Introduction
National income explains how an economy’s total output is created and distributed. In simple terms, it shows where income comes from and where it goes. Firms produce goods and services by using factors of production, mainly labor and capital. Labor includes workers’ time and effort, while capital includes machines, buildings, equipment, and technology.
A central question in macroeconomics is: How do firms decide how many workers to hire and how much capital to use? The answer depends on productivity. Firms compare the extra output created by one more worker or one more unit of capital with the cost of using that factor.
This article explains the firm’s demand for factors, the marginal product of labor, the marginal product of capital, the division of national income, and the Cobb-Douglas production function in a clear and practical way.
The Firm’s Demand for Factors
Firms need labor and capital to produce output. A bakery, for example, needs workers, ovens, ingredients, and a kitchen. A technology company needs employees, computers, software, and office space. In each case, the firm chooses the amount of labor and capital that helps maximize profit.
A profit-maximizing firm does not hire workers randomly. It asks: Will one more worker increase revenue more than cost? The same logic applies to capital. A firm rents or buys additional capital only if the extra output is worth more than the cost.
Marginal Product of Labor Explained
What Is the Marginal Product of Labor?
The marginal product of labor, often called MPL, is the extra output a firm gets from hiring one additional unit of labor while keeping capital fixed.
In formula form:
MPL = F(K, L + 1) − F(K, L)
This means the firm compares output with one more worker to output with the current number of workers. The difference is the additional production created by that worker.
Example of Marginal Product of Labor
Imagine a bakery with a fixed number of ovens. If the bakery hires one more worker, it may produce more bread. At first, each new worker may add a lot of extra output because there is enough equipment and space.
However, as more workers are added to the same kitchen, the space becomes crowded. Workers may wait for ovens, share tools, or get in each other’s way. As a result, each additional worker adds less output than the previous worker.
This is called diminishing marginal product.
Diminishing Marginal Product
Why Marginal Product Declines
The principle of diminishing marginal product says that, holding one factor fixed, the extra output from adding more of another factor eventually decreases.
For labor, this means that if capital stays fixed, adding more workers will eventually produce smaller increases in output. For capital, this means that if labor stays fixed, adding more machines will eventually produce smaller increases in output.
Why This Matters for Firms
Diminishing marginal product helps explain why firms do not hire unlimited workers or rent unlimited capital. Even if each worker adds output, the extra output may become too small to justify the cost.
A firm keeps hiring only as long as the additional worker increases profit.
From Marginal Product of Labor to Labor Demand
How Firms Decide How Much Labor to Hire
A firm compares the extra revenue from hiring one more worker with the wage paid to that worker.
The extra revenue from labor depends on two things:
The price of the product and the marginal product of labor.
If the product price is P and the marginal product of labor is MPL, then the extra revenue from one more worker is:
P × MPL
The extra cost is the wage, represented by W.
So the firm hires labor until:
P × MPL = W
This can also be written as:
MPL = W / P
Real Wage and Labor Demand
What Is the Real Wage?
The real wage is the wage measured in units of output rather than dollars. It is written as:
W / P
For example, suppose a worker earns $20 per hour and bread sells for $2 per loaf. The real wage is 10 loaves of bread per hour. This means the worker must produce at least 10 loaves per hour for the bakery to justify hiring that worker.
Why Real Wage Matters
The real wage shows the true cost of labor to the firm. If workers become more productive, firms can afford to pay higher real wages. If productivity falls, hiring additional workers becomes less profitable.
This is why labor productivity is closely connected to real wages.
Marginal Product of Capital Explained
What Is the Marginal Product of Capital?
The marginal product of capital, or MPK, is the extra output a firm gets from using one additional unit of capital while keeping labor fixed.
In formula form:
MPK = F(K + 1, L) − F(K, L)
This means the firm compares output with one more unit of capital to output with the current amount of capital.
Example of Marginal Product of Capital
Think again about the bakery. The first oven may greatly increase bread production. The second oven may also help a lot. But if the bakery keeps adding ovens without hiring more workers, eventually the extra ovens may not be used efficiently.
At that point, each new oven adds less output than the previous one. This is diminishing marginal product applied to capital.
Capital Demand and the Real Rental Price
How Firms Decide How Much Capital to Use
A firm compares the extra revenue from one more unit of capital with the rental cost of that capital.
If the product price is P, the marginal product of capital is MPK, and the rental price of capital is R, then the firm rents capital until:
P × MPK = R
This can also be written as:
MPK = R / P
What Is the Real Rental Price of Capital?
The real rental price of capital is the rental price measured in units of goods rather than money. It shows how much output the firm must produce to justify renting one more unit of capital.
The Division of National Income
How Income Is Divided Between Labor and Capital
National income is divided among the factors of production. Workers receive wages, and capital owners receive returns on capital.
If firms are competitive and profit-maximizing, each factor is paid according to its marginal product. This means:
Labor income equals:
MPL × L
Capital income equals:
MPK × K
Economic profit is what remains after labor and capital are paid.
So total output can be expressed as:
Y = (MPL × L) + (MPK × K) + Economic Profit
Economic Profit vs. Accounting Profit
Economic profit is different from accounting profit.
Economic profit is what remains after paying all factors of production, including labor and capital. Accounting profit is the profit commonly reported by firms, and it often includes the return to capital owned by the firm’s owners.
In many macroeconomic models, if the production function has constant returns to scale and firms are competitive, economic profit becomes zero. This does not mean firms have no accounting profit. It means that all output is distributed to labor and capital.
Constant Returns to Scale
What Constant Returns to Scale Means
A production function has constant returns to scale when increasing labor and capital by the same percentage increases output by that same percentage.
For example, if a firm doubles both labor and capital and output also doubles, the production function has constant returns to scale.
This idea is important because it helps explain why total output can be fully divided between labor and capital payments.
Euler’s Theorem in Simple Terms
Euler’s theorem says that when a production function has constant returns to scale, total output can be exactly divided between payments to labor and payments to capital.
In simple terms:
F(K, L) = (MPK × K) + (MPL × L)
This supports the idea that national income is distributed according to marginal productivity.
The Cobb-Douglas Production Function
What Is the Cobb-Douglas Production Function?
The Cobb-Douglas production function is one of the most important models in economics. It explains how labor and capital combine to produce output.
The function is usually written as:
F(K, L) = A K^α L^(1−α)
In this formula:
A represents technology or productivity.
K represents capital.
L represents labor.
α represents capital’s share of income.
1 − α represents labor’s share of income.
Why the Cobb-Douglas Function Matters
The Cobb-Douglas production function is useful because it shows a stable relationship between output, labor, and capital. It also helps explain why labor and capital shares of national income can remain relatively stable over time.
If capital receives a share α of total income, then labor receives the remaining share 1 − α.
For example, if capital’s share is about 30%, labor’s share is about 70%.
Labor Productivity and Real Wages
The Link Between Productivity and Wages
One of the most important lessons from this topic is that real wages depend strongly on labor productivity.
Labor productivity is commonly measured as output per worker or output per hour. When workers produce more output per hour, firms can pay higher real wages while still remaining profitable.
This relationship can be summarized simply:
Higher productivity supports higher real wages.
Why Productivity Growth Improves Living Standards
When labor productivity rises, workers can produce more goods and services in the same amount of time. Over the long run, this helps increase incomes and living standards.
Technological progress, better education, improved equipment, stronger management, and efficient production systems can all increase productivity.
This is why productivity growth is one of the main drivers of economic improvement.
The Black Death and Factor Prices
A Historical Example of Labor Scarcity
A historical example helps show how factor prices respond to changes in supply. During the Black Death in fourteenth-century Europe, the labor force fell sharply. With fewer workers available, the marginal product of labor increased.
As a result, real wages rose.
At the same time, land became less productive at the margin because there were fewer workers to farm it. This reduced the return to land. The example shows how changes in the supply of labor or capital can affect wages, rents, and income distribution.
Key Takeaways
The firm’s demand for labor and capital depends on marginal productivity. A firm hires workers until the marginal product of labor equals the real wage. It rents capital until the marginal product of capital equals the real rental price.
Diminishing marginal product explains why each additional worker or unit of capital eventually adds less output when the other factor is fixed.
National income is divided mainly between labor income and capital income. In competitive markets, each factor earns its marginal product.
The Cobb-Douglas production function helps explain how labor and capital contribute to output and why income shares can remain relatively stable over time.
Most importantly, labor productivity is a key determinant of real wages. When workers become more productive, real wages and living standards tend to rise over time.
Conclusion
National income is not created randomly. It comes from the productive use of labor, capital, and technology. Firms decide how much labor and capital to use by comparing marginal products with factor prices. Workers are paid according to the marginal product of labor, and capital owners receive returns based on the marginal product of capital.
The Cobb-Douglas production function provides a simple but powerful way to understand this process. It connects production, income distribution, productivity, wages, and long-run economic growth. For students of economics, finance, and business, understanding these ideas is essential for analyzing how economies grow and how income is distributed.
