Productivity and Economic Growth

Prerequisites: Firm Costs, Pricing, and Competition · Income, Wealth, and Inequality · GDP and National Income · Interest, Saving, and Investment

If an economy's total output increases by 20%, does that mean everyone is richer? If both population and working hours also increase by 20%, output per person and per hour might remain completely unchanged. The starting point for long-term growth is to first fix the basis of comparison, and then track how capital, technology, and organization improve sustainable productive capacity.

This article first calculates labor productivity, then lets two scenarios undergo multi-period changes from the same capital starting point. Finally, it distinguishes between a one-time level increase and sustained growth, returning to distribution, environment, and measurement. All numbers are set for teaching purposes; the "periods" in the interaction are not predictions for any real economy's years.

What Total, Per Capita, and Per Hour Each Answer

In the second row scenario, total growth does not raise these two average indicators; only the third row raises both by 20%. Labor productivity is the ratio of output to hours worked, not a direct measure of employee diligence; better machines, organization, skills, and technology all affect it.

Price changes should also be deducted first. If nominal income increases merely because the listed prices of the same products have risen, it cannot be called a proportional increase in actual productive capacity. Cross-regional comparisons require further handling of purchasing power, working hours, and statistical scope.

A Single Good Can Be Both Consumed and Turned into Capital

Establish a simplified economy: population is fixed, everyone is identical, and a single composite good can be either consumed or invested in; one unit of investment yields one additional unit of capital in the next period. Capital per person is $k$, and output per person per period is y=A√k, where $A$ aggregates the productivity conditions given in the model. Capital depreciates by 10% per period, and the investment rate is $s$.

The change in capital is:

Current output y = A√k
Investment = s×y
Consumption = (1−s)×y
Next-period capital = 0.9×current capital + Investment

First, set $A=2$, $s=25\%$, and $k=25$. Then $y=10$, investment is 2.5, which exactly covers depreciation of 2.5, and consumption is 7.5. In the next period, $k$ remains 25; this is called a steady state. It does not mean there is no production activity, but rather that the renewal investment each period exactly maintains the capital stock.

The implication of diminishing marginal returns to capital can be seen by comparing output increasing from 4 to 6 when $k=4\to 9$, with output increasing from 6 to 10 when $k=9\to 25$: the more capital there is, the smaller the additional output brought by increasing the same unit of capital further. Note that this does not mean total output will decrease due to increased capital.

Average additional output per unit of extra capital is 2/5=0.4 over the first interval and 4/16=0.25 over the second. Compare output per added capital unit, not just the total output increases of 2 and 4.

What Is Lost First When Raising the Investment Rate from the Same Starting Point

Now, only raise the investment rate from 25% to 40%, while capital still starts from 25 and $A$ remains 2. In the change period, output is still 10, but investment rises from 2.5 to 4, and consumption drops from 7.5 to 6, decreasing by 1.5 initially. Capital will not jump immediately to the new long-run level just because a higher investment rate is announced.

In the next period, capital is 0.9×25+4=26.5, output is approximately 10.296, and consumption is approximately 6.177; capital accumulates gradually, and future output and consumption only then increase.

The last row of the table is the limit, not period 3. Directly comparing old steady-state consumption of 7.5 with new steady-state 9.6 misses the transitional cost of initially reducing consumption; drawing a time path from the same starting point allows one to see who pays the cost and when.

First, keep $A=2$ and change the investment rate from 25 to 40. The gray baseline is the original scenario's consumption of 7.5 per period; new consumption starts below it and may later exceed it. The figure only plots 60 periods; the value cards list the positive steady state separately; not approaching the limit within a finite time does not mean the steady-state formula is wrong.

Next, adjust the investment rate back to 25, and change $A$ alone to observe that current output also changes. This differs from the initial effect of only changing the savings share. Changing only one condition at a time makes it easier to identify where the changes come from.

How to Calculate the Steady State, and Why It Is Not Permanent Growth

The positive steady state satisfies sA√k=0.1k, so k=(sA/0.1)². When $s=0.4$ and $A=2$, $k=64$ and $y=16$; when $s=0.25$ and $A=2$, $k=25$ and $y=10$. There is also the mathematical boundary point of $k=0$, but since this lesson's path starts from positive capital, it is not treated as a target steady state.

A higher investment rate causes capital and output to grow during the transition period; as it approaches the new steady state, net investment approaches zero, and the per capita output growth rate also approaches zero. It raises the long-run level but does not create a permanent positive growth rate in a model with fixed technology and fixed population.

A one-time increase of $A$ from 2 to 2.5 raises output at the same capital and triggers new capital adjustment; as long as $A$ remains fixed thereafter, it eventually tends toward another steady state. To explain sustained technological progress, one needs to let productivity conditions change over time and explain how they form, rather than treating a one-time parameter adjustment as progress that happens every year.

Does More Investment Always Mean Higher Consumption

In the positive steady state of this model, output is sA²/0.1, and consumption is (1−s)sA²/0.1. Increasing $s$ raises future output but leaves a smaller share available for consumption; the resulting effects on consumption need not have the same sign.

When $A=2$, the steady-state consumption for $s=25\%$ is 7.5, for 40% it is 9.6, and for 50% it is 10. Only under these special assumptions does $s=50\%$ maximize steady-state per capita consumption; this does not directly provide the optimal real-world investment rate, as transitional consumption, intergenerational weights, risk, population, and different types of capital have not yet been included.

A scenario with "higher long-term numbers" may also impose unacceptable costs on families with tight current resources. Evaluation needs to put time and distribution back in, rather than just picking the maximum value in the last row.

Returning from A to Real Production Mechanisms

Education, health, R&D, organization, infrastructure, and institutions may all affect productive capacity. $A$ is merely an aggregate of factors not detailed item-by-item in the model; it is not a directly measured "number of inventions," nor can all unexplained growth be attributed to a single type of technology.

Observing that high-income regions possess a certain institution does not prove that the institution alone causes high income: history, geography, reverse causality, and common causes may all be involved. Evaluating an improvement requires specific mechanisms, comparable objects, time, and implementation evidence.

Capital itself has differences in quality and composition. Idle machines do not necessarily form effective productive capacity; adding equipment but lacking skills, energy, or reliable supply may result in actual outcomes deviating from a single $k$ model. Adding one important complementary condition to a case usually has more explanatory power than simply stacking a larger capital number.

Final Transfer Task for the Course

A city's real GDP grows by 20%, its population grows by 10%, and working hours grow by 20%. Calculate the changes in output per person and per hour, and explain what additional information is needed to evaluate residents' well-being.

Someone sees the new steady-state $y=16$ after raising the investment rate and says, "Starting next year, production will be 6 more each period than the previous one, and currently no one pays the cost." Please correct this statement using this lesson and previous lessons.

Sources and Further Reading

These references support concepts and statistical definitions; the numerical cases and diagrams are original synthetic teaching examples.

OpenStax · Labor Productivity and Economic Growth