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Euler’s Theorem in Economics
1. Introduction
Euler’s Theorem is a mathematical concept that plays a key role in economic theory, especially in the study of production functions and income distribution. It helps analyze whether total output is completely exhausted when factors of production are paid their marginal products.
In economics, Euler’s theorem is used to solve the Adding-Up Problem—i.e., whether wages, rents, and profits together fully account for total output.
2. Statement of Euler’s Theorem
Euler’s theorem states that:
👉 If a production function is homogeneous of degree nn, then: Y=f(L,K)=L⋅∂f∂L+K⋅∂f∂KY = f(L, K) = L \cdot \frac{\partial f}{\partial L} + K \cdot \frac{\partial f}{\partial K}
where:
- Y=f(L,K)Y = f(L, K) is the total output (GDP).
- LL and KK are labor and capital inputs.
- ∂f∂L\frac{\partial f}{\partial L} is the marginal product of labor (MPL).
- ∂f∂K\frac{\partial f}{\partial K} is the marginal product of capital (MPK).
This means that if each factor is paid according to its marginal product, total output will be exactly distributed—with no surplus or deficit.
📌 Key Special Case:
- If the production function is homogeneous of degree 1 (i.e., constant returns to scale), then:
f(L,K)=L⋅MPL+K⋅MPK=Yf(L, K) = L \cdot MPL + K \cdot MPK = Y
This confirms that all of the output is fully distributed as wages and capital earnings.
3. Application in Economics
🔹 (1) Euler’s Theorem and the Adding-Up Problem
One of the biggest questions in income distribution is:
🔹 Does total output fully distribute to factors of production?
✔ If firms pay workers their Marginal Product of Labor (MPL) and pay capital its Marginal Product of Capital (MPK), then total income should exactly equal total output.
✔ This is why the marginal productivity theory of income distribution is based on Euler’s theorem.
🔹 (2) Proof of Euler’s Theorem for a Cobb-Douglas Production Function
A commonly used production function is Cobb-Douglas: Y=ALαKβY = A L^\alpha K^\beta
where:
- AA = Technology constant
- α\alpha = Output elasticity of labor
- β\beta = Output elasticity of capital
✔ If α+β=1\alpha + \beta = 1 (constant returns to scale), then:
Marginal product of labor (MPL): MPL=∂Y∂L=αALα−1KβMPL = \frac{\partial Y}{\partial L} = \alpha A L^{\alpha – 1} K^\beta
Marginal product of capital (MPK): MPK=∂Y∂K=βALαKβ−1MPK = \frac{\partial Y}{\partial K} = \beta A L^\alpha K^{\beta – 1}
Multiplying MPL by LL and MPK by KK: L⋅MPL+K⋅MPK=αY+βY=(α+β)YL \cdot MPL + K \cdot MPK = \alpha Y + \beta Y = (\alpha + \beta) Y
✔ If α+β=1\alpha + \beta = 1, then Y=L⋅MPL+K⋅MPKY = L \cdot MPL + K \cdot MPK, meaning all output is exactly distributed.
✔ If α+β>1\alpha + \beta > 1, firms cannot pay all factors without exceeding output.
✔ If α+β<1\alpha + \beta < 1, total output is not fully distributed, leaving a residual (e.g., monopoly profits).
🔹 (3) Euler’s Theorem in Returns to Scale
🔹 Constant Returns to Scale (CRS): f(L,K)=L⋅MPL+K⋅MPKf(L, K) = L \cdot MPL + K \cdot MPK
✔ Wages and capital earnings exhaust total output.
🔹 Increasing Returns to Scale (IRS): f(L,K)>L⋅MPL+K⋅MPKf(L, K) > L \cdot MPL + K \cdot MPK
✔ Output increases faster than inputs, creating economic profits.
🔹 Decreasing Returns to Scale (DRS): f(L,K)<L⋅MPL+K⋅MPKf(L, K) < L \cdot MPL + K \cdot MPK
✔ Some output remains undistributed, implying under-utilization of resources.
4. Importance of Euler’s Theorem in Economic Theory
✔ Explains Factor Pricing – Ensures that factors are paid according to their marginal productivity.
✔ Validates Competitive Market Theories – Perfect competition ensures full distribution of output.
✔ Influences Tax and Wage Policies – Governments use this principle to justify wage policies and capital taxation.
✔ Helps in Growth Models – Used in Solow’s Growth Model to study capital accumulation.
5. Criticism of Euler’s Theorem in Economics
❌ Real-World Markets are Imperfect – Monopoly power, labor unions, and minimum wage laws distort income distribution.
❌ Income Inequality is Ignored – Even if total income is distributed, wage gaps and wealth inequality remain.
❌ Does Not Account for Profits in Non-Competitive Markets – Monopoly and oligopoly firms earn excess profits, violating Euler’s theorem.
❌ Government Intervention Affects Distribution – Taxes, subsidies, and public services modify factor earnings.
📌 Example:
- Big Tech firms (Amazon, Google, Apple) generate higher profits than their capital share predicts, due to intellectual property and market power.
- Gig economy workers (Uber, DoorDash) often receive less than their MPL, violating the assumption of competitive wages.
6. Conclusion
✔ Euler’s Theorem proves that under constant returns to scale, total income is exactly distributed among labor and capital.
✔ It is used in marginal productivity theory, factor pricing models, and growth economics.
✔ Real-world deviations occur due to market power, government policies, and wage distortions.