Why Bigger Packages Lower Per‑Gram Costs: A Simple Economic Model

Consider products such as flour, detergent, or jam that are sold in various package sizes. Larger packages often have a lower price per gram, which many attribute to savings on packaging and operating costs. This article asks whether that is the main reason and proposes a simple model to analyze other factors.

The study focuses on how product cost changes with package size, examining wholesale prices (the price retailers pay) which include production cost, packaging cost, transportation cost, and material cost. Ignoring competition and scale effects, the model assumes product cost is proportional to the quantity produced, denoted by a constant multiplied by product quality.

Packaging cost depends on the time required for packing, sealing, and preparing for shipment, which is roughly proportional to volume (and thus to quality). For a given range of volumes, the latter two time components are similar, leading to constants k1 and k2 in the cost expression.

Transportation cost may depend on both weight and volume; because volume is proportional to the weight of a filled package, the model simplifies this relationship. Material cost of packaging is more complex, depending on the costs incurred by the packaging producer. Assuming the container cost is negligible, each package’s material cost depends on its weight and volume. If the range of volumes is small, the material used per package can be considered constant, making material cost proportional to the surface area of the package.

Assuming packages are geometrically similar, volume scales with the cube of a linear dimension and surface area with the square. Therefore, surface area is proportional to the two‑thirds power of volume, and material cost per package is proportional to the two‑thirds power of the product’s mass.

Combining these relationships, the wholesale cost per gram is expressed as a decreasing function of package mass. As the package gets larger (i.e., the mass of product per package increases), the cost per gram falls.

The rate of cost reduction (the derivative of cost with respect to mass) is itself a decreasing function, meaning that while larger packages continue to save money per gram, the incremental savings diminish for very large packages. Consequently, the total savings rate also decreases.

This model can be extended to retail pricing, where retail cost depends on wholesale price, sales cost, and warehousing cost, which have similar functional forms. Thus, the conclusions also apply to retail prices.

For different items, real data can be used to validate and refine the model, improving its precision.

Shen Wenxuan, Yang Qingtiao. "Mathematical Modeling Attempts"