Applications & case studies

Chapter 3 — what the model says about real policy, and what the evidence says back

This page is written for the 🔵 Advanced reader: advanced undergraduates and master’s students in public policy. It assumes the Summary.

Why this matters for policy

The neoclassical growth model answers a question every finance ministry, central bank and climate negotiator faces: how should we weigh the present against the future? In the model, the answer is compressed into one parameter, the rate of time preference \(\rho\), and one condition, the modified golden rule \(f'(k^*) = \rho\).

That makes the model a disciplined lens on two live debates:

  • whether a rate of return above the growth rate (\(r > g\)) must concentrate wealth;
  • how heavily governments should discount the future when they evaluate long-lived investments such as climate policy.

Case study 1 — Is Piketty right that \(r > g\) drives inequality?

Context. In Capital in the Twenty-First Century (2014), Thomas Piketty argues that when the return on capital \(r\) exceeds the growth rate \(g\), inherited wealth grows faster than income, so wealth and income concentrate at the top. He calls \(r > g\) the “fundamental force for divergence”. The argument shaped debates on wealth and inheritance taxation.

The model’s lever. On the balanced growth path output grows at \(g = n\), and in the market economy \(r = f'(k^*) = \rho\). Since the model requires \(\rho > n\), \(r > g\) always holds in the NGM. It is the model’s dynamic-efficiency condition (summary §3.1).

What the model predicts. By itself, nothing explosive. The representative household earns \(r\) on its capital and a wage, and factor shares depend on technology. With Cobb–Douglas production the labour share is constant. With CES production it moves with the elasticity of substitution \(\varepsilon\): if \(\varepsilon < 1\), labour becomes more essential as capital accumulates and the labour share rises (summary, “A digression on inequality”).

What the evidence shows.

  • The chapter reproduces Schmelzing’s (2019) eight centuries of real interest rates: declining, but mostly above growth, without the plutocracy Piketty’s logic would predict.
  • Acemoglu and Robinson (2015) find in cross-country regressions that the \(r - g\) gap does not explain historical patterns of top income shares. They argue that institutions and technology shape the distribution.
  • The charts below show the two quantities the debate turns on.

Chart: Our World in Data, CC BY 4.0. Data: World Inequality Database (WID.world).

Chart: Our World in Data, CC BY 4.0. Data: International Labour Organization.

Guided questions.

  1. In the NGM, \(r = \rho\) and \(g = n\) on the balanced growth path. Use the model to explain why \(r > g\) is required for the household’s problem to be well defined.
  2. Pick two countries in the top-1% chart whose trends diverge after 1980. Does a single “\(r > g\)” force explain both? What else might?
  3. Suppose capital is mostly human capital, evenly spread across the population. What happens to Piketty’s mechanism? Which policy conclusion changes?

A strong answer:

  • notes that \(\rho > n\) is needed for finite utility, so \(r > g\) is an efficiency property here, not a pathology;
  • uses the charts to show that countries facing similar global interest rates have very different top-income trends, which points to institutions, taxation and labour-market policies;
  • explains that if “capital” is largely human capital earned through work, the returns accrue to workers and the divergence channel weakens;
  • connects the elasticity-of-substitution result (\(\varepsilon < 1\)) to the labour-share chart, and is honest that measured labour shares involve measurement choices (self-employment income, for instance).

Case study 2 — How much should a society save? The social discount rate

Context. Governments discount the costs and benefits of long-lived projects, such as infrastructure, pensions and climate policy. The 2006 Stern Review on the Economics of Climate Change called for sharp and immediate cuts in emissions. William Nordhaus (2007) showed that its conclusions depend decisively on a near-zero time discount rate. He argued they would not survive assumptions consistent with market real interest rates and saving rates.

The model’s lever. The rate of time preference \(\rho\). The chapter notes that Ramsey himself considered it unethical for a planner to discount future utility. Caplin and Leahy (2004) argue that standard discounting imposes a “tyranny of the present” from a planner’s point of view.

What the model predicts. The modified golden rule \(f'(k^*) = \rho\) ties the economy’s long-run capital stock to \(\rho\). A planner with a lower \(\rho\) accumulates more capital (and, by analogy, invests more in the future) at the cost of consumption today. The key graphs trace that transition: consumption falls on impact and rises later. In the model, the market real interest rate also equals \(\rho\) in steady state, which is why Nordhaus anchors the discount rate to observed returns.

What the evidence shows. There is no single “correct” \(\rho\). Market interest rates reveal how people do trade off the present against the future. Ethical arguments concern how a planner should. The policy answer moves a lot with that choice, which is exactly Nordhaus’s point about the Stern Review.

Guided questions.

  1. Using the modified golden rule, explain why a planner who sets \(\rho\) near zero recommends much more investment for the future than one who sets \(\rho\) equal to observed real interest rates.
  2. In the NGM, the market’s \(r\) equals the planner’s \(\rho\) in steady state. Why does this equality support Nordhaus’s position? What would a defender of near-zero discounting reply?
  3. Who bears the cost of a lower social discount rate today, and who gains? Relate this to the impact dip in consumption after a permanent fall in \(\rho\).

A strong answer:

  • separates the positive question (what discount rate do markets reveal?) from the normative one (what should a planner use?);
  • uses \(f'(k^*) = \rho\) to show the size of the effect: steady-state capital is very sensitive to \(\rho\) (see Exercise 3);
  • notes that the costs fall on today’s generations and the gains on future ones;
  • recognises that the NGM’s single infinitely-lived household hides exactly this intergenerational conflict, which the overlapping-generations models of Chapter 8 bring into the open.

Caveats & limits

  • One household, no distribution. The representative household cannot speak directly to inequality. The chapter’s inequality discussion relies on extra structure (factor shares, human capital, mobility).
  • Infinite horizon. Barro’s dynastic altruism justifies it, but real generations differ. Chapter 8 shows that oversaving (dynamic inefficiency) can arise with overlapping generations.
  • No technological progress. Without exogenous technical change the model has no long-run per-capita growth, so its steady state says nothing about the growth policies that matter most (Chapters 5–7).
  • Devil’s advocate. “Optimal saving rules out oversaving everywhere.” True for an infinitely-lived planner, but not in economies of overlapping generations, where the market can accumulate capital beyond the golden rule.

Discussion prompts

  1. Should a democratic government’s discount rate reflect citizens’ revealed impatience, or a planner’s ethical judgement? Who gets to choose \(\rho\)?
  2. If a wealth tax lowers the after-tax return on capital, what does the NGM predict for the steady-state capital stock and wages? Who might lose from a policy aimed at the rich?
  3. The model says the market reproduces the planner’s allocation. Name one real-world feature (externality, missing market, borrowing constraint) that would break that equivalence. What would it change?
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