17.2.1 Definitions and Formulas

Demand Function

Expresses consumer demand for a product in terms of its unit price p and the number of units x that consumers will buy at price p. The demand function D(x) creates a deceasing (downhill) curve because fewer products will be sold if p is larger; conversely, smaller prices will create more demand.

Consumers’ Surplus

Let p be a fixed established price for a commodity and x be the number of units bought at p. The consumers’ surplus is the difference between what consumers would be willing to pay for a commodity and what they actually pay for it. The formula for consumers’ surplus is: CS = ∫ ₀^xD(x) dx -px, where D(x) is the demand function.

Supply Function

Expresses producers willingness to supply x units of a commodity at price p. The supply function S(x) creates an increasing (uphill) curve because producers are willing to put more of the commodity on the market if the unit price is higher.

Producer’s Surplus

Let p be a fixed established price for a commodity and x be the number of units that producers are willing to supply at price p. The producers’ surplus is the difference between what the suppliers actually receive x and what they would be willing to receive at price p. The formula for producers’ surplus is: PS = px -∫ ₀^xS(x) dx where S(x) is the supply function.

Market Equilibrium, equilibrium quantity, equilibrium price

The point (x,p) where the demand curve and the supply curve intersect, i.e. the point at which market equilibrium occurs. This is the highest price consumers are willing to pay for what producers are willing to supply. x is called the equilibrium quantity and p is called the equilibrium price.

Income Stream

Created when a business generates a stream of income R(t), where t is in years, over a period of time T years and this income is invested at an annual rate r compounded continuously. R(t) could be a constant stream or a variable stream but it is invested, nonetheless, on a continuing basis over the T years.

Future Value of an Income Stream

The total amount of money that will be accumulated when an income stream R(t) has been invested at an annual rate r compounded continuously for T years. The formula for the future value of an income stream R(t) is: FV = e^rt ∫ ₀^T R(t)e^-rt dt.

Present Value of an Income Stream

The principal P that would have to be invested at an annual rate r compounded continuously over T years in order to equal the accumulated value of an income stream over the same period T and the same rate r. The formula for the future value of an income stream R(t) is: PV = ∫ ₀^T R(t)e^-rt dt.