A Nonconvex Approach to Cost–Volume–Profit Analysis with Variable Price

Textbook break-even analysis asks one firm, at a fixed price, how much it must sell to stop losing money. But when several producers supply the same market, the price is not handed to them — it emerges from what they jointly produce. Ask the break-even question of all of them at once and the answer stops being a point. It becomes a region, and the region need not be convex.

Abstract

Classical cost–volume–profit (CVP) analysis usually provides point-wise break-even or target-profit calculations under fixed-price assumptions. This paper developed a nonconvex feasible–profit framework for multi-producer CVP analysis in an oligopolistic market where the common price is endogenously determined by the joint production vector. The framework was applied to a three-country copper export market consisting of Mongolia, Chile, and Peru, using annual data for 2013–2025. For three reduced-form price environments, the price coefficient was estimated by least squares, feasible–profit membership was tested, and a penalty method was used to compute local stationary points. The results showed that feasible–profit membership depends strongly on the assumed price mechanism.

Everything below is computed live. The least-squares price coefficients, the feasibility tests, the region boundaries and the sensitivity sweep all run in your browser from the paper's input table. Move a control and every figure re-solves.

The market

Three countries export copper to China. Each supplies x_j million tonnes, pays a variable cost c per tonne and a fixed cost F, and receives the same reference price. Profit is the usual CVP expression — with one change that spoils everything:

profit Π_j(x) = p(x)·x_j − c·x_j − F

The price p(x) depends on everyone's output. That single substitution turns a linear break-even condition into a nonlinear one, and the set of jointly profitable allocations — the feasible–profit region Δ — into something that need not be convex.

Input data. Volumes in million tonnes, price and costs in USD per tonne.
Year	Mongolia	Chile	Peru	Q	p	c	F

Costs are calibrated rather than observed: c₂₀₁₃ = 2000 USD/t, scaled each year with the copper price, and F = 0.38c. They are applied identically to all three countries, which is a limitation the paper is explicit about — but it also makes the structure unusually clear, as the next section shows.

One threshold, three producers

Because c and F are common, the profitability condition rearranges into something very simple. Every producer faces the same minimum scale:

the binding condition Π_j(x) ≥ 0 ⇔ x_j ≥ F p(x) − c

So joint profitability reduces to a race between the smallest exporter and a moving bar. Mongolia ships between 0.57 and 2.07 million tonnes against Chile's 2.8–9.5, so Mongolia is always the one that binds. Every infeasible year in the paper is a year Mongolia fell below the bar — never Chile, never Peru.

Figure 1 · Mongolia's exports against the break-even threshold

Where a threshold line rises above Mongolia's export line, that price environment declares the observed year infeasible. Case 2 sits far below throughout — its inverse price is high when output is low, so the bar is never demanding. Cases 1 and 3 bite in the early years, when total market volume, and therefore the modelled price, was small.

The region itself

Fixing Peru at its observed volume, the figure below sweeps Mongolia and Chile across the plane and shades every pair where all three producers clear the bar and total exports stay within China's demand. This is the feasible–profit region in cross-section.

Figure 2 · Feasible–profit region, Peru held at its observed volume

Year 2013

Cost scale 100%

Price mechanism

all three profitable observed exports capacity limit Σx ≤ Q

Three price mechanisms

The paper does not claim to know how the copper price forms. Instead it tests three reduced-form environments and asks how much the verdict depends on that choice. Each coefficient a is fitted by least squares to the 2013–2025 sample.

Price environments and fitted coefficients — recomputed in-browser
Case	p(x)	â (live)	â (published)	Observed feasible

The verdict moves a lot. Under the inverse specification the observed export vector is inside the region in every single year; under the pairwise specification it sits outside for the first three. Same data, same profitability inequality — the only difference is the assumed link between output and price.

Figure 3 · Feasibility by year and price environment

observed vector inside Δ outside Δ penalty stationary point outside Δ

Filled cells are the observed export vector; the outlined cells in Case 2 for 2024–2025 mark years where the observed vector is feasible but the penalty method's stationary point is not. Feasibility of the data and feasibility of the computed point are separate questions.

How fragile is the verdict?

The 2025 observation is re-tested while each parameter is moved by ±5%, ±10% and ±20% in turn. Drag through the perturbations and watch which environments hold.

Figure 4 · 2025 sensitivity sweep

Parameter perturbed

Bars show the modelled 2025 price at each perturbation; a red marker means the observed 2025 export vector leaves the feasible–profit region. Only the price coefficient under the inverse specification breaks it, and only at −20%.

What it means

The headline result is negative, and usefully so: whether a country's exports look profitable depends on a modelling choice that is rarely made explicit. Under one price mechanism Mongolia was losing money in 2013; under another it never was. Neither specification is a structural demand system, and the paper is careful to say so.

What survives that ambiguity is the geometry. Once price responds to joint output, break-even stops being a point and becomes a region with curved, possibly nonconvex boundaries — and a small producer sitting near the boundary can be pushed across it by decisions it does not control. That is a different kind of risk from the one classical CVP analysis is built to measure.