Understanding the Solow Growth Model: A Step-by-Step Guide to ECO202 Extra Question F6

Introduction: Why the Solow Model Matters Today

In June 2026, as economists debate the long-term growth prospects of AI-driven automation, the Solow growth model remains a fundamental tool for understanding how capital accumulation, population growth, and technology shape a nation's output per capita. This tutorial walks you through the key steps of ECO202 Extra Question F6, using the fictional economies of Mont Plaisant and Mont-Tremblant to illustrate core concepts. By the end, you'll be able to convert aggregate equations to per-person terms, find steady-state values, and compare growth rates across economies.

Part (a): Converting Capital Accumulation to Per-Person Terms

The aggregate capital accumulation equation is: K_{t+1} = (1 - δ)K_t + I_t. Since the economy is closed with no government, investment equals savings: I_t = sY_t. Population grows at rate n: L_{t+1} = (1+n)L_t. To convert to per-person terms, divide by L_t. Let k_t = K_t/L_t and y_t = Y_t/L_t. Then:

K_{t+1}/L_t = (1-δ)k_t + s y_t

But we want k_{t+1} = K_{t+1}/L_{t+1}. Since L_{t+1} = (1+n)L_t, we have:

k_{t+1} = [ (1-δ)k_t + s y_t ] / (1+n)

Subtract k_t from both sides to get Δk_{t+1}:

Δk_{t+1} = k_{t+1} - k_t = [s y_t - (n+δ)k_t] / (1+n)

For small n, this approximates Δk ≈ s y_t - (n+δ)k_t, the standard per-person capital accumulation equation.

Part (b): Steady-State Capital and Output Per Capita

In steady state, Δk = 0, so s y* = (n+δ)k*. Given the production function Y = K^α (AL)^(1-α), but here A is constant (no tech growth). Assuming Y = K^α L^(1-α), per-person output is y = k^α. Substitute:

s (k*)^α = (n+δ)k*

Solve for k*:

k* = [s/(n+δ)]^(1/(1-α))

Then y* = (k*)^α = [s/(n+δ)]^(α/(1-α)).

Part (c): Effect of an Increase in the Savings Rate

If the savings rate s increases, the steady-state capital per capita k* rises because more output is invested. From the formula, a higher s increases the numerator, raising k* and thus y*. This is why policies promoting savings (e.g., tax incentives) can boost long-run output per capita, though the transition may involve temporary lower consumption.

Part (d): Comparing Growth Rates: Mont-Tremblant vs. Mont Plaisant

Mont Plaisant is in steady state, so its output per capita y is constant (growth rate 0%). Mont-Tremblant has identical parameters but lower output, meaning it is below its steady state. According to the Solow model, an economy below its steady state grows faster than one at steady state due to diminishing returns: capital is more productive when scarce. Therefore, Mont-Tremblant's output per capita growth rate is positive and higher than Mont Plaisant's zero growth. This convergence property explains why poorer economies often grow faster, a phenomenon observed in many developing nations today.

Part (e): Solow Diagram

Imagine a graph with k on the x-axis. The investment line is s y = s k^α, an increasing concave curve. The depreciation line is (n+δ)k, a straight line from the origin. The intersection gives k*. For Mont Plaisant, the economy is at k*. For Mont-Tremblant, its current k is lower, so s y > (n+δ)k, meaning capital per person rises (positive Δk). The diagram shows Mont-Tremblant moving right along the investment line toward k*.

Part (f): Time Graphs for Y and y

On a ratio scale (log scale), constant growth appears as a straight line. For Mont Plaisant (steady state), total output Y grows at rate n (population growth), so log Y increases linearly with slope n. Output per capita y is constant, so log y is horizontal. For Mont-Tremblant, Y grows faster than n initially because y is rising; log Y has a steeper slope that gradually decreases to n. Log y starts lower and rises asymptotically to the steady-state level, with the slope declining to zero.

Real-World Connections: AI and Automation

In 2026, the Solow model helps analyze how AI investments (a form of capital) affect growth. If AI boosts the savings rate (s) or effectively increases TFP, steady-state output per capita rises. However, the model reminds us that without technological progress, long-run growth per capita is zero—a sobering thought for the AI hype. Understanding these dynamics is crucial for students in economics and related fields.