Solow Growth Model
What is Solow Growth Model?
The Solow growth model shows diminishing returns to capital push an economy to a steady state, so lasting growth per worker needs technological progress.
Output per worker depends on capital per worker through a production function with diminishing returns, so each extra machine adds less than the one before it. Saving adds a fraction s of output to the capital stock each period, while depreciation and a growing workforce subtract from capital per worker, and the two forces meet at a steady state where capital per worker stops changing. Because a steady state is a level and not a growth rate, a higher saving rate makes a country richer without making it permanently faster growing: once the new level is reached, output per worker again grows only at the rate of technological progress. The model predicts conditional convergence, meaning two countries sharing the same saving rate, depreciation, population growth and technology end up at the same income per worker, and the one starting poorer grows faster along the way. Its weak point is that the single thing producing lasting growth, technology, is assumed rather than explained, which is the gap endogenous growth models were built to close.
Solow Growth Model: a worked example
Let output per worker be y = k^0.5, with saving rate s = 0.3, depreciation delta = 0.05 and workforce growth n = 0.01. The steady state solves 0.3 x k^0.5 = 0.06 x k, so k^0.5 = 0.3 / 0.06 = 5 and k* = 25, giving y* = 25^0.5 = 5 and consumption per worker of 0.7 x 5 = 3.5. Check it: investment is 0.3 x 5 = 1.5 and the capital lost to depreciation and workforce growth is 0.06 x 25 = 1.5, so capital per worker holds steady. Now raise saving to s = 0.4: k^0.5 = 0.4 / 0.06 = 6.67, so k* = 44.4, y* = 6.67 and consumption is 0.6 x 6.67 = 4.0. Output per worker is 33 percent higher than before, but the economy grows quickly only while it travels from k = 25 to k = 44.4 and then stops again, which is the model's whole point: saving moves the level, not the long-run growth rate.
The mistake students make with solow growth model
The usual error is concluding that a country which saves more grows faster forever. A higher saving rate raises the steady-state level of capital and output per worker, and growth is faster only during the transition to that level. Once there, output per worker grows at the rate of technological progress, which the saving rate does not touch. A related slip is reading convergence as unconditional: the model says each country converges to its own steady state, so one with low saving and high population growth stays poorer however long you wait.
Solow Growth Model questions
What is the steady state in the Solow model?
It is the level of capital per worker at which new investment exactly covers depreciation plus the capital needed to equip a growing workforce, so capital per worker stops changing. Output per worker is then constant as well, unless technology is improving. An economy below its steady state grows toward it and one above it falls back toward it, which is what makes the steady state stable.
What is the golden rule saving rate?
It is the saving rate that maximizes consumption per worker in the steady state, found where the marginal product of capital equals depreciation plus population growth. With a Cobb-Douglas production function it equals capital's share of income, so if capital earns 50 percent of output the golden rule saving rate is 0.5. Saving more than that leaves a country with more capital but less consumption, a situation the model calls dynamic inefficiency.
Why does the Solow model treat technology as exogenous?
The model was built to isolate what capital accumulation alone can achieve, and the answer is that it cannot deliver permanent growth in output per worker. Technology therefore enters as a term growing at a rate the model does not explain, which is a deliberate simplification rather than a claim about reality. Endogenous growth models drop that assumption and let research, education and the returns firms expect from innovating determine the growth rate.
Formula / Example
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