time of the shock reaches the maximum attainable age (see figure 20).
Looking at the consumption side of the economy we see a clear redistribution effect. At shock impact working agents reduce consumption because of reduced labour income which is not entirely compensated by a higher government redistribution. For retired agents consumption rises; those who would have worked, had there been no shock, are more than compensated for their loss of income (see figure 22). We see a clear saddle-like dip in consumption for those agents that are still in the labour force or are about to enter the labour force in a couple of years. Again the largest changes occur for agents who are alive during the shock (see figure 21).
hence consuming less. Agents born just before the shock exhibit a small peak in welfare; they do not supply much labour, earn a relatively large wage because of the high capital stock and receive large sums in redistribution. Agents born after shock impact have regressively lower welfare (see figure 18).
4.2.2 Anticipated shock
At time t = 1 a labour tax increase of 10% in period t = 11 is announced.
Older working agents start cutting back their labour supply prior to the shock. They are compensated for their loss in labour income by relatively high redistribution after the shock. Young agents generally supply more labour before the shock, showing a (local) maximum in labour supply in the period before the shock. After the shock a similar ridge is created as under the unanticipated scenario, but this ridge is not as voluptuous (see figure 25).
All distinct minimums and maximums in assets can of course be explained by changes in labour supply. Logically a dip in individual assets is exhibited for agents who start cutting back their labour supply early. Agents who increase labour supply, logically also create a ridge in assets. Again a local minimum is exhibited for agents who postpone labour market entry at shock impact, as they were indebted before the shock on which they had to pay a relatively high interest rate, a situation from which they recover by supplying relatively much labour after the shock, which is why the local minimum in assets dies out after retirement (see figure 26).
The most striking effect of the announcement for consumption is that the hump shape exhib-ited at the unannounced shock is less pronounced. Agents can smooth consumption optimally because of the announcement (see figure 27).
dips in welfare by the announcement of the shock, however, final welfare is still much lower than pre-shock welfare (see figure 29).
4.3
Scenario 3: tax swap
This scenario examines the effect of a tax swap starting at situation (d) and ending up at situation (e). The goal of the government is to keep tax income stable, while at the same time lowering labour income tax, resulting in an upward adjustment of consumption tax. We abstain in this scenario from anticipation effects since this would dominate the transitional dynamics and would blur our view of what we are really interested in. At time t = 1 the government lowers labour income tax with 10% (from 40% to 30%). To keep tax income stable at 0.0168 it increases consumption tax to 28.93% at first. Due to rising labour supply (and therefore rising labour tax income) the consumption tax gradually falls back to 22.10% (see figure 30).
A striking result is that young working agents at first cut back their labour supply. For these agents the consumption tax increase and lower wages outweigh the effect of a lower labour tax. After cutting back their labour supply, young agents start to increase their labour supply as the consumption tax falls. However, the cost of total future consumption does not increase as dramatically for the young agents as for older agents who face a relatively high consumption tax for the period in which they were planning to consume the most. These agents have to increase their labour supply. The labour tax cut results in a rise in retirement age to 46. The oldest working agent at the time of the shock even keeps working until he/she is 49 (see figure 32). After the first surge in labour supply after the shock the labour supply rises slowly as consumption tax drops slowly (see figure 31).
The agent born at shock impact creates a dip in assets, while supplying relatively much labour, since he/she faces a relatively high interest rate for the first, borrowing, part of his/her life. The working population has rising assets. All retired agents do not have an increase in assets since they cannot adjust labour supply; they can therefore only adjust consumption to meet the solvency condition in the end (see figure 33).
All younger agents increase consumption. The main effect in consumption dies out with the new born agent at the time of the shock (see figure 34).
For welfare we see a drop for those who are already retired at the time of the shock or are close to retirement. We see a rise in welfare for those who are already in the labour market and see their labour become more lucrative. But for younger agents alive at the time of the shock welfare drops because they face a relatively high interest rate in the phase of their lives in which they borrow, for which they have to compensate by relatively much labour supply with a relatively low wage while facing a relatively high consumption tax. But still they have a higher welfare than pre-shock agents. Post-shock agents’ welfare rises steadily as consumption tax decreases and the interest rate and wages stabilise (see figure 31).
4.4
Scenario 4: redistribution towards the young
This scenario takes as its starting point situation (d) and examines the effect of a labour tax increase in combination with a redistribution scheme to return the capital level to the old state, eventually ending up in situation (f). In order to return the capital level to the pre-shock level, the tax redistribution is skewed towards the young (i.e. φ > 0 in (T5)). Knowing the behaviour of agents, the government can calculate the size of φ beforehand. At t = 1 the government increases labour tax to 40% and at the same time sets φ = 0.0089.
Interestingly, most aggregate variables adjust very quickly in this scenario, which implies that after the shock most shifts occur between generations, not between periods. Only assets have to rebound from an initial drop (see figures 36 and 37).
At first we see a drop in labour supply because of the labour tax increase. However, agents alive during the shock supply relatively much labour compared to younger generations. Because the redistribution is skewed towards the young we see a sideways shift in working ages, the entry age shifts from 6 to 8 and the retirement age shifts from 45 to 46 (see figure 38).
labour market (see figure 39).
For consumption we see a direct decrease for the retired because of the skewed redistribution. Furthermore, we see an extra decrease for the working population because the higher redistri-bution for some ages does not outweigh the negative impact of a higher labour tax. The agents who are in their pre-entry phase obviously consume more, since they are not yet affected by the higher labour tax but receive a higher redistribution. The largest changes brought about by the shock die out with the newborn generations. The higher labour tax creates a saddle-like path for those who are in the labour market at the time of the shock (see figure 40).
Total consumption decreases but total labour supply decreases to a greater extent creating a higher welfare in the end. Agents at the end of their career are worst off, we can also see that particular effect in the net transfers (see figure 41). All young agents have a higher welfare than they would have had before the shock (see figure 37).
5
Conclusion
Every tax increase, decrease or swap has a very different impact on different agents participating in the economy. After the change in taxes it is always labour supply that adjusts the quickest, this quick adjustment in the end brings about an adjustment in consumption and assets.
A consumption tax increase has the biggest impact on those who are already retired. They are not able to adjust their labour supply anymore and are therefore hit hardest by the tax reform. Young agents at the time of the shock are best off, they can significantly reduce labour supply and have a relatively high capital stock to work with and large redistributions. In the end a consumption tax could increase expected welfare depending on the size of the increase and other taxes in place. Announcing the shock has the effect that agents who will be retired at shock impact can increase their labour supply before the shock to compensate for higher cost of consuming after the shock. Announcement increases overall welfare.
floating their way. A labour tax increase is in the end detrimental for welfare, although the shock does produce a higher welfare for the elderly. Announcing the shock has the effect that the advantage for the retired population is not as pronounced as under the unanticipated scenario. Welfare smoothes out and is on an aggregate level higher than under the unannounced case. An interesting result is that the announcement has a clear redistributional effect.
A labour tax increase does not alway have to be detrimental to welfare and the capital stock. If the government aims at returning the capital stock at the pre-shock level, through a redistribution scheme benefiting the young, welfare can actually rise under a labour tax increase. Consumption is cut back for the retired because of lower redistribution and the working population also cuts back consumption because of the higher labour tax (and for some also lower redistribution). However, for the young welfare rises because labour supply is cut back significantly; they do not need to supply as much labour because their capital gains are much higher under the redistribution scheme. Therefore, only the retired and nearly retired are worse off.
A tax swap (higher consumption tax, lower labour tax) has a negative impact on the retired. They face dearer consumption for the rest of their lives while not being able to benefit from lower labour taxes. All retired agents therefore cut back consumption. Very young working agents actually cut back labour supply in the face of the shock because of lower wages and more expensive consumption. The tax swap has the clear redistributional effect of benefiting the young at the expense of the old. Total welfare eventually rises, just as under the redistribution scheme only the retired (or agents close to retirement) are worse off.
Overall, for all fiscal changes we always see a clear discrepancy between the retired, the working population and the pre-entry population. Therefore, apart from their permanent effect, fiscal changes also have a clear redistributional effect for the population alive at the time of the shock or at the time of the announcement.
In this thesis we have investigated the effect of fiscal changes in a detailed OLG model. For future research it would be interesting to see how changing demographics affect an economy and how fiscal changes have an effect in a dynamic population. Moreover, it would be interesting to see various forms of endogenous growth incorporated.
this system reacts to fiscal changes through time. Next we can try to apply this knowledge on a more sophisticated version of OLG models.
6
Appendix
6.1
Efficiency and demographics
(a) Efficiency (Eu) (b) Relative cohort size (pu)
0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 AGE (u) 0 10 20 30 40 50 60 70 80 0 0.005 0.01 0.015 0.02 0.025 AGE (u)
(c) Mortality rate (µu) (d) Surviving fraction of cohort (Qui=0[1 − µi])
0 10 20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 AGE (u) 0 10 20 30 40 50 60 70 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 AGE (u)
6.2
Steady states
(a) Consumption (Cu) (b) Assets (Au)
0 10 20 30 40 50 60 70 80 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 AGE (u) C a C d 0 10 20 30 40 50 60 70 80 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 AGE (u) A a A d
(c) Labour supply (Lu) (d) Net transfers (N T Ru)
0 10 20 30 40 50 60 70 80 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 AGE (u) L a L d 0 10 20 30 40 50 60 70 80 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 0.025 AGE (u) NTR a NTR d
Figure 2: Steady states (a) and (d)
6.3
Graphs scenario 1: Consumption tax increase (unanticipated)
(a) Taxes (taxt) (b) Production (yt)
0 50 100 150 200 250 300 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02 TIME (t) 0 50 100 150 200 250 300 0.0345 0.035 0.0355 0.036 0.0365 0.037 0.0375 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.0388 0.039 0.0392 0.0394 0.0396 0.0398 0.04 0.0402 TIME (t) 0 50 100 150 200 250 300 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −22.65 −22.6 −22.55 −22.5 −22.45 −22.4 TIME (t) 0 50 100 150 200 250 300 0.027 0.0272 0.0274 0.0276 0.0278 0.028 0.0282 0.0284 0.0286 0.0288 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.0675 0.068 0.0685 0.069 0.0695 0.07 0.0705 0.071 0.0715 0.072 0.0725 TIME (t) 0 50 100 150 200 250 300 0.134 0.136 0.138 0.14 0.142 0.144 0.146 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 TIME (t) AGE (u) (b) Zoom 41 42 43 44 45 46 47 48 49 50 0 10 20 30 40 50 60 70 80 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 AGE (u) TIME (t)
(a) Total 0 10 20 30 40 50 60 70 80 0 50 100 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 AGE (u) TIME (t) (b) Zoom 41 42 43 44 45 46 47 48 49 50 0 10 20 30 40 50 4 5 6 7 8 9 10 x 10−3 AGE (u) TIME (t)
6.5
Graphs scenario 1: Consumption tax increase (anticipated)
(a) Taxes (taxt) (b) Production (yt)
0 50 100 150 200 250 300 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 0.02 TIME (t) 0 50 100 150 200 250 300 0.0345 0.035 0.0355 0.036 0.0365 0.037 0.0375 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.0386 0.0388 0.039 0.0392 0.0394 0.0396 0.0398 0.04 0.0402 TIME (t) 0 50 100 150 200 250 300 0.184 0.186 0.188 0.19 0.192 0.194 0.196 0.198 0.2 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −22.65 −22.6 −22.55 −22.5 −22.45 −22.4 −22.35 −22.3 TIME (t) 0 50 100 150 200 250 300 0.027 0.0272 0.0274 0.0276 0.0278 0.028 0.0282 0.0284 0.0286 0.0288 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.067 0.068 0.069 0.07 0.071 0.072 0.073 TIME (t) 0 50 100 150 200 250 300 0.134 0.136 0.138 0.14 0.142 0.144 0.146 0.148 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 TIME (t) AGE (u) (b) Zoom 30 32 34 36 38 40 42 44 46 48 50 0 10 20 30 40 50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 AGE (u) TIME (t)
(a) Total 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 AGE (u) TIME (t) (b) Zoom 40 42 44 46 48 50 0 5 10 15 20 25 30 35 40 45 50 4 5 6 7 8 9 10 x 10−3 TIME (t) AGE (u)
6.6
Welfare difference anticipated/unanticipated
0 50 100 150 200 250 300 −22.65 −22.6 −22.55 −22.5 −22.45 −22.4 −22.35 −22.3 TIME (t) EL EL106.7
Graphs scenario 2: Labour tax increase (unanticipated)
(a) Taxes (taxt) (b) Production (yt)
0 50 100 150 200 250 300 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 TIME (t) 0 50 100 150 200 250 300 0.031 0.032 0.033 0.034 0.035 0.036 0.037 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.0375 0.038 0.0385 0.039 0.0395 0.04 0.0405 0.041 TIME (t) 0 50 100 150 200 250 300 0.18 0.185 0.19 0.195 0.2 0.205 0.21 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −23.1 −23 −22.9 −22.8 −22.7 −22.6 −22.5 −22.4 TIME (t) 0 50 100 150 200 250 300 0.024 0.025 0.026 0.027 0.028 0.029 0.03 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.06 0.062 0.064 0.066 0.068 0.07 0.072 TIME (t) 0 50 100 150 200 250 300 0.12 0.125 0.13 0.135 0.14 0.145 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 50 0 10 20 30 40 50 60 70 80 90 100 0 0.05 0.1 0.15 0.2 0.25 TIME (t) AGE (u) (b) Zoom 30 35 40 45 0 10 20 30 40 50 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
(a) Total 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 AGE (u) TIME (t) (b) Zoom 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 −4 −2 0 2 4 6 8 10 12 14 x 10−3 TIME (t) AGE (u)
6.8
Graphs scenario 2: Labour tax increase (anticipated)
(a) Taxes (taxt) (b) Production (yt)
0 50 100 150 200 250 300 0.016 0.0165 0.017 0.0175 0.018 0.0185 0.019 0.0195 TIME (t) 0 50 100 150 200 250 300 0.031 0.032 0.033 0.034 0.035 0.036 0.037 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.037 0.0375 0.038 0.0385 0.039 0.0395 0.04 0.0405 0.041 TIME (t) 0 50 100 150 200 250 300 0.18 0.185 0.19 0.195 0.2 0.205 0.21 0.215 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −23.1 −23 −22.9 −22.8 −22.7 −22.6 −22.5 −22.4 0 50 100 150 200 250 300 0.024 0.025 0.026 0.027 0.028 0.029 0.03 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.06 0.062 0.064 0.066 0.068 0.07 0.072 0.074 TIME (t) 0 50 100 150 200 250 300 0.12 0.125 0.13 0.135 0.14 0.145 0.15 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 TIME (t) AGE (u) (b) Zoom 30 35 40 45 0 10 20 30 40 50 60 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 AGE (u) TIME (t)
(a) Total 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 AGE (u) TIME (t) (b) Zoom 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 −0.01 −0.005 0 0.005 0.01 0.015 TIME (t) AGE (u)
6.9
Welfare difference anticipated/unanticipated
(a) 10 years anticipation
0 50 100 150 200 250 300 −23.1 −23 −22.9 −22.8 −22.7 −22.6 −22.5 −22.4 TIME (t) EL EL 10 (b) 20 years anticipation 0 10 20 30 40 50 60 70 −22.505 −22.5 −22.495 −22.49 −22.485 −22.48 −22.475 −22.47 −22.465 −22.46 −22.455 TIME (t) EL EL 20
6.10
Graphs scenario 3: Tax swap
(a) Consumption tax (θC,t) (b) Production (yt)
0 50 100 150 200 250 300 0.18 0.2 0.22 0.24 0.26 0.28 0.3 TIME (t) 0 50 100 150 200 250 300 0.036 0.037 0.038 0.039 0.04 0.041 0.042 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.0396 0.0398 0.04 0.0402 0.0404 0.0406 0.0408 0.041 0.0412 TIME (t) 0 50 100 150 200 250 300 0.174 0.176 0.178 0.18 0.182 0.184 0.186 0.188 0.19 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −22.55 −22.5 −22.45 −22.4 −22.35 −22.3 −22.25 −22.2 −22.15 −22.1 TIME (t) 0 50 100 150 200 250 300 0.0285 0.029 0.0295 0.03 0.0305 0.031 0.0315 0.032 0.0325 0.033 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.07 0.072 0.074 0.076 0.078 0.08 0.082 TIME (t) 0 50 100 150 200 250 300 0.142 0.144 0.146 0.148 0.15 0.152 0.154 0.156 0.158 0.16 0.162 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 TIME (t) AGE (u) (b) Zoom 20 25 30 35 40 45 50 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 AGE (u) TIME (t)
(a) Total 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 AGE (u) TIME (t) (b) Zoom 25 30 35 40 45 50 55 60 0 5 10 15 20 25 30 35 40 45 50 −0.015 −0.01 −0.005 0 0.005 0.01 TIME (t) AGE (u)
6.11
Graphs scenario 4: Redistribution scheme
(a) Tax (taxt) (b) Production (yt)
0 50 100 150 200 250 300 0.016 0.017 0.018 0.019 0.02 0.021 0.022 0.023 0.024 0.025 TIME (t) 0 50 100 150 200 250 300 0.0356 0.0358 0.036 0.0362 0.0364 0.0366 0.0368 TIME (t)
(c) Interest rate (rt) (d) Wage (wt)
0 50 100 150 200 250 300 0.037 0.0375 0.038 0.0385 0.039 0.0395 0.04 0.0405 0.041 TIME (t) 0 50 100 150 200 250 300 0.185 0.19 0.195 0.2 0.205 0.21 0.215 TIME (t)
(a) Welfare (EΛt−80,t−80) (b) Consumption (ct) 0 50 100 150 200 250 300 −22.6 −22.5 −22.4 −22.3 −22.2 −22.1 −22 −21.9 TIME (t) 0 50 100 150 200 250 300 0.0278 0.028 0.0282 0.0284 0.0286 0.0288 0.029 TIME (t)
(c) Assets (at) (d) Labour supply (nt)
0 50 100 150 200 250 300 0.0714 0.0715 0.0716 0.0717 0.0718 0.0719 0.072 0.0721 TIME (t) 0 50 100 150 200 250 300 0.12 0.125 0.13 0.135 0.14 0.145 TIME (t)
(a) Total 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 100 0 0.05 0.1 0.15 0.2 0.25 TIME (t) AGE (u) (b) Zoom 25 30 35 40 45 50 0 10 20 30 40 50 0 0.05 0.1 0.15 0.2 0.25 AGE (u) TIME (t)
(a) Total 0 10 20 30 40 50 60 70 80 0 20 40 60 80 100 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 AGE (u) TIME (t) (b) Zoom 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 40 45 50 −0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02 TIME (t) AGE (u)
References
Altig, D., Auerbach, A. J., Kotlikoff, L. J., Smetters, K. A., and Walliser, J. (2001). Simulating fundamental tax reform in the united states. American Economic Review, 91(3):574–595. Blanchard, O. J. (1985). Debt, deficits, and finite horizons. Journal of Political Economy,
93(2):223–47.
Bouzahzah, M., de la Croix, D., and Docquier, F. (2002). Policy reforms and growth in computable OLG economies. Journal of Economic Dynamics and Control, 26:2093–2113.
Davidoff, T., Brown, J. R., and Diamond, P. A. (2003). Annuities and individual welfare. (9714). Franklin, B. and Bigelow, J. (1887). The Works of Benjamin Franklin, volume 2. G.P. Putnam’s
Sons, New York and London.
Hansen, G. (1993). The cyclical and secular behavior of the labour input: Comparing efficiency units and hours worked. Journal of Applied Econometrics, 8:71–80.
Heijdra, B. J. and Ligthart, J. (2010). The transitional dynamics of fiscal policy in small open economies. Macroeconomic Dynamics, 14(01):1–28.
Heijdra, B. J. and Mierau, J. O. (2010). Growth effects of consumption and labour income taxation in an overlapping-generations life-cycle model. Forthcoming: Macroeconomic Dynamics, 14. Heijdra, B. J., Mierau, J. O., and Reijnders, L. S. (2010). The tragedy of annuitization. CESifo
Working Paper, July(3141).
Lucas, R. E. (1988). On the mechanics of economic development. Journal of Monetary Economics, 22(1):3 – 42.
Romer, P. M. (1989). Capital accumulation in the theory of long-run growth. Modern Business Cycle Theory, pages 51–127. editor Barro, R.J.
