Country A devotes 15% of its time endowment to education; Country B devotes 5%. Both start with identical human and physical capital. What does the Lucas model predict about their long-run income trajectories?
ABoth countries grow at the same long-run rate, since growth ultimately depends on the initial capital stock
BCountry A initially grows faster but both converge to the same income level as human capital equalizes through trade
CCountry A grows permanently faster and diverges from Country B in income level without bound
DCountry B grows faster in the short run because its workers spend more time producing output rather than studying
In the Lucas model, the growth rate of human capital is ḣ = δ(1 − u)h, where (1 − u) is the time devoted to education. A higher education time allocation means a permanently higher growth rate of human capital — and therefore a permanently higher GDP growth rate. Two countries that differ in education time allocations will diverge in income levels without ever converging, because the faster-growing country compounds its advantage every period. This is the key difference from Solow/Ramsey models, where countries converge to the same steady-state income level regardless of policy differences.
Question 2 Multiple Choice
What structural feature of the Lucas model prevents human capital accumulation from facing diminishing returns, unlike physical capital in the Solow model?
AInternational trade allows countries to export excess human capital, preventing saturation of the domestic market
BGovernment education subsidies maintain a constant return to schooling regardless of the existing stock of skills
CThe education sector uses existing human capital to produce new human capital — more skilled people learn faster — so the growth rate of h does not fall as h rises
DHuman capital depreciates rapidly, keeping the effective stock low and preventing diminishing returns from setting in
In Solow/Ramsey models, adding more physical capital yields diminishing marginal returns because capital competes for fixed factors. In Lucas's model, the human capital accumulation equation is ḣ = δ(1 − u)h — the growth RATE of h is constant (determined by u and δ), not declining. More skilled people learn faster because knowledge is applied to knowledge production. This self-reinforcing structure eliminates diminishing returns and sustains growth permanently. It is analogous to the AK model where the accumulated factor has constant rather than diminishing returns.
Question 3 True / False
In the Lucas model, two countries starting with the same physical and human capital but different education time allocations will eventually converge to the same income level as the less-educated country catches up.
TTrue
FFalse
Answer: False
The Lucas model predicts permanent divergence, not convergence. Because human capital grows at a rate proportional to the time invested in education — with no diminishing returns — a country that devotes more time to education maintains a permanently higher growth rate. The income gap between countries with different education policies widens without bound. This contrasts sharply with the Solow and Ramsey models, where all countries with the same technology and preferences converge to the same steady-state income per capita regardless of their starting point.
Question 4 True / False
The external effect of human capital in the Lucas model implies that individual workers benefit not only from their own skills but from the average skill level of those around them, creating a social return to education that exceeds the private return.
TTrue
FFalse
Answer: True
Lucas introduces a knowledge spillover: the average human capital in the economy h̄ raises everyone's productivity, not just the individual who acquired it. Programmers surrounded by skilled colleagues produce more; workers in high-skill cities benefit from dense knowledge networks. Because individuals only internalize their own skill gains when deciding how much to study, they ignore the positive spillover they generate for others. This wedge between private and social returns means individuals underinvest in education from society's perspective — a classic justification for education subsidies.
Question 5 Short Answer
Why does the Lucas model predict that differences in education investment across countries lead to permanent income divergence, when the Solow and Ramsey models predict convergence?
Think about your answer, then reveal below.
Model answer: In the Solow and Ramsey models, capital accumulation faces diminishing returns — as capital per worker rises, its marginal product falls, slowing growth until the economy converges to a steady state. All countries with the same parameters converge regardless of initial conditions. In Lucas's model, human capital has no diminishing returns: the growth rate of h is δ(1 − u), which depends only on the time allocation u, not on the level of h. Countries with higher u grow faster forever. The compounding of this permanent rate difference produces income levels that diverge without bound — a structurally different prediction that better matches observed persistent inequality across nations.
The key mathematical difference is whether the accumulated factor's marginal product declines. Physical capital's marginal product falls as K rises, pulling growth to zero. Human capital's 'marginal product' in the education equation is constant — the growth rate of h is always δ(1-u) regardless of h. This constant-returns structure is what makes growth endogenous and sustained. The external spillover amplifies this: high-skill economies attract more skilled workers, boosting average h̄ and further raising productivity — a self-reinforcing advantage that prevents convergence.