Learning By Doing and Knowledge Accumulation

Why Production Experience Matters

In some production processes, making more of a product helps workers and firms discover faster, more reliable, or less wasteful ways to make it. These improvements are called learning by doing: productivity rises through cumulative production experience, not solely through formal research or new physical capital. A process developed at one firm may also benefit other producers if knowledge is shared, observed, or otherwise spills over.

The key distinction is between a one-time increase in the level of accumulated experience and a continuing increase in its growth rate. More experience can leave productivity permanently higher, even after the period in which experience grew rapidly has passed. But learning by doing does not, by itself, guarantee a constant long-run growth rate. That depends on how experience accumulates and how strongly productivity responds to it.

The model below is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It treats experience as a source of productivity and makes the spillover from one firm's production to other firms explicit.

A Learning-by-Doing Model

Consider firm $i$, which produces output using capital $K_i$, labor $L_i$, and productivity $A_i$. Let $E_i$ denote the firm's cumulative production experience, and let $Z$ denote a stock of shared knowledge or economy-wide production experience. Both stocks are positive. To compare stocks with different units, choose positive reference levels $E_0$ and $Z_0$. Assume that experience at the start of a period determines productivity during that period.

Definition: A learning-by-doing productivity function with a knowledge spillover is $$ A_i=A_0\left(\frac{E_i}{E_0}\right)^\lambda \left(\frac{Z}{Z_0}\right)^\phi, $$ where $A_0$ is baseline productivity, $\lambda\geq 0$ is the elasticity of productivity with respect to the firm's own experience, and $\phi\geq 0$ is the elasticity with respect to shared knowledge. If $\phi>0$, an increase in shared knowledge raises productivity even for a firm whose own experience has not changed.

A common production specification is $Y_i=A_iK_i^\alpha L_i^{1-\alpha}$, with $0<\alpha<1$. Holding $A_i$ fixed, this production function has constant returns to the firm's rival inputs, capital and labor. The productivity function adds a separate channel: production experience changes $A_i$. In a simple discrete-time formulation, $E_{i,t}=E_{i,t-1}+Y_{i,t-1}$; a shared stock might evolve as $Z_t=Z_{t-1}+\sum_iY_{i,t-1}$. These laws assume no depreciation or forgetting. Other assumptions about what counts as experience, or whether knowledge depreciates, would change the model.

The exponents are elasticities, not percentage-point effects. For example, with other inputs fixed, a 1% increase in $E_i$ produces approximately a $\lambda$% increase in $A_i$ for a small change. For finite changes, the power function must be evaluated rather than multiplying the percentage change by the elasticity.

Calculating Productivity Gains

When experience stocks change by known factors, divide the new productivity formula by the old one. The baseline $A_0$ and the reference stocks cancel, leaving a product of the own-experience factor and the shared-knowledge factor. Each factor must be raised to its own elasticity.

Worked Example: Combining Own Experience and a Spillover

A firm's productivity is $A=10(E/E_0)^{1/2}(Z/Z_0)^{1/4}$. Its own experience doubles, while shared experience increases by a factor of 16. Find the productivity factor and percentage increase.

Method. Compare productivity after the changes with productivity before them. The own-experience ratio is 2, and the shared-experience ratio is 16, so

\frac{A_{\mathrm{new}}}{A_{\mathrm{old}}} =2^{1/2}16^{1/4} =\sqrt{2}\cdot 2 =2\sqrt{2} \approx 2.8284.

Productivity is about 2.8284 times its previous level. The percentage increase is $(2.8284-1)\times100\%=182.84\%$, rounded to two decimals. The factor is not 4: that would incorrectly treat the two powered terms as if their exponents were both 1.

Check. Since $16^{1/4}=2$ and $2^{1/2}\approx1.4142$, their product is approximately $2.8284$, consistent with $2\sqrt{2}$.

Why the Growth-Rate Formula Holds

The productivity function also gives a direct relationship between experience growth and productivity growth. Assume the power-function model above, with positive, differentiable experience stocks and constant elasticities $\lambda$ and $\phi$. Taking logarithms turns products into sums. Differentiating with respect to time then gives the proportional growth rate of productivity.

Take logarithms.
$\ln A_i=\ln A_0+\lambda(\ln E_i-\ln E_0)+\phi(\ln Z-\ln Z_0)$. The reference stocks and $A_0$ are constants.

Differentiate over time.
For any positive differentiable quantity $Q$, the derivative of $\ln Q$ is $\dot Q/Q$. Applying this to productivity and the two experience stocks gives the growth equation below.

\frac{\dot A_i}{A_i} =\lambda\frac{\dot E_i}{E_i} +\phi\frac{\dot Z}{Z}

This is an exact equation for continuous-time proportional growth under the stated function and constant-elasticity assumptions. It says that productivity growth is the sum of the own-experience growth contribution and the spillover contribution. If experience stocks are constant, the equation implies no productivity growth from learning by doing. If experience rises but the growth rates of the stocks later fall to zero, productivity stops rising, though its accumulated level can remain above its earlier level. With discrete percentage changes over a finite period, the exact productivity factor is instead $(E_{\mathrm{new}}/E_{\mathrm{old}})^\lambda(Z_{\mathrm{new}}/Z_{\mathrm{old}})^\phi$.

Worked Example: Adding Learning to Output Growth Accounting

A firm has $\lambda=0.3$ and $\phi=0.2$. In continuous time, its own experience grows at 4% per year and shared knowledge grows at 2% per year. Its production function is $Y=AK^{0.4}L^{0.6}$. If capital grows at 2% per year and labor at 1% per year, find the growth rates of productivity and output.

Method. Use the log-derivative growth equations. All rates here are continuous proportional growth rates, so the growth contributions add directly.

First calculate productivity growth:

g_A =0.3(4\%)+0.2(2\%) =1.2\%+0.4\% =1.6\%.

For the production function, taking logarithms gives $\ln Y=\ln A+0.4\ln K+0.6\ln L$. Differentiating gives $g_Y=g_A+0.4g_K+0.6g_L$. Substituting the data:

g_Y =1.6\%+0.4(2\%)+0.6(1\%) =1.6\%+0.8\%+0.6\% =3.0\%.

Answer. Productivity grows at 1.6% per year and output grows at 3.0% per year under these assumptions. The calculation does not infer that experience must grow at those rates in every model; they are given rates for this example.

Check. Capital and labor contribute 1.4 percentage points in total, and adding the 1.6% productivity contribution gives 3.0%.

From Firm Experience to Persistent Growth

Cumulative experience can make a productivity gain persistent because past production remains in the experience stock in the no-depreciation model. A temporary period of rapid production can therefore leave $E_i$, or shared $Z$, at a permanently higher level than it otherwise would have been. If the stock remains higher, the productivity function continues to imply a higher productivity level, even if experience stops growing.

Persistent growth is a stronger claim than a persistent increase in the productivity level. The growth equation requires experience stocks to keep growing to generate continuing productivity growth. Whether they do so at stable rates depends on the model's production and accumulation conditions. For example, simply setting $E_i$ equal to cumulative output does not establish that output or productivity will grow at a constant rate. Learning by doing can reinforce growth, but its presence alone does not prove a balanced growth path.

A spillover is important because a firm's accumulated experience may raise other firms' productivity through $Z$, even if those firms did not produce the same quantity themselves. In this way, the consequences of production experience can extend beyond the firm that first acquired it. The distinction between a firm's own experience and shared knowledge is essential: the first affects its productivity directly, while the second can affect many producers.

Worked Example: Separating Direct and Spillover Gains

Two hypothetical firms each initially have productivity $A=1$ and produce $Y_i=A_i\sqrt{K_iL_i}$. Each has $K_i=L_i=100$. Firm 1's own experience doubles, and the shared knowledge stock quadruples. Firm 2's own experience is unchanged. Both firms have $\lambda=1/2$ and $\phi=1/4$. Find each firm's new output and the total output gain, holding capital and labor fixed.

Method. First calculate the productivity factor for each firm, then apply the production function. Initially, $\sqrt{100\cdot100}=100$, so each firm produces $1\cdot100=100$; total output is $200$.

For firm 1, both experience channels change:

\frac{A_{1,\mathrm{new}}}{A_{1,\mathrm{old}}} =2^{1/2}4^{1/4} =\sqrt{2}\sqrt{2} =2.

Thus $A_{1,\mathrm{new}}=2$ and $Y_{1,\mathrm{new}}=2\cdot100=200$. For firm 2, own experience is unchanged, so its own-experience factor is $1$. It receives only the shared-knowledge gain:

\frac{A_{2,\mathrm{new}}}{A_{2,\mathrm{old}}} =1^{1/2}4^{1/4} =\sqrt{2} \approx1.4142.

Firm 2's new output is $100\sqrt{2}\approx141.42$, rounded to two decimals. Total new output is $200+141.42=341.42$. The total gain is $341.42-200=141.42$, consisting of a 100-unit gain at firm 1 and a $141.42-100=41.42$-unit gain at firm 2.

Check. Firm 1's productivity doubles, while firm 2's rises by a factor of about 1.4142. Applying both factors to their initial outputs of 100 gives the stated total. The gain at firm 2 illustrates the spillover, not an increase in its own experience.

Common Mistakes and Interpretation

Multiplying experience factors without applying elasticities. In the first example, the correct factor is $2^{1/2}16^{1/4}=2\sqrt{2}$, not $2\cdot16$ or 4. Each stock change must be raised to its own exponent.
Confusing a productivity level with a growth rate. A stock of experience that remains high can keep productivity at a higher level; it does not imply that productivity keeps growing when the stock stops growing.
Omitting the spillover for other firms. When $\phi>0$, a firm with unchanged own experience can still gain productivity if $Z$ increases.
Adding ordinary finite percentage changes as if they were exact. The continuous-time growth equation adds proportional growth rates exactly under its assumptions. For finite discrete changes, use the powered stock ratios to find the exact factor.

Key takeaway: In a power-function learning model, own experience and shared knowledge raise productivity according to their respective elasticities. Accumulated experience can preserve a higher productivity level, while continuing productivity growth requires continuing growth in experience stocks.

Check Your Understanding

For each problem, apply the stated learning function and keep level changes distinct from growth rates.

A firm's productivity is $A=3(E/500)^{1/4}(Z/2000)^{1/2}$. Its own experience rises by a factor of 16 and shared knowledge rises by a factor of 4. By what factor does productivity change? (Answer: $16^{1/4}4^{1/2}=2\cdot2=4$.)
A firm has $\lambda=0.2$, $\phi=0.3$, continuous own-experience growth of 5% per year, and continuous shared-knowledge growth of 1% per year. Its production function is $Y=AK^{0.4}L^{0.6}$, with capital growing at 1.5% and labor at 0.5%. Find productivity growth and output growth. (Answer: $g_A=0.2(5\%)+0.3(1\%)=1.3\%$; $g_Y=1.3\%+0.4(1.5\%)+0.6(0.5\%)=2.2\%$ per year.)
Firm B has unchanged own experience, while shared knowledge increases ninefold. Its productivity function has $\lambda=0.5$ and $\phi=0.5$. By what factor does B's productivity change? (Answer: $1^{0.5}9^{0.5}=3$.)
Two firms each initially produce 80 and 120 units, respectively. Their production functions have fixed inputs during the comparison. Firm A's own experience doubles, the shared stock quadruples, and $\lambda=0.25,\ \phi=0.5$ for both firms. Firm B's own experience is unchanged. Find each firm's new output and total output. (Answer: A's factor is $2^{0.25}4^{0.5}\approx2.3784$, so its output is about 190.27; B's factor is $4^{0.5}=2$, so its output is 240; total output is about 430.27.)
In a no-depreciation model, experience stops growing after a temporary production boom. Explain what happens to the productivity level and to productivity growth after the boom, assuming positive $\lambda$ or $\phi$ for the stock that remains higher. (Answer: The higher experience stock can leave productivity at a permanently higher level than before the boom, but once the relevant experience growth rates are zero, the learning contribution to productivity growth is zero.)

What You'll Learn