Crops of all types are typically produced for profit by a producer. Profit can be defined as the value of products produced minus the value of inputs used in production:

n = py ■ qy — px ■ qx (15.1)

where n = profit, py = price of product, qy = quantity of product, px = price of input, and qx = quantity of input.

Producer behavior is often represented as profit maximization in economic models. The amount of profit that producers can make is limited by several constraints and exogenous variables. Constraints include technology and quantities of resources or inputs available to the producer. Exogenous variables include prices for inputs and outputs on which the producer’s decisions have no significant market effect. Exogenous variables also include inputs such as temperature, sunlight, and rainfall that have no price, are beyond the control of the producer, and can have large effects on production and profit. Control variables (those affected by the producer’s decisions) include which type and variety of crop to produce and what quantities of variable inputs to use (area of land, machinery use, seed, fertilizers, crop protection and other chemicals, irrigation water, labor, management, insurance, other risk management aids, fuel and lubricants, other supplies, custom services, and others). Control variables also include intermediate and long-term variables, such as the type and quantity of land to own or rent, the type and capacity of machinery to own or lease, the type and amount of debt to incur, the type of human capital to acquire and maintain, and others. Included among control variables, too, are management decisions such as planting and harvesting dates that do not have specific prices attached but can have large effects on production and profit.

The Profit Maximizing Crop Producer’s Objective Function is:

Y X

Maximize: n = ^(py ■ qy) — ^(px ■ qx) (15.2)

y=1 x=1

Subject to: qy < f (qx) the technology constraint

qx < q*x constraints on the availability of some inputs py = p*y, px = p*x prices determined exogenously

where n, p, and q are profit, price, and quantity as in Equation 15.1, is the summation over all products (y) or over all inputs (x), f(qx) is a function of quantities of inputs (x) used that defines the maximum amount of product (y) that can be produced with current technology, and q* and p* are fixed quantities and prices that are exogenous to the producer.

The technology constraint embodies the biological relationships that define crop growth and product yield. Genetic stock of the plants, via seed, determines the maximum growth and yield possible. Most plants and animals are limited in their growth and yield by insufficient supply of one or more of the inputs or conditions for optimal growth. Much of modern agriculture and silviculture is dedicated to identifying those deficiencies and correcting them. Examples include various fertilizers, irrigation water and drainage, chemicals for plant protection from disease and weeds and pests, mechanical treatment of soils, and temperature and light control in greenhouses. Another important part of modern agriculture and silviculture is dedicated to development of genetically superior plants and animals. Sustainability in crop production is inseparable from production decisions and is discussed in a subsequent section.