Growth Theory II

The AK Model, Romer (1986), and Lucas (1988)

Alan Lujan

Johns Hopkins University

April 14, 2026

The Problem with Exogenous Growth

Where Does Growth Come From?

Last lecture established that in the RCK model, all long-run per-capita growth comes from labor-augmenting technological progress at rate \(g\). But the model treats \(g\) as exogenous.

Why can’t a perfectly competitive model explain \(g\)? With constant returns to scale, Euler’s theorem says:

\[Y = F_K \, K + F_L \, L\]

All output is paid to capital and labor. The firm has no money left over to finance R&D, patenting, or any deliberate investment in technology.

The Endogenous Growth Program

Romer (1986) showed the way forward: relax constant returns to scale in accumulable factors. Three approaches emerged:

Model	Mechanism	Key assumption
Rebelo AK	Constant returns to capital	\(\alpha = 1\), no labor in production
Romer (1986)	Knowledge externalities	Aggregate IRS via spillovers
Lucas (1988)	Human capital	Time allocated to study vs. work

We also examine two extensions to the RCK framework: finite horizons (Blanchard 1985) and adjustment costs (q-Ramsey).

Rebelo’s Unifying Framework

The General Cobb-Douglas Model

Rebelo (1991) identified the essential structure underlying all endogenous growth models. Consider the general form:

\[Y_t = A \, K_t^\alpha \, L_t^\beta\]

where \(\alpha + \beta\) need not equal one. With a constant saving rate \(s\):

\[\dot{K}_t = s \, A \, K_t^\alpha \, L_t^\beta - \delta K_t\]

Per-Capita Accumulation

Defining \(k = K/L\) and using \(\dot{L}/L = n\):

\[\frac{\dot{k}_t}{k_t} = s \, A \, k_t^{\alpha - 1} L_t^{\alpha + \beta - 1} - (\delta + n)\]

If a steady-state growth rate \(\gamma = \dot{k}/k\) exists, the RHS must be constant. Differentiating:

\[0 = (\alpha - 1) \dot{k}/k + (\alpha + \beta - 1) \dot{L}/L\]

Steady-State Growth

Substituting \(\dot{k}/k = \gamma\) and \(\dot{L}/L = n\):

\[\boxed{0 = (\alpha - 1) \gamma + n(\alpha + \beta - 1)}\]

This single equation catalogs every possible growth outcome.

A Catalog of Growth Possibilities

Case	Returns	Result
\(\alpha + \beta = 1\), \(\alpha < 1\)	CRS, diminishing returns to \(K\)	\(\gamma = 0\) (Solow)
\(\alpha + \beta = 1\), \(\alpha = 1\) (\(\beta = 0\))	CRS, constant returns to \(K\)	\(\gamma > 0\) (Rebelo AK)
\(\alpha + \beta > 1\), \(\alpha < 1\), \(n > 0\)	IRS	\(\gamma = \frac{\alpha+\beta-1}{1-\alpha} n\)
\(\alpha + \beta > 1\), \(\alpha < 1\), \(n = 0\)	IRS, no pop. growth	\(\gamma = 0\)

Only two configurations generate sustained per-capita growth without exogenous technology.

Why Solow Cannot Generate Perpetual Growth

With \(\alpha < 1\), the marginal product of capital falls as \(k\) grows:

\[f'(k) = \alpha \, k^{\alpha - 1} \to 0 \quad \text{as} \quad k \to \infty\]

Eventually the gross return to capital falls below depreciation, so further accumulation is unprofitable. Capital accumulation alone cannot sustain growth when returns diminish.

There are only three sources of growth: technology, labor, and capital. If technology is fixed and capital has diminishing returns, growth must eventually stop.

Diminishing vs. Constant Returns to Capital

The Rebelo AK Model

Setup

A social planner maximizes discounted CRRA utility:

\[\max \int_0^\infty \frac{C^{1-\rho}}{1-\rho} \, e^{-\delta_t \, t} \, dt \quad \text{s.t.} \quad \dot{K} = AK - C\]

where we set population growth and depreciation to zero for clarity.

The production function is \(Y = AK\), so \(f'(K) = A\) always. No diminishing returns.

The Euler Equation in the AK Model

The Hamiltonian is:

\[\mathcal{H} = u(C) + \lambda(AK - C)\]

First condition: \(u'(C) = \lambda\), giving \(C^{-\rho} = \lambda\).

Second condition: \(\dot{\lambda} = \delta_t \lambda - A\lambda\), giving \(\dot{\lambda}/\lambda = \delta_t - A\).

Combining:

\[\boxed{\frac{\dot{C}}{C} = \frac{1}{\rho}(A - \delta_t)}\]

Growth is constant and immediate. There are no transitional dynamics.

Saving and Growth in the AK Model

From the resource constraint \(\dot{K}/K = A - C/K\), steady-state growth requires a constant \(C/K\) ratio:

\[C/K = A - \gamma \quad \Longrightarrow \quad \frac{S}{AK} = \frac{\gamma}{A}\]

The saving rate is constant, and the growth rate is directly proportional to it:

\[\gamma = A \cdot s\]

A well-defined solution requires \(\gamma < A\), i.e., \(\rho^{-1}(A - \delta_t) < A\). This impatience condition (analogous to the consumption model) also ensures the transversality condition holds.

AK vs. RCK: No Convergence

In the AK model, poor and rich countries grow at the same rate forever (no convergence). In the RCK model, countries below the steady state grow faster, implying convergence. Data shows conditional convergence, somewhere between the two predictions.

The AK Model as Reinterpretation

The Rebelo AK model is the perfect foresight infinite-horizon consumption model reinterpreted as a growth model:

Feature	Consumption model	AK growth model
Interest rate	\(R\) (exogenous)	\(A\) (MPK, constant)
Growth rate	\(\rho^{-1}(R - \delta_t)\)	\(\rho^{-1}(A - \delta_t)\)
Labor income	Yes	No (\(\beta = 0\))
Impatience condition	Growth \(< R\)	Growth \(< A\)

The key difference: the consumption model has both labor and capital income, while the AK model rules out labor income by assumption.

The Romer (1986) Model

Knowledge as a Byproduct of Investment

Romer (1986) provides microfoundations for increasing returns. Each firm \(j\) has a CRS production function:

\[y_j = k_j^\alpha \, \ell_j^{1-\alpha} \, \Xi^\eta\]

where \(\Xi\) is the aggregate stock of knowledge. Romer assumes no capital depreciation, so cumulative investment equals the capital stock. Knowledge equals this aggregate:

\[\Xi_t = \int_{-\infty}^t I_v \, dv = K_t\]

Each firm takes \(\Xi\) as given (it is an externality).

Why CRS at the Firm Level Matters

At the firm level, the production function is CRS in \((k_j, \ell_j)\) holding \(\Xi\) fixed. This means:

Firms earn zero profits (Euler’s theorem applies)
Perfect competition is consistent
No need for monopolistic competition or patents

At the aggregate level, substituting \(\Xi = K\) and normalizing \(L = 1\):

\[Y = K^{\alpha + \eta}\]

If \(\alpha + \eta > 1\), there are increasing returns at the aggregate level, even though each firm sees CRS.

The Private Euler Equation

Households maximize CRRA utility. Each household ignores its trivial contribution to \(\Xi\). The private marginal product of capital is \(\alpha \, k^{\alpha-1} \Xi^\eta\), so the individual Euler equation is:

\[\dot{c}_{t,i}/c_{t,i} = \rho^{-1}(\alpha \, k_{t,i}^{\alpha-1} \Xi_t^\eta - \delta_t)\]

With identical households and \(\Xi = K\), the aggregate Euler equation becomes:

\[\frac{\dot{c}}{c} = \frac{1}{\rho}(\alpha \, k^{\alpha + \eta - 1} - \delta_t)\]

Balanced Growth Path

A balanced growth path exists when \(\alpha + \eta = 1\). Then \(k^{\alpha+\eta-1} = 1\), and:

\[\boxed{\frac{\dot{c}}{c} = \frac{1}{\rho}(\alpha - \delta_t)} \qquad \text{(private optimum)}\]

Growth is perpetual and constant, just as in the AK model. The effective “broad” exponent on capital is \(\alpha + \eta = 1\).

The Externality Wedge

The gap between social and private growth rates is:

\[\Delta \gamma = \frac{\eta}{\rho}\]

This is the growth cost of the knowledge externality. Each firm’s investment generates social benefits (more knowledge for everyone) that the firm does not capture.

Policy implication: Capital accumulation should be subsidized to close the gap between private and social returns. Equating the after-subsidy private MPK to the social MPK, \((1+s^*)\alpha = \alpha + \eta\), gives \(s^* = \eta/\alpha\).

R&D Spending and Growth

The Lucas (1988) Model

Human Capital and Growth

Lucas (1988) introduces a time allocation decision between working and studying:

\[y = A \, k^\alpha \, (u \, h)^{1-\alpha}\]

where \(u\) is the fraction of time spent working and \(h\) is human capital per worker.

Human capital accumulates according to:

\[\dot{h}/h = \phi(1 - u)\]

Time spent studying (\(1 - u\)) builds human capital at rate \(\phi\), with no diminishing returns. The exponent on \(h\) in the accumulation equation is exactly one.

Mankiw’s Distinction and Interpretations

Mankiw (1995): knowledge is the total stock of discoveries; human capital is the subset transmitted into brains via studying. Since lifetimes are finite, human capital cannot grow without bound, making it an implausible source of perpetual growth. The Lucas model is best interpreted as a normative tool.

Interpretation	\(u\)	\(1-u\)	\(h\)
Aggregate	Pop. fraction producing	Pop. fraction researching	Stock of knowledge
Individual	Time working	Time studying	Personal human capital

The Solow Version

Fix the saving rate at \(s\). Capital accumulation becomes:

\[\frac{\dot{k}}{k} = s \, A \, k^{\alpha-1} (u \, h)^{1-\alpha} - (n + \delta)\]

For a steady-state growth rate to exist, \(k/h\) must be constant, so \(k\) and \(h\) must grow at the same rate.

Broad Capital

Define “broad capital” as \(\kappa \equiv k^\alpha \, h^{1-\alpha}\). Then:

\[y = \hat{A} \, \kappa \qquad \text{where} \quad \hat{A} = A \, u^{1-\alpha}\]

If \(u\) is constant and \(k\) and \(h\) grow at the same rate, broad capital has an exponent of one. The model reduces to Rebelo’s AK structure.

The engine of growth is human capital accumulation, which itself has a linear law of motion: \(\dot{h} = \phi(1-u) h\).

The Optimal Growth Rate

The social planner solves for optimal paths of both \(k\) and \(h\). The steady-state growth rate is:

\[\boxed{\frac{\dot{c}}{c} = \frac{1}{\rho}(\phi - \delta_t)}\]

The key parameter is \(\phi\) (efficiency of human capital accumulation), playing the role of \(A\) in the Rebelo model.

Human Capital Externalities

Lucas also considers an externality version where productivity depends on average human capital \(\bar{h}\):

\[y_i = A \, k_i^\alpha \, (u_i \, h_i)^{1-\alpha} \, \bar{h}^\psi\]

The externality \(\bar{h}^\psi\) is like an increase in the return to saving. Its effect on growth depends on the balance between income and substitution effects.

The Decentralized Growth Rate

Working through the Hamiltonian for the decentralized economy:

\[\gamma_h = \frac{\rho^{-1}(\phi - \delta_t)}{1 + \psi(1 - 1/\rho)/(1-\alpha)}\]

Three cases:

\(\rho = 1\) (log utility): income and substitution effects cancel; \(\gamma_h = \phi - \delta_t\) regardless of \(\psi\)
\(\rho < 1\): substitution effect dominates; externality raises growth
\(\rho > 1\): income effect dominates; externality lowers growth

Externality Implications

Individual consumers do not internalize the benefit their human capital has on others (via \(\bar{h}\)). The social planner, who accounts for the externality’s effect on aggregate productivity, chooses more investment in human capital than the decentralized equilibrium provides.

Lucas derives an optimal subsidy to education that corrects the externality and induces households to invest the socially optimal amount in human capital.

Growth Rate and the Externality

Human Capital and Income

Extensions to the RCK Framework

Blanchard (1985): Finite Horizons

Blanchard (1985) introduces agents who face a constant probability of death \(p\) each instant. Expected remaining life is \(p^{-1}\) (e.g., \(p = 1/50\) gives a 50-year horizon).

Key modifications:

Effective discount rate: \(\hat{\delta_t} = \delta_t + p\) (mortality risk makes the future less valuable)
Annuity market: insurance pays \(p \cdot w\) while alive in exchange for assets at death, raising the effective return to \(\hat{r} = r + p\)
Individual consumption: \(c(s,t) = \hat{\delta_t}\bigl(w(s,t) + h(s,t)\bigr)\), where \(\hat{\delta_t} = \delta_t + p\)

Individual consumption growth is \(\dot{c}/c = \hat{r} - \hat{\delta_t} = r - \delta_t\): mortality terms cancel.

Aggregate Dynamics and Ricardian Equivalence

The twist comes in aggregation. Newly born agents enter with zero wealth, pulling down the aggregate:

\[\dot{C} = (r - \delta_t)C - p(p + \delta_t)W\]

The extra term \(-p(p+\delta_t)W\) is absent when \(p = 0\) (the RCK case).

Consequences:

Steady-state capital \(\bar{K} < K^*_{\text{RCK}}\): finite-horizon agents discount more heavily and save less
Ricardian equivalence fails: a tax cut financed by future taxes transfers resources from the unborn to the living; without altruistic links across generations, current consumers do not offset this by saving more

The q-Ramsey Model: Adjustment Costs

The q-Ramsey model adds quadratic capital adjustment costs to the Ramsey framework. The cost of adjusting the capital stock by \(I\) when the stock is \(K\) is:

\[j(I, K) = \frac{\xi}{2}\,K\left(\frac{I}{K}\right)^2\]

The Euler equation is modified: the effective interest factor now depends on both the marginal product of capital and the marginal adjustment cost. When \(\xi = 0\), the model reduces to the standard Ramsey model.

Adjustment Costs and Convergence Speed

Key results from the q-Ramsey model:

Same steady state as standard Ramsey (no investment at SS, so \(j = 0\))
Slower convergence: costly to adjust \(K\) quickly, so the economy moves gradually toward equilibrium
Larger interest rate fluctuations: \(R\) reflects both MPK and adjustment cost terms, giving the model a fighting chance at matching business cycle interest rate variation
Both higher \(\xi\) (investment smoothing) and higher \(\rho\) (consumption smoothing) slow convergence: they have similar effects through different channels

Synthesis

Policy Implications

Model	Market outcome	Policy lever
RCK (M11)	Efficient	None needed
Romer	Under-invests in capital	Subsidize R&D (\(s^* = \eta/\alpha\))
Lucas	Under-invests in education	Subsidize schooling
Blanchard	Under-accumulates capital	Fiscal caution (debt crowds out \(K\))

Both Romer and Lucas predict the decentralized economy under-invests relative to the social optimum. R&D tax credits, public research funding, and education subsidies each target a different externality.

Comparing Models and the Common Thread

Model	Engine	Dynamics	Market failure
Rebelo AK	CRS in \(K\)	None	None
Romer (1986)	Knowledge spillovers	None (BGP)	Externality
Lucas (1988)	Human capital	None (BGP)	Externality (\(\psi>0\))
Blanchard (1985)	Finite horizons	Convergence	Non-Ricardian
q-Ramsey	Adjustment costs	Slow convergence	None

The common thread: perpetual growth requires constant returns to accumulable factors. Rebelo assumes it directly (\(Y = AK\)); Romer achieves it via spillovers (\(\alpha + \eta = 1\)); Lucas via linear human capital accumulation (\(\dot{h}/h = \phi(1-u)\)).

Key Takeaways

Exogenous growth models cannot explain why technology improves: CRS + perfect competition exhausts income on factor payments
Rebelo’s taxonomy: \(0 = (\alpha-1)\gamma + n(\alpha+\beta-1)\) catalogs all growth possibilities
AK, Romer, Lucas all reduce to constant returns in accumulable factors; Romer and Lucas predict under-investment due to externalities
Blanchard finite horizons lower steady-state capital and break Ricardian equivalence
q-Ramsey adjustment costs slow convergence without changing the steady state