The Solow Lab

Each extra unit of capital adds less output than the previous one: diminishing MPK. Investment still rises as the economy gets richer, but it rises more slowly than it would without diminishing returns. Meanwhile, replacing worn-out capital requires δ times the entire capital stock. Eventually investment just covers replacement.

The first machine can transform someone’s work. The hundredth machine assigned to that same worker may add very little. This is diminishing returns to capital with labor and technology fixed, even though the production function has constant returns to scale when both K and L rise together.

c* = f(k*) − δk*

Steady-state consumption is output left after replacing worn-out capital. One extra unit of capital adds MPK units of output but requires δ more units of replacement investment each year. Below the Golden Rule, the extra output exceeds the extra replacement cost. Above it, the extra replacement cost is greater.

The Golden Rule maximizes consumption across steady states. It is not a claim that everyone prefers the transition, or that consumption should always be postponed. The economy moves toward the steady state implied by its saving rate; it does not automatically choose the Golden Rule.

Output and replacement at the Golden Rule

For this Cobb–Douglas example, the Golden Rule saving rate equals α = ½. That equality is a result of this production function, not the definition of the Golden Rule.

With technological progress, “steady” refers to capital and output per effective worker. E keeps increasing, so the same y translates into more output per actual worker: Y/L = Ey. Capital per actual worker K/L rises too. Investment continues to replace depreciating capital and equip both additional workers and additional effective labor.

A higher saving rate can lift the economy’s path, but with diminishing returns it cannot permanently steepen the path’s growth rate. A one-time improvement in technology raises a level; a continuing positive g sustains growth in living standards.

Δk = sf(k) − (δ + n + g)k

Timing note: this extension solves the continuous-time form of Mankiw’s growth equation and displays annual observations. Rates n and g are continuously compounded; n + g is the model’s total-output growth rate. The Chapter 8 table separately uses the exact annual recurrence printed in the slides.

Y = AK^αL^1−α

ΔA/A ≈ ΔY/Y − α(ΔK/K) − (1−α)(ΔL/L)

This is the usual small-growth-rate approximation, with constant factor shares. A is total factor productivity. It is distinct from labor efficiency E in Chapter 9.

With the same Cobb–Douglas production function, writing Y = K^α(EL)^1−α implies A = E^1−α. In continuous growth rates, ΔA/A = (1−α)g. A 2% rise in labor efficiency therefore does not mean 2% TFP growth.

On the balanced-growth path, TFP growth and capital deepening both contribute in growth accounting. Yet continuing technological progress is what sustains the model’s growth in output per worker. Accounting for growth and explaining its long-run driver are related but different exercises.

This is a closed economy with Y = C + I, constant returns to scale, diminishing marginal product of capital, fixed saving rates within each scenario, and no government purchases or net exports. The basic model holds L and technology fixed. Production is normalized to y = k^α; all levels are illustrative model units.

Chapter 8 reproduces the slides’ annual step Δk = sf(k) − δk, using unrounded values internally. The Golden Rule transition uses the same annual step. Chapter 9 uses the continuous-time equation with population and technology and samples its exact Cobb–Douglas solution once each year. The different timing conventions avoid introducing cross-term errors while preserving the slide table exactly. Displayed rounding does not affect the model.

The Golden Rule is a criterion for maximum steady-state consumption, not a complete social-welfare calculation. Growth accounting is an approximate decomposition, not a causal identification strategy. Changes made in one section do not silently alter the assumptions in another.