# Substitute goods

This page is about substitution for goods for consumers. For substitution in the context of factors of production to a production process, see substitute factors of production.

## Contents

## Definition

### In terms of partial derivative of individual utility function

Two goods are said to be substitutes in terms of the utility function of an individual or household if the second-order mixed partial derivative of personal utility with respect to the quantities of the two goods consumed is negative. Note that under reasonable assumptions, this definition is equivalent to the other two definitions.

### In terms of effect on consumption

Two goods are said to be **substitute goods** if an increase in consumption of either one of them leads to a decrease in demand (i.e., a contraction in the demand curve) for the other.

### In terms of price and quantity demanded

Assuming the law of demand, the definition in terms of substitution is equivalent to the condition that:

- A decrease in the unit price of one of the goods should cause a
*decrease*in demand for the other, i.e., reduce the quantity demanded for the other good holding its unit price constant. - Equivalently, an increase in the unit price of one of the goods should cause an
*increase*in demand for the other, i.e., increase the quantity demanded for the other good holding its unit price constant.

## Algebra of substitution

### Reflexivity

We generally do not talk of goods as being substitutes for themselves. However, this does make conceptual sense when interpreted appropriately: a good is self-substituting if the second derivative of the utility function with respect to the quantity consumed is negative (i.e., diminishing returns from consumption). Most goods are self-substituting in this sense, at least in the consumption range at which people would finally be consuming the good.

### Symmetry

The general conceptual definition as well as its mathematical formulations imply that the relation of being substitute goods is *symmetric*, i.e., if *A* is a substitute good for *B*, then *B* is a substitute good for *A*. While it is possible to come up with contrived counterexamples to symmetry, the assumption of symmetry is practically harmless.

### Transitivity

Transitivity is the statement that if *A* and B* are substitutes and *B* and *C* are substitutes, then *A* and *C* are substitutes.*

The mathematical formulation of substitute goods does not say anything conclusive about transitivity. From a conceptual perspective, there is a strong reason to believe that the relation of being substitute goods is generally transitive, and more so if at least one of the substitution relations is close. One common reason why goods are substitutes is that they meet the same common need. In this case, the relation would be transitive (for instance, the case where *A*, *B*, and *C* are all different brands of wine).

The reason why transitivity might fail is that the "common need" that *A* and *B* fulfill might differ from the "common need" that *B* and *C* fulfill. For instance, *A* might be a *phone Internet access plan*, *B* might be a *home broadband Internet and cable TV connection*, *C* might be a *pure cable connection without Internet.* *A* and *B* fulfill a common need for Internet access, whereas *B* and *C* fulfill a common need for cable TV, but *A* and *C* do not have a common need that they fulfill.

## Combined analysis of substitute goods

If two goods are perfect substitutes, then the prices for the goods are likely to converge (with the appropriate unit conversion factors). In such cases, it is possible to do an analysis of them as a single good. Combined analysis may also serve as a first-order approximation for goods that are fairly close but not perfect substitutes.