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September 21, 2020, Christopher D. Carroll

LucasGrowth

The Lucas Growth Model
Lucas (1988) presents a growth model in which output is generated via a production
function of the form
Y = AK α (`hL)1−α(1)where Y, A, and K are as usually defined and 0 < α < 1, where ` is defined as theproportion of total labor time spent working, and h is what Lucas calls the stock of‘human capital.’
The production function can be rewritten in per-capita terms as
y = Ak α (`h)1−α(2)which is a constant returns to scale production function in k and `h.
Capital accumulation proceeds via the usual differential equation,k̇ = y − c − (ξ + δ)k,(3)

while h accumulates according toḣ = φh(1 − `)ḣ/h = φ(1 − `).

(4)

1 Discussion
Before analyzing the model, an aside. Mankiw (1995) has persuasively argued fordefining ‘knowledge’ as the sum total of technological and scientific discoveries (what iswritten in textbooks, scholarly journals, websites, and the like), and defining ‘humancapital’ as the stock of knowledge that has been transmitted from those sources intohuman brains via studying.
Recall Rebelo (1991)’s key insight about endogenous growth models: In order toproduce perpetual growth, there must be a factor or a combination of factors that canbe 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 anindividual can accumulate. Thus, while increasing human capital (with more years ofschooling, for example) may be able to extend the length of the transition period in agrowth 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 Lucasmodel. One is at the aggregate level. Here we can think of ` as the fraction of thepopulation engaged in useful work to produce goods and services, while proportion 1 − `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 bethought of as operating his or her own production function of the form in (2), where(1 − `) is now interpreted as the proportion of the time this individual spends studying
and ` 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 abouthuman capital because h can be accumulated without bound, and without diminishingreturns, neither of which makes sense for an individual. It can’t be about generalizedknowledge, 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 atool for normative than for positive analysis.

2 The Solow VersionWe analyze first the ‘Solow’ version of the model, in which the saving rate is exogenously
fixed at s. Thus the capital accumulation equation becomesk̇ = sy − (ξ + δ)kk̇/k = s(y/k) − (ξ + δ)= sk α−1 (`h)1−α − (ξ + δ)
= s(k/h)α−1 `1−α − (ξ + δ).

(5)This equation tells us that the steady-state growth rate in this model (if one exists)requires a constant ratio of k to h. Thus, k and h must be growing at the same rate inequilibrium.
Further insight can be obtained by defining Â = A`1−α and rewriting the per-capita
production function as
y = Âk α h1−α .

(6)

If we define a measure of ‘broad capital’ as the combination of physical capital andhuman capital,κ ≡ k α h1−α ,(7)y = Âκ.

(8)

the model becomes

So if ` is constant and if k and h are growing at the same rate, then the exponent on‘broad capital’ is 1, and we are effectively back at the usual Rebelo AK model.

2The key assumption that permits this to work is the accumulation equation for humancapital, which is itself like an AK model, in the sense that the exponent on humancapital in the accumulation equation for human capital is one. Human capital can beaccumulated without bound and without diminishing returns.

3 The 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 foresightpaths of the two state variables k and h. It’s not worth going through the math here;
I’ll just present the conclusion, which is that the steady-state growth rate isċ/c = ρ−1 (φ − θ).

(9)Note that this confirms the crucial role of the CRS accumulation equation for humancapital: The key parameter that corresponds to the interest rate is φ, the parameterthat 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 humancapital. The idea here is that each person is more productive if they are surrounded byother people with high levels of human capital. Specifically, in this version of the model
the individual’s production function isyi = Akiα (`i hi )1−α h̄ψ(10)where h̄ is average human wealth in the population (and the other variables reflect thevalues for the individual).
Working through the decentralized solution, Lucas shows that the steady-state growthrate of human capital for an individual consumer will beρ−1 (φ − θ)
(11).
γh =
1 + ψ(1 − 1/ρ)/(1 − α)Since every individual is assumed to be identical, the growth rate of aggregate humancapital (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 ψ = 0), this solution is identical to (9). If there is an externality, its effect depends onwhether 1/ρ 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 h̄ externality is like an increase in
the interest rate, and thus its effect will be determined by the balance between the incomeand substitution effects. We know that for ρ = 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 ρ = 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/ρ) is greater than 1), then the externality
boosts saving and therefore growth. If consumers have an intertemporal elasticity of less3

than one, the income effect outweighs the substitution effect, saving falls, and growth isslower.
Lucas also shows that this decentralized solution is suboptimal, because individualconsumers do not obtain the full benefits to society of increasing their own stock ofknowledge. Devoting more time to hi accumulation they increase h̄, which benefits allothers in the economy in addition to themselves. Lucas shows, unsurprisingly, that thesocially optimal solution requires greater investment in human capital accumulation thanis obtained in the decentralized model. He also derives an optimal subsidy to humancapital accumulation that corrects the externality and induces households to invest thesocially optimal amount in human capital.

References
Lucas, Robert E. (1988): “On the Mechanics of Economic Development,” Journal ofMonetary Economics, 22, 3–42.
Mankiw, N. Gregory (1995): “The Growth of Nations,” Brookings Papers on
Economic Activity, 1995(1), 275–326.
Rebelo, Sergio T. (1991): “Long-Run Policy Analysis and Long-Run Growth,”
Journal of Political Economy, 99(3), 500–521.

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