The Lucas Growth Model - Intertemporal Choice

Lucas (1988) presents a growth model in which output is generated via a production function of the form

\YLev = \PtyLev \Kap^{\kapShare} (\labor \hLucas \Labor)^{1-\kapShare}

(1)

where $\YLev, \PtyLev,$ and $\KLev$ are as usually defined and $0 < \kapShare < 1$ , where $\labor$ is defined as the proportion of total labor time

spent working, and $\hLucas$ is what Lucas calls the stock of ‘human

capital.’

The production function can be rewritten in per-capita terms as

\yRat = \PtyLev \kRat^\kapShare (\labor \hLucas)^{1-\kapShare}

(2)

which is a constant returns to scale production function in

$\kap$ and $\labor \hLucas$ .

Capital accumulation proceeds via the usual differential equation,

\dot{k} = \yRat - \cRat - (\popGro+\depr) \kap,

(3)

while $\hLucas$ accumulates according to

\begin{gathered}\begin{aligned} \dot{\hLucas} & = \phi \hLucas (1-\labor) \\ \dot{\hLucas}/h & = \phi (1-\labor). \end{aligned}\end{gathered}

(4)

1Discussion¶

Before analyzing the model, an aside. Mankiw (1995) has persuasively

argued for defining ‘knowledge’ as the sum

total of technological and scientific discoveries (what is

written in textbooks, scholarly journals, websites, and the like),

and defining ‘human capital’ as the stock of knowledge

that has been transmitted from those sources into human brains via

studying.

Recall Rebelo (1991)’s key insight about endogenous growth models: In order

to produce perpetual growth, there must be a factor or a combination

of factors that can be accumulated indefinitely without diminishing

returns. Mankiw points out that since lifetimes are finite, there is a

maximum limit to the amount of human capital that an individual can accumulate.

Thus, while increasing human capital (with more years of schooling,

for example) may be able to extend the length of the transition period

in a growth model, human capital accumulation cannot be the source of

perpetual growth. It is more plausible to think that scientific knowledge can be

accumulated indefinitely (though presumably there is some limit even

to the accumulation of knowledge). These considerations

suggest that models of endogenous growth should focus more on

understanding the process of fundamental research and technological

development than on human capital accumulation as Mankiw defines it.

With this distinction in mind, there are (at least) two

interpretations of the Lucas model. One is at the aggregate level.

Here we can think of $\labor$ as the fraction of the population

engaged in useful work to produce goods and services, while proportion

$1-\labor$ is not working in conventional boring jobs that require

asking customers questions like “Would you like

fries with that?” but instead is producing ‘knowledge’ by

conducting scientific and technological research.

The other interpretation is at the level of an individual agent. Such

an agent can be thought of as operating his or her own production

function of the form in (2), where $(1-\labor)$ is now interpreted as the proportion of the time this individual spends studying and $\labor$ is the time spent working.

From the point of view of

Mankiw’s distinction, it is hard to interpret Lucas’s model as being

either about human capital accumulation or about knowledge. It can’t

be about human capital because $\hLucas$ can be accumulated without bound,

and without diminishing returns, neither of which makes sense for an

individual. It can’t be about generalized knowledge, because the

optimization problem reflects the return for an individual, while only

a trivial proportion of total knowledge (in Mankiw’s sense) is

contributed by any single individual.

Given these considerations, it probably makes more sense to think of the

model as a tool for normative than for positive analysis.

2The Solow Version¶

We analyze first the ‘Solow’ version of the model, in which

the saving rate is exogenously fixed at $s$ . Thus the

capital accumulation equation becomes

\begin{gathered}\begin{aligned} \dot{\kap} & = \save \yRat - (\popGro+\depr) \kap \\ \dot{\kap}/\kap & = \save (\yRat/\kap) - (\popGro+\depr) \\ & = \save \kRat^{\kapShare - 1} (\labor \hLucas)^{1-\kapShare}-(\popGro+\depr) \\ & = \save (\kap/\hLucas)^{\kapShare-1} \labor^{1-\kapShare} -(\popGro+\depr). \end{aligned}\end{gathered}

(5)

This equation tells us that the steady-state growth

rate in this model (if one exists) requires a constant

ratio of $\kap$ to $\hLucas$ . Thus, $\kap$ and $\hLucas$ must be growing

at the same rate in equilibrium.

Further insight can be obtained by defining $\hat{A} = \PtyLev \labor^{1-\kapShare}$

and rewriting the per-capita production function as

\begin{gathered}\begin{aligned} \yRat & = \hat{\PtyLev} \kRat^{\kapShare} \hLucas^{1-\kapShare}. \end{aligned}\end{gathered}

(6)

If we define a measure of ‘broad capital’ as the combination

of physical capital and human capital,

\kappa \equiv \kRat^\kapShare \hLucas^{1-\kapShare},

(7)

the model becomes

\yRat = \hat{\PtyLev} \kappa.

(8)

So if $\labor$ is constant and if $\kap$ and $\hLucas$ are growing at the same rate,

then the exponent on ‘broad capital’ is 1, and we are effectively back

at the usual Rebelo $\PtyLev \Kap$ model.

The key assumption that permits this to work is the accumulation

equation for human capital, which is itself like an $\PtyLev \Kap$ model, in the

sense that the exponent on human capital in the accumulation equation

for human capital is one. Human capital can be accumulated without

bound and without diminishing returns.

3The Ramsey/Cass-Koopmans Version¶

Lucas does not examine the Solow version of the model with a constant

saving rate, but instead the version in which a social planner solves

for the optimal perfect foresight paths of the two state variables $\kap$

and $\hLucas$ . It’s not worth going through the math here; I’ll just

present the conclusion, which is that the steady-state growth rate is

\dot{c}/c = \CRRA^{-1}(\phi - \theta).

(9)

Note that this confirms the crucial role of the CRS accumulation

equation for human capital: The key parameter that corresponds to

the interest rate is $\phi$ , the parameter that determines the efficiency

of human capital accumulation in equation (4).

Lucas also solves a version of the model in which there is an

externality to human capital. The idea here is that each person

is more productive if they are surrounded by other people with high

levels of human capital. Specifically, in this version of the model

the individual’s production function is

\yRat_i = \PtyLev \kap_{i}^{\kapShare} (\labor _{i}\hLucas_{i})^{1-\kapShare} \bar{\hLucas}^{\psi}

(10)

where $\bar{\hLucas}$ is average human wealth in the population (and the

other variables reflect the values for the individual).

Working through the decentralized solution, Lucas shows that the

steady-state growth rate of human capital for an individual consumer

will be

\begin{gathered}\begin{aligned} % % \gamma_\hLucas & = \left(\frac{(\phi-\theta)(1-\kapShare)}{\CRRA(1+\psi-\kapShare)-\psi}\right) %\\ \gamma_\hLucas & = \left(\frac{(\phi-\theta)(1-\kapShare)}{\CRRA(1+\psi-\kapShare-\psi/\CRRA)}\right) %\\ \gamma_\hLucas & = \left(\frac{\CRRA^{-1}(\phi-\theta)(1-\kapShare)}{1-\kapShare+\psi(1-1/\CRRA)}\right) \\ \gamma_\hLucas & = \left(\frac{\CRRA^{-1}(\phi-\theta)}{1+\psi(1-1/\CRRA)/(1-\kapShare)}\right) . \end{aligned}\end{gathered}

(11)

Since every individual is assumed to be identical, the growth rate

of aggregate human capital (and everything else) is the same as

the rate for this individual.

It is easy to see that if there is no externality to human capital

accumulation (that is, if $\psi = 0$ ), this solution is identical to

(9). If there is an externality, its effect depends on

whether $1/\CRRA$ is greater than, equal to, or less than 1. This is

because the effect depends on whether the externality causes the

saving rate to rise or to fall (since in endogenous growth models,

saving is the source of all growth). The $\bar{\hLucas}$ externality is like an

increase in the interest rate, and thus its effect will be determined

by the balance between the income and substitution effects. We know

that for $\CRRA=1$ the income and substitution effects exactly offset

each other, leaving consumption unchanged in response to an

increase in the interest rate, which is why (11)

collapses to (9) for $\CRRA=1$ . If consumers are very

willing to cut current consumption in exchange for higher future

consumption (that is, if the intertemporal elasticity of substitution

$(1/\CRRA)$ is greater than 1), then the externality boosts saving

and therefore growth. If consumers have an intertemporal elasticity

of less than one, the income effect outweighs the substitution effect,

saving falls, and growth is slower.

Lucas also shows that this decentralized solution is suboptimal,

because individual consumers do not obtain the full benefits to

society of increasing their own stock of knowledge. Devoting more

time to $\hLucas_i$ accumulation they increase $\bar{\hLucas}$ , which benefits all

others in the economy in addition to themselves. Lucas shows,

unsurprisingly, that the socially optimal solution requires greater

investment in human capital accumulation than is obtained in the

decentralized model. He also derives an optimal subsidy to human

capital accumulation that corrects the externality and induces

households to invest the socially optimal amount in human capital.

Scraps/draft material

rewrite (11) as

\begin{gathered}\begin{aligned} % to \gamma_\hLucas & = \left(\frac{(\phi-\theta)(1-\kapShare)}{\CRRA(1-\kapShare)+\psi(\CRRA-1)}\right) \\ & = \left(\frac{\CRRA^{-1}(\phi-\theta)}{1+\psi(\CRRA-1)/(\CRRA(1-\kapShare))}\right) \end{aligned}\end{gathered}

(12)

and it is obvious that if $\CRRA>1$ as we usually assume,

The bottom line is that there are only two configurations of the model

that are capable of generating perpetual growth in a way that makes any

sense: 1.b. (the Rebelo $AK$ model) and 2.a.ii

What this means is that any model that aims to permit perpetual

long-run growth must ultimately boil down to a structure in which

there are constant returns to scale for some set of factors that can

jointly be accumulated forever.

Consider a social planner maximizing the discounted sum of utility

in an economy with an $AK$ production function:

\max_{\{c\}} \int_0^{\infty} u(C) e^{-\theta t} dt

(13)

subject to

\dot{K} = \PtyLevK - C,

(14)

where we have assumed zero population growth and zero depreciation

to make the analysis less cluttered.

This problem can be solved with the same Hamiltonian apparatus we used

to solve the Ramsey/Cass-Koopmans model. In particular, with CRRA

utility with risk aversion $\CRRA$ one can show that optimal behavior

requires

\begin{gathered}\begin{aligned} \dot{\CLev}/\CLev & = \CRRA^{-1}(\PtyLev - \theta). \end{aligned}\end{gathered}

(15)

Note that this equation comes about because the marginal product of

capital in this model is always $A = r$ , because $f'(K) = \PtyLev$ . Note

further that according to this equation, the growth rate of consumption

is always the same; unlike the Cass-Koopmans model with a normal

production function, this model has no transitional dynamics.

We can also use (14) to obtain an expression for the

steady-state growth rate:

\begin{gathered}\begin{aligned} \dot{K}/K & = \PtyLev - C/K. \end{aligned}\end{gathered}

(16)

If the model has a steady-state growth rate of $\gamma = \dot{K}/K$ ,

this equation implies that

\begin{gathered}\begin{aligned} C/K & = \PtyLev - \gamma. \end{aligned}\end{gathered}

(17)

Thus, this is a model with a constant saving rate, because

\begin{gathered}\begin{aligned} S/AK & = (\PtyLev K - C)/AK \\ & = 1 - \PtyLev^{-1} (C/K) \\ & = \gamma/A \\ \gamma & = \PtyLev (S/AK). \end{aligned}\end{gathered}

(18)

Thus, the steady-state growth rate in a Rebelo economy is

directly proportional to the saving rate.

A further requirement for the Rebelo model to have a well-defined

solution is that

\begin{gathered}\begin{aligned} \CRRA^{-1}(\PtyLev -\theta) & < \PtyLev. \end{aligned}\end{gathered}

(19)

Recalling that $A$ is effectively the real interest rate

in this model, this equation can be interpreted as the

‘impatience’ condition that we imposed in the infinite

horizon perfect foresight consumption model. In fact,

the Rebelo $AK$ model is essentially just a way of

reinterpreting the perfect foresight infinite horizon

consumption problem as a model for economic growth.

The principal distinction is that we usually use the

perfect foresight infinite horizon model to analyze

circumstances where the agent has both labor and

capital income, whereas the Rebelo model rules out

labor income by assumption.

References¶

Lucas, R. E. (1988). On the Mechanics of Economic Development. Journal of Monetary Economics, 22(1), 3–42. 10.1016/0304-3932(88)90168-7
Mankiw, N. G. (1995). The Growth of Nations. Brookings Papers on Economic Activity, 1995(1), 275–326. 10.2307/2534576
Rebelo, S. T. (1991). Long-Run Policy Analysis and Long-Run Growth. Journal of Political Economy, 99(3), 500–521. 10.1086/261764