Abstract
When two inputs are combined as a product of powers and the output price and input prices are fixed, one can ask what input level maximises profit. This paper finds the profit maximum by formula and by numerical search for four exponent pairs summing to $0.5$, $0.8$, $0.9$ and $0.95$, counts the shares of cost and profit in revenue and the sensitivity to price, and compares cases where the exponents sum to $1$ and $1.1$. No new theorem or law is claimed.
First, when the exponents sum to less than 1, one profit-maximising input level exists: for all four pairs, the maximum from the formula and from the numerical search agreed within a relative $10^{-6}$ (Section 2).
Second, then each input’s cost is exactly its exponent’s share of revenue, and profit is 1 minus the sum: with exponents $0.3$ and $0.6$, costs were $0.300000$ and $0.600000$ of revenue and profit $\mathbf{0.100000}$. With sums $0.5$, $0.8$ and $0.95$, profit was $0.500000$, $0.200000$ and $0.050000$ (Section 2).
Third, the closer the sum is to 1, the more sensitive the optimal scale is to price: raising price by $1$ per cent raised output by $1.000000$, $4.000000$, $9.000000$ and $\mathbf{19.000000}$ per cent for sums $0.5$, $0.8$, $0.9$ and $0.95$ (Section 3).
Fourth, at a sum of exactly 1, the choice splits into producing without limit or not at all: with exponents $0.5$ and $0.5$, scaling by $t$ gave profit of $+0.5$ per unit $t$ at price $2.5$ ($500.000000$ at $t=1000$) and $-0.5$ per unit $t$ at price $1.5$, with no maximum in between in either case (Section 4).
Fifth, the separator is whether scale has an upper limit: with the sum at $1.1$, scaling by $1$, $10$, $100$ and $1000$ gave profits of $0.500000$, $11.473135$, $196.223298$ and $\mathbf{2988.155787}$, still growing. A profit maximum exists only when widening scale stops the growth of earnings, that is, when the sum is below $1$ (Section 5).
Keywords: profit maximisation, product-of-powers production, returns to scale, cost shares, price sensitivity, production function.
Scope (what is not claimed)
No new theorem or law is claimed: the product-of-powers production function, the conditions for maximising profit at fixed prices, that cost shares equal the exponents, and that there is no profit maximum when the exponents sum to $1$ or more are all known in economics. No real economy is claimed: the exponent pairs and prices are set values for comparison and correspond to no particular firm or industry. Prices are held fixed: selling and input prices are assumed not to change with the quantity produced. Market equilibrium, monopoly and taxes are not treated. Time is not included: the time to add capacity, borrowing and risk are not treated. Relation to earlier papers: Paper 934, in its scope note, did not treat prices and costs and did not treat how much to invest or the quantity that maximises profit; this paper sits in that seat (no earlier occupant). What is added: checking the formula maximum against numerical search for four exponent pairs, finding the shares of cost and profit in revenue from the exponents, finding the sensitivity to price for each sum, showing by varying scale that sums of $1$ and $1.1$ have no maximum, and placing the separator on “whether scale has an upper limit”.
1. Notation and the Question
Setting: two inputs $x$ and $y$ produce $Q=x^{\alpha}y^{\beta}$, sold at price $p$. With input prices $r$ and $w$, profit is:
$$\pi=pQ-rx-wy,\qquad s=\alpha+\beta$$For $s<1$, output and inputs at the profit maximum are
$$Q^*=\left[\left(\frac{p\alpha}{r}\right)^{\alpha}\left(\frac{p\beta}{w}\right)^{\beta}\right]^{1/(1-s)},\qquad x^*=\frac{p\alpha Q^*}{r},\qquad y^*=\frac{p\beta Q^*}{w}$$The check sets $p=r=w=1$ and also finds the maximum by a search that raises profit on $\log x$ and $\log y$ while halving the step.
Question: when does a profit-maximising input level exist, how is revenue divided there, and how sensitive is the optimal scale to price?
2. Profit Maximum and the Division of Revenue
Computation 2.1: profit maximum at $p=r=w=1$.
| exponents $\alpha$, $\beta$ | sum $s$ | output $Q^*$ | profit (formula) | profit (search) | profit/revenue |
|---|---|---|---|---|---|
| $0.3$, $0.2$ | $0.5$ | $0.255085$ | $0.127542$ | $0.127542$ | $0.500000$ |
| $0.4$, $0.4$ | $0.8$ | $0.025600$ | $0.005120$ | $0.005120$ | $0.200000$ |
| $0.3$, $0.6$ | $0.9$ | $1.259712\times10^{-3}$ | $0.000126$ | $0.000126$ | $\mathbf{0.100000}$ |
| $0.45$, $0.5$ | $0.95$ | $7.389459\times10^{-7}$ | — | — | $0.050000$ |
For all four pairs the search profit agreed with the formula within a relative $10^{-6}$ (for the sum $0.95$ the values are small, so only the ratio is shown). Cost/revenue was $0.300000$, $0.400000$, $0.300000$ and $0.450000$ for $x$ and $0.200000$, $0.400000$, $0.600000$ and $0.500000$ for $y$, equal to the exponents.
Observation 2.2 at the maximum, revenue is paid to the inputs in the shares of the exponents, and the remaining 1 minus the sum is profit. With exponents $0.3$ and $0.6$, $0.300000$ of revenue goes to $x$, $0.600000$ to $y$, and profit is $0.100000$: the closer the sum is to $1$, the smaller the profit share even at the maximum; at a sum of $0.95$ it was $0.050000$.
3. Sensitivity to Price
Computation 3.1: change in output (per cent) when price rises from $p=1$ to $1.01$, and $s/(1-s)$.
| sum of exponents | change in output | $s/(1-s)$ |
|---|---|---|
| $0.5$ | $1.000000$ | $1.000000$ |
| $0.8$ | $4.000000$ | $4.000000$ |
| $0.9$ | $9.000000$ | $9.000000$ |
| $0.95$ | $\mathbf{19.000000}$ | $19.000000$ |
Observation 3.2 the closer the sum is to 1, the more sensitive the optimal scale is to price. A $1$ per cent rise in price raises output by $1$ per cent at a sum of $0.5$ but by $19$ per cent at $0.95$: the change in output matched $s/(1-s)$, which grows without bound as the sum approaches $1$.
4. Sums of Exactly 1 and Above 1
Computation 4.1: profit with $x=y=t$ at scale $t=1$, $10$, $100$ and $1000$.
| exponents, price | $t=1$ | $t=10$ | $t=100$ | $t=1000$ |
|---|---|---|---|---|
| $0.5$, $0.5$, price $2.5$ | $0.500000$ | $5.000000$ | $50.000000$ | $500.000000$ |
| $0.5$, $0.5$, price $1.5$ | $-0.500000$ | $-5.000000$ | $-50.000000$ | $-500.000000$ |
| $0.5$, $0.6$, price $2.5$ | $0.500000$ | $11.473135$ | $196.223298$ | $\mathbf{2988.155787}$ |
Observation 4.2 at a sum of 1, profit is proportional to scale and there is no maximum in between. At price $2.5$ profit is $+0.5$ per unit $t$ and grows as scale widens; at price $1.5$ it is $-0.5$ per unit $t$ and producing nothing is best: at a sum of $1.1$, profit per unit $t$ also kept growing with scale.
5. The Separator
Observation 5.1 the separator is whether scale has an upper limit. With the sum below $1$, the growth of earnings stops as scale widens and a single profit maximum exists: with the sum at $1$ the growth is constant, and depending on price the answer is “without limit” or “not at all”; above $1$ the growth accelerates and there is no maximum. The difference in the sum of exponents that Paper 934 counted appeared here as the presence or absence of a profit maximum.
Observation 5.2 what is not said. Real firms, prices that change with quantity, market equilibrium, time and risk are not treated. What is shown is that with product-of-powers production at fixed prices, whether a profit maximum exists, how revenue is divided and the sensitivity to price are set by the sum of exponents.
6. Roll Call and Closing
- product-of-powers production and profit (Section 1): sum of exponents $s$.
- formula and search maxima agree (Computation 2.1): a maximum exists for $s<1$.
- profit/revenue $\mathbf{0.100000}$ (sum $0.9$) (Computation 2.1): costs in exponent shares.
- output up $\mathbf{19.000000}$ per cent for a $1$ per cent price rise (sum $0.95$) (Computation 3.1): $s/(1-s)$.
- $\pm0.5$ per unit $t$ at sum $1$, $\mathbf{2988.155787}$ at $1.1$ (Computation 4.1): no maximum.
- whether scale has an upper limit (Observation 5.1): the separator; the scope note of Paper 934.
Closing: A profit-maximising input level exists only when the exponents sum to less than 1, and profit is 1 minus that sum of revenue. For four pairs summing to $0.5$, $0.8$, $0.9$ and $0.95$, the formula and search maxima agreed, costs took the exponents’ shares of revenue, and profit was $0.500000$, $0.200000$, $0.100000$ and $0.050000$: raising price by $1$ per cent raised output by $1$, $4$, $9$ and $19$ per cent; at a sum of $1$ profit was proportional to scale with no maximum, and at $1.1$ it kept growing. The separator is whether scale has an upper limit. Placed among the earlier papers: the sum of exponents that Paper 934 counted on the production side alone was counted, with prices and costs added, as the presence or absence of a profit maximum. To be honest: the practice of deciding how much to produce is not claimed. In a model with fixed prices, how the sum of exponents sets where the profit maximum lies and its sensitivity, and no further.
References
- Cobb, C. W., & Douglas, P. H. (1928). A theory of production. The American Economic Review.
- Varian, H. R. (1992). Microeconomic Analysis (3rd ed.). W. W. Norton.
- Yamagishi, Y. (2026). Diminishing Returns Answers What Happens When One Input Grows; Scale All Inputs Together and Output Can More Than Double [Paper 934]. YOI: https://www.yamagishiyuuki.com/yoi/934
Reference
Yamagishi, Y. (2026). A Profit-Maximising Input Level Exists Only When the Exponents Sum to Less Than 1, and Profit Is 1 Minus That Sum of Revenue [Paper 948]. YOI: https://www.yamagishiyuuki.com/yoi/948
ORCID: 0009-0006-8587-3450
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The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 948. This paper is a preprint and has not been peer reviewed.