8. Growth: Solow and Beyond

In micro, one firm's production and one household's choices were settled separately; in macro, we pile up every firm's machines and every household's saving and run the sum for decades. This chapter shows why stacking capital alone makes growth stop (Solow), and how dynamic programming pulls the household's choices out from behind the saving rate (Ramsey).

• A Country Cannot Grow on Machines Alone — In micro, one firm's production function settled that firm's choices. In macro, we add that function up to a whole country and run it for decades. More ⚙️ machines raise output, but each one adds less. Worn-out machines, though, grow in a straight line with the pile. Growth stops where the two flows meet. That meeting point is the steady state in the formula below. Poor and rich countries both end up gathering at the same spot. For example, picture a village that plants 🌱 seeds. At first, each extra seed makes the harvest jump. Once the field is crowded, extra seeds add only a little. Some seeds also rot every year. When new planting just matches the rotting, the harvest stops growing.
• Where the Saving Rate Comes From — In micro, splitting consumption between two periods was decided once and finished. In macro, that choice repeats every period, linked to tomorrow again and again. The saving rate is not a number handed down to anyone. Each household chooses how much to spend today and leave for tomorrow. A higher interest rate or more patience changes what gets left. The formula below adds today's satisfaction to tomorrow's value. Modern macro models run on exactly this rule. For example, say you get 10,000 won of 💰 allowance. You could spend it all today. But picturing the toy you want tomorrow, you drop some into the 🐷 piggy bank. Weighing tomorrow's fun alongside today's is exactly how the formula thinks.