Why the AK Model Changes the Growth Mechanism
In the Solow model with diminishing returns, adding capital per worker eventually produces smaller increments of output per worker. The AK model asks what happens if the relevant stock of reproducible capital does not have diminishing returns. If each additional unit continues to raise output by the same amount, saving and investment can keep expanding productive capacity without the marginal product of capital declining. That possibility makes the AK model a simple mechanism for sustained per-capita growth.
The model is an aggregate simplification, not a claim that every individual firm has constant returns to its own machines. The capital stock can be interpreted broadly to include reproducible physical capital and other accumulable productive assets. In an interpretation involving knowledge spillovers, the return to capital for the economy as a whole can exceed the return a private investor receives directly.
Production and Accumulation Assumptions
The production equation says that output is proportional to capital. Its average product is \(Y_t/K_t=A\), and the marginal product of capital is also \(A\). Thus, increasing capital by one unit always raises output by \(A\) units within this model. The same relationship holds per person: a given proportional increase in \(k_t\) produces the same proportional increase in \(y_t\).
To describe how the stock changes, use a discrete-time model. Periods are indexed by \(t\). Assume the saving rate \(s\) is fixed, where \(0\leq s\leq1\); all saving becomes investment; a fraction \(\delta\) of existing capital depreciates each period, where \(0\leq\delta\leq1\); and the population grows at a constant rate \(n\geq0\). Productivity \(A\) does not change over time. There is no separate source of technological progress in this setup.
Under these assumptions, investment is \(I_t=sY_t=sAK_t\), and the surviving part of the existing capital stock is \((1-\delta)K_t\). Therefore, next period’s aggregate capital is the sum of those two components. Population growth must also be accounted for when converting aggregate capital into capital per person.
The first equation does not say that all output becomes capital: only the fixed share \(s\) is invested. Nor does it say that capital per person rises whenever aggregate capital rises. If the population grows, aggregate capital must grow enough to keep pace with the larger population as well as replace depreciated capital.
Deriving the Law of Motion per Person
The central step is to divide next period’s aggregate capital by next period’s population. This gives a precise link between the production, saving, depreciation, and population assumptions.
Substitute \(Y_t=AK_t\) into the accumulation equation: \(K_{t+1}=(1-\delta+sA)K_t\).
By assumption, \(L_{t+1}=(1+n)L_t\).
Divide the aggregate stock by population and use \(k_t=K_t/L_t\):
This division explains why population growth appears in the denominator of the discrete-time expression: each unit of next-period aggregate capital is spread across \(1+n\) times as many people as the current population. The exact factor multiplying \(k_t\) is therefore a ratio, not simply the numerator with \(n\) subtracted.
For example, \(sA>\delta+n\) means that investment per unit of current capital is enough to offset both depreciation and the capital needed to keep up with population growth. Because the return to capital does not diminish as the capital stock expands, a positive gap can persist rather than being eliminated by a falling marginal product. The model thus provides an endogenous accumulation mechanism: the saving decision and the return to reproducible capital help determine whether per-person capital can continue to expand.
Worked Example: Reading the AK Production Function
A hypothetical economy has \(A=0.40\) units of output per unit of capital and \(K=250\) units of capital. Calculate output, the output added by a 25-unit capital increase, and output after the capital stock doubles. State what the calculations show about returns to capital.
Method. Apply \(Y=AK\) separately to each capital stock. This keeps the productivity parameter fixed while changing capital; it avoids treating \(A\) as a percentage growth rate.
Initial output. \(Y=0.40(250)=100\) units of output.
Output increase from 25 additional units of capital. The new stock is \(250+25=275\), so new output is \(0.40(275)=110\). The increase is \(110-100=10\) units. Equivalently, the marginal product calculation gives \(A\Delta K=0.40(25)=10\).
Output after capital doubles. The doubled stock is \(2(250)=500\), so output is \(0.40(500)=200\). This is twice the initial output of 100.
Check and interpretation. In both calculations, each additional capital unit raises output by \(0.40\) units. The constant marginal product is the feature that distinguishes this production relationship from one with diminishing returns to capital.
Worked Example: Computing Capital per Person Next Period
A hypothetical economy has \(A=0.30\), \(s=0.25\), \(\delta=0.04\), \(n=0.02\), and current capital per person \(k_t=40\) thousand dollars. Find the exact per-person capital factor, next period’s capital per person, and output per person in both periods.
Method. Use the discrete-time law of motion derived above, then use \(y_t=Ak_t\). Keep unrounded values through the calculations; this avoids errors from premature rounding.
Capital factor. The numerator is \(1-\delta+sA=1-0.04+0.25(0.30)=0.96+0.075=1.035\). The denominator is \(1+n=1.02\). Thus the factor is \(1.035/1.02=1.01470588\ldots\).
Next-period capital per person. \(k_{t+1}=40(1.035/1.02)=41.4/1.02=40.58823529\ldots\) thousand dollars. Rounded only at the final step to two decimal places, \(k_{t+1}=40.59\) thousand dollars per person.
Output per person. Current output is \(y_t=0.30(40)=12.00\) thousand dollars. Next-period output is \(y_{t+1}=0.30(40.58823529\ldots)=12.17647059\ldots\), or \(12.18\) thousand dollars to two decimal places.
Check and common mistake. Since \(sA=0.075\) exceeds \(\delta+n=0.06\), capital per person should rise. The computed factor \(1.01470588\ldots\) is greater than 1, and \(40.59>40\), confirming the direction. A tempting but incorrect shortcut is to use \(1+sA-\delta-n=1.015\) as the exact factor. That is not the model’s discrete-time ratio: the exact factor is \(1.035/1.02\), approximately \(1.0147\).
Worked Example: Comparing Capital Accumulation Regimes
Two hypothetical economies share \(A=0.40\), \(\delta=0.04\), \(n=0.02\), and initial capital per person of 50 thousand dollars. Economy H saves \(s=0.15\), while Economy J saves \(s=0.20\). For each economy, determine whether capital per person rises, stays constant, or falls, and calculate next period’s capital and output per person.
Method. Compare \(sA\) with \(\delta+n\), then verify the conclusion by calculating the capital factor. This avoids deciding from the saving rate alone, without accounting for the productivity of capital and the population and depreciation requirements.
Common threshold. Depreciation plus population growth is \(\delta+n=0.04+0.02=0.06\).
Economy H. Its investment return term is \(sA=0.15(0.40)=0.06\), equal to the threshold. Its factor is \((1-0.04+0.06)/(1+0.02)=1.02/1.02=1\). Therefore \(k_{t+1}=50\) thousand dollars per person, and \(y_{t+1}=0.40(50)=20\) thousand dollars per person. Both equal their current per-person values.
Economy J. Its investment return term is \(sA=0.20(0.40)=0.08\), which exceeds 0.06. Its factor is \((1-0.04+0.08)/(1+0.02)=1.04/1.02=1.01960784\ldots\). Therefore \(k_{t+1}=50(1.04/1.02)=50.98039216\ldots\), or 50.98 thousand dollars per person. Output is \(y_{t+1}=0.40(50.98039216\ldots)=20.39215686\ldots\), or 20.39 thousand dollars per person. Current output is \(0.40(50)=20\) thousand dollars per person.
Check. Economy H is exactly at the threshold and has no change in capital per person. Economy J is above the threshold, and its calculated capital and output per person both exceed their current values.
What the AK Mechanism Does—and Does Not—Claim
The AK model’s growth mechanism depends on a special production assumption. In the model, \(Y=AK\) gives a constant marginal product of capital, so more capital does not itself make the return on additional capital fall. With a constant saving rate, investment remains proportional to the capital stock. If that investment is sufficient to offset depreciation and population growth, per-person capital can keep rising.
This differs from the standard Solow mechanism with diminishing returns to capital per worker. There, a growing capital stock eventually produces smaller additional output, which weakens investment relative to the amount needed to maintain the stock. The AK specification removes that diminishing-return force for the broad reproducible capital stock. One possible economic rationale is that accumulating capital also creates knowledge or other benefits that are not fully captured by an individual investor.
The model does not prove that every economy will grow in this way. Its conclusion depends on the assumptions: a fixed saving rate, constant productivity, the specified accumulation rule, and a broad capital stock with a non-diminishing return. If \(sA\leq\delta+n\), the basic model does not imply rising capital per person. If productivity changes, saving behavior varies, or the relevant return diminishes, the simple AK calculation no longer describes the economy without modification.
- Do not confuse aggregate and per-person growth. Aggregate capital can rise while capital per person falls if population grows faster.
- Do not call \(A\) the growth rate. \(A\) is output per unit of capital in \(Y=AK\); accumulation depends on \(s\), depreciation, and population growth as well.
- Do not use the exact discrete-time formula as a continuous-time approximation. The denominator \(1+n\) follows from the discrete population equation and must be retained in exact calculations.
Check Your Understanding
For each problem, state any comparison or calculation that supports your final result. Money amounts are in thousands of dollars where specified.
- A hypothetical production system has \(A=0.25\) and capital \(K=320\) units. Calculate output, then calculate output if capital increases by 40 units. What does the change imply about the marginal product in this model? (Answer: Initial output \(=80\); new output \(=90\); the 40-unit increase adds 10 units of output, so the marginal product is \(0.25\).)
- An economy has \(A=0.35\), \(s=0.20\), \(\delta=0.05\), \(n=0.01\), and \(k_t=60\) thousand dollars per person. Calculate the capital factor, next-period capital per person, and next-period output per person, rounding amounts to two decimals. (Answer: Factor \(=1.02/1.01\approx1.0099\); \(k_{t+1}\approx60.59\) thousand dollars; \(y_{t+1}\approx21.21\) thousand dollars.)
- An economy has \(A=0.30\), \(s=0.20\), \(\delta=0.04\), \(n=0.02\), and \(k_t=75\). Determine whether per-person capital changes next period and calculate current and next-period output per person. (Answer: \(sA=\delta+n=0.06\), so the factor is 1; \(k_{t+1}=75\), and current and next-period output per person are both \(22.50\).)
- For an economy with \(A=0.50\), \(\delta=0.03\), and \(n=0.02\), find the saving rate at which capital per person is unchanged. Then, if \(s=0.12\) and \(k_t=40\), calculate next-period capital per person, rounding to two decimals. (Answer: The unchanged-capital saving rate is \(s=0.10\); at \(s=0.12\), the factor is \(1.03/1.02\), and \(k_{t+1}\approx40.39\).)
- Two economies each have \(A=0.40\), \(\delta=0.03\), \(n=0.02\), and \(k_t=100\). Economy X saves 10% and Economy Y saves 15%. For each, calculate the capital factor, next-period capital per person, and next-period output per person. Round per-person amounts to two decimals. (Answer: X: factor \(1.01/1.02\), \(k_{t+1}\approx99.02\), \(y_{t+1}\approx39.61\); Y: factor \(1.03/1.02\), \(k_{t+1}\approx100.98\), \(y_{t+1}\approx40.39\).)