Quantities Magnitudes and Relationships

We can slart by thinking of how we measure things in economics. Numbers represent quantities and ultimately i( is this circumstance that makes it possible to use mathematics as an instrument for economic modeling. When we discuss market activity, for example, we are concerned with the quantity traded and the price at which (he trade occurs. This is so whether the "quantity" is automobiles, bread, haircuts, shares, or treasury hills. These items possess cardinality, which means lhat we can place a definite number on the quantity we observe. Cardinality is absolute but is not always nc-cessary for comparisons. Orilinality is also a property of numbers but refers only to the ordering of items. The difference between these two number concepts may be illustrated by the following two statements.

1, Last year, the economy 's growth rate was 3%.

2. The economy's output last year was greater than the year before.

Both of these statements convey quantitative information. The first of these is a cardinal property of the change in output. We are able to measure the change and put a definite value on it. The second is an ordinal statement about economic activity in die past year. Last year's output is higher than the year before. This of course is an implication of the first statement, but the first statement cannot be inferred from the second statement.

However, there is a greater difference between cardinality and ordinaJity, because we can also decide on a ranking of items based on their qualitative properties rather than on their quantifiable ones. Most statements about preferences are ordinal in (his sense, so to say: "I prefer brand A to brand B, and brand B to brand C" is an ordinal statement about how one person subjectively evaluates three brands of a good. If we let larger numbers denote more preferred brands, then we could associate brand A with the number 3, brand B with the number 2, and brand C with the number 1. However, the numbers 10, 8, and 0 would serve equally well in the absence of any other information. This statement may provide useful information, and certain logical and mathematical consequences may follow from it, but it is not a statement about quantities.

Variables and Parameters

When we start to build an economic model, we know thai we are not going to be able to explain everything. Some things must be treated as given or as data for our problem. These are the exogenous variables and the parameters of the model. The endogenous variables, then, are ihose that are going to be explained by the model, A simple example will illustrate.

Suppose dial we are irying to determine the equilibrium price and quantity in a market for a homogeneous good. We hypothesize that the quantity demanded of some good may be represented as qn — a — bp + cy

which is a simple linear demand function. Each time price, p. increases by one dollar, the quantity demanded. ql), falls by b times one dollar. A rise in income, y, of one dollar increases quantity demanded by < units. This demand curve may be chosen purely for simplicity, or if nuy be known, by looking at market data, thai the demand curve for this good does lake this simple form Now suppose that the supply of this good is tixed at some amount which we will call qs, and suppose that we believe that the prevailing price in this market is the price that equates demand with supply. Then

So here, p is the endogenous variable, a and h are the exogenously given parameters, and f/-v and v are exogenous variables. For instance, demand is determined by tastes, weather, and many other environmental and social factors, all of which are captured here by a. b. and c. All supply-side considerations are, in this particularly simple case, summarized by the quantity qs. Parameters may also incorporate the effects of exogenous variables, which we do not wish to specify explicitly. For example, a may incorporate the effects of prices of others goods on the demand for this one. Finally, in this example, there is one further endogenous variable. Since quantity demanded, q°. depends partly on /> (which is endogenous) it too is endogenous: therefore q° is only known when the price is known. Substituting equation (1.2) into equation (1.1) gives simply qD = qs as the value of demand.

In general, as in this simple example, we can use relationships between economic variables and background parameters to reach conclusions or predictions based upon the mathematical solutions of those relationships.

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