Chapter Economic Models As mentioned before, any economi.: theory is necessarily an abstraction from the will world. For one thing, the immense complexity of the real economy makes it imposiblc for us to understand all the imerrelationships at once; nor, for that matter, are all these intern:- lationships of equal importance lor the understunding afthe particular economic phenom- enon under study. The sensible procedure is, therefore, to pick out what appeals to OUT reason to be the primary factors and relationships rdavant to our problem and to focus our attention on these alone. Such a deliberately simplified analytical framework is called an economic nlndel. since it is only a skeletal and rough rcprcscnMion of the actual economy. 2.1 Ingredients of a Mathematical Model An economic model is merely a theoretical framework, and there is no inherent reason why it must be mathematical. If the model is mathematical, however, it will usually consist ()fa sct of equations designed to describe the structure of the model. By relating a number uf Iwiables to one another in certain “ays, these equations give mathematical form to the sci of analytical assumptions adopted. Theo, through application of the relevant mathematical operations to these equations, we may seck t) derive a set of conclusions which logicl111y foli(lw from those assumptions. Variables, Constants, and Parameters A variable is something whose magnitude can cbange, i.e., something that can take 011 dif ferent values. Variables frequently used in economics include pric(:, profit, revenue, cost, national income, consumption, investment, imports, and exports. Since each variable can assume various values, it must be represented by a symbol inRtcad of i1 specific number. For example, we may represent price by P, profit by IT, revenue by R, cost by C, national in- come by Y, and so forth. When we vrite P :; 3 or C = 18, h()wcvcr we are “freezing“ these variables at specific values (in appropriately chosen units). Properly constructed, an economic model can be solved to give us tht: so/Illion values of a (:crrain set M variables, such as the market-clearing level (If price, Of the profit ma,(imizing level of output, Such variables, whose solution values we seck from the model, are known as endogenous variahfjoint sets. A fourth type ofn:latiollship occurs when two sets have some elements in common but ome elemenh peculiar to each, In that event, the two sets are neither equal nor dioint; also, neither set is a subset of the other. Operations on Sets When we add, subtract, multiply, divide, or tuke the square fQot of some numbers, we are performing mathematical operations. Although 5et5 are different from numbers. one can similarly perform certain mathematical operations on them. Three principal operations to be discussed here involve the union, imcrsecthm, and complement of sets. To take the union oftwo sets A and B means to foon a new set containing those elements (and only those elements) belonging to A, or tl 8, or to both A and H. The union set is sym- bolized by A U B (read: “;1 union B“). If A (3,5,71 and B (2, 3,4,8), thee AI., B = 2,3,4,5,7,81 This example, incidentally, illustrates the l:ac in w“hich two sets.4 and H ,Ire neither equal nor disjoint and in which neither is a subset nrthc other. Again referring to Fig. 2.1, we see that the union of the set of all integers and the set of all fractions is the set of all rational numbers. Similarly, the union of the rational-number set and the irrational-number set yields the set of all real numbers. The intersectiun of two sets A and B, on the othcr hand, is a new set which contains those elements (and only those elements) belonging to hothA and B. The intersection set is sy1l1 boli7ed by A n R (read: “A intersection B“). From the sets A and B in Example 1, we can write An8lll If A “ (-3,6,10) and B = 9,2, 7,4), then An B “ 0. SetA and set Bare disjoint; there- fore their intersection is the empty setno element is common to A and B. It s obviQUS that intersection i:; a mOTe restrictiv concept than union. n the fonner, only the elements commfJn to A and B arc acceptable, whcreas in the latter, membership in either A or B is sufficient to establish membership in the union set. The operator symbols n and Uwhich, inCIdentally, have the saDiC kind of general statuti as the symbols .,1, +, 7, etC.-therefore have the connotations “and“ and “or,“ n:spcctivdy, This point can be better appreciated by comparing the following formal definitions of intcrection and 1f1 ion: Union: An B:; (x I x E A and x E 13) AU B = Ix I x E A or x E B 2 Part One llJln;ilm:lion FIGURE 2.2 Example 5 Example 6 Unl0Jl AUB A A 8 (Iii Whrtt about the complement of a set? To explain this, let us first introdut:t: the concept of the universal stf. In a particular context of JiScllssion, if the only numbCrf:! llsed are the set of the first seven posit