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Assumption: Tax Revenues Are an Increasing Function of Aggregate Income • E.G

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Assumption: Tax Revenues Are an Increasing Function of Aggregate Income • E.G Chapter 1 Mathematics as a Language Mathematics is a deductive science{ie conclusions are drawn from what have been assumed. While doing this, mathematics uses symbols which represent di®erent things in di®erent contexts. But symbols have to be put together or operated consistently in order to make meaningful statements about some given premises. Therefore, mathematics has its own rules to convey information and meaning. In this class we will learn about these rules as well as logical processes to make meaningful statements in the context of economics. Mathe- matics greatly helps with the process of building theories. 1.1 Building Theories What is a theory? It is a thing with the following three parts: 1. de¯nitions 2. assumptions (have to be consistent with one another) 3. hypotheses (\if ... then " statements; that is, predictions). These follow from the assumptions. For example one might have a theory to explain the e®ects of ¯scal policy. The following is not a theory, just examples of de¯nitions, assumptions, and predictions. ² e.g. of a de¯nition: aggregate income is the $ value of everything produced in the year ² e.g. of anwww.allonlinefree.com assumption: tax revenues are an increasing function of aggregate income ² e.g. of a prediction: if government expenditures, G, increase, aggregate income, Y, will increase. A theory must have all three parts and the hypotheses must follow logically (can be deduced) from the de¯nitons and assumptions. The above example de¯nition, assumption and hypothesis is not a theory, the hypothesis does not follow from the de¯nition and one assumption. 1 Consider the following example of a theory. De¯nitions of men, cry and Rambo. Assume all men cry. Also assume Rambo is a man. What logically follows? Rambo cries. This is a theory. Note that predictions do not follow from one and only one assumption, it is just a restatement of that assumption. For example, the statement \it will rain tomorrow" is not a theory. Consider the Rambo example again. de¯ne Rambo, men, cry, etc. Assumptions: (1) All men cry (2) Rambo is a man prediction: Rambo cries This is a theory{note that the prediction required more than just one of the assumptions. Consider the alternative theory. Assumption: Rambo cries. Prediction: Rambo cries. Is this a theory? No, because it is just a restatement of the assumption. Each prediction must require more than one assumption. Is the following statement a hypothesis (prediction in scienti¯c sense of the word)? \The end of the world is coming" Consider now the process of deriving predictions from the assumptions and de¯nitions. The process of deduction is often di±cult, but can be made eas- ier by describing the assumptions mathematically. In which case, the body of mathematical logic can be applied to the problem. 1. mathematical symbolism is precise 2. things are expressed neatly and compactly, so it's easier not to get confused 3. there are all these math theorems out there to help us mathematically deduce the predictions E.g. of an assumption in words: the aggregate level of consumption increases as the level of aggregate income increases. To mathematically express this as- sumption we will use the symbol C to represent aggregate consumption; Y to represent aggregate income. So mathematically we will write our assumption as dC C = f(Y ); = f 0(Y ) > 0 dY 1.2 Necessary and Su±cient Conditions An understanding of these two terms is a necessary condition for understanding economic modelswww.allonlinefree.com and theories. Consider two things denoted by A and B. A is necessary for B if the existence of B requires the existence of A. Said in symbols: (A is necessary for B) , (notA ) notB) , (B ) A) , (B is su±cient for A). What does ) mean? It means implies. , means the implication goes both directions. If A , B, A and B are equivalent. Said another way, (A , B) , A iff B. A is su±cient for B if the existence of B requires the existence of A. Said in symbols: (A is su±cient for B) , (notB ) notA) , (A ) B) , (B is necessary for A). 2 1.3 Set Theory A \set" is a collection of things. We use uppercase letters to denote a set. For example, S ´ fthe set of all dogs who live on the Yeditepe campusg: Some symbols are reserved for special sets: Â denotes the empty set (a set with no members), and ­ denotes the universal set (the set that includes everything). Now consider the following set that might be important to a ¯rm that uses only two inputs, l and k, to produce its output: A ´ f(l; k): wl + rk · m; l ¸ 0; k ¸ 0g where l is the quantity of labor, w, the wage rate, is its price, k is the quantity of capital, r is the rental price of capital, and m is some amount of money. De¯ne this set in words \all those bundles of labor and capital that the ¯rm can purchase for $m or less". Graph the set. What do we call this set? To picture 1 this set, solve wl + rk = m. Solution is: k = r (m ¡ lw) if r 6= 0. Grahp this set 1 for values m = 100; w = 25, and r = 40; k = 40 (100 ¡ l25) Now, we are going to look at a more general input requirement set. Assume a ¯rm produces product x using the inputs l and k. Further assume the production function x = f(k; l) which identi¯es the maximum number of units of output that can be produced using k units of capital and l units of labor. Consider the following sets I(x) = f(l; k): f(k; l) ¸ x; l; k ¸ 0g Contrast the above set with Ix(x) = f(l; k): f(k; l) = x; l; k ¸ 0g This set is the isoquant for output level x. De¯ne the isoquant in words. For example, if x = f(k; l) = kl:5 There are operations with sets such as union and intersection but I assume the reader is already familiar with all these. 1.4 Functions What is a function?www.allonlinefree.com What does it mean to say that y = f(x), where x and y are variables. A function associated with each value of the variable x a unique value of the variable y. We will mostly stick with the variables x and y but keep in mind that nothing is special about those letters. We could write a = h(b). To de¯ne a function we need three things. (1) The domain of the function, say X; (2) the range of the function, say Y ; (3) a rule that associates with each member of X a unique member of Y , say f. Hence, y = f(x) is a mapping from X to Y , and the value that y takes for a given x is called the image of x 3 shown by f(x). If the mapping is not a unique mapping, it is not a function. For example, y = f(x) = x2 is a function. In contrast, ½ 3 if x · 5 y = f(x) = is not a function 7 if x ¸ 5 All functions are relationships, but all relationships are not functions. Being a function is su±cient to be a relationship. Being a relationship is necessary but not su±cient to be a function. For example, f(x; y): y · xg is a relationship, but does not de¯ne a function y = f(x). Why? Graph this relationship with x on the horizontal axis and y on the vertical axis. Does this relationship de¯ne a function x = g(y)? How about x as a function of y? Is the following a function? f(x; y): y = 5 if x · 5 and y = 2x if x > 5g . www.allonlinefree.com 4 Chapter 2 Matrix Algebra 2.1 Matrices Consider a system of m linear equations in n unknowns: y1 = a11x1 + a12x2 + ¢ ¢ ¢ + a1nxn; y2 = a21x1 + a22x2 + ¢ ¢ ¢ + a2nxn; (2.1) . yn = am1x1 + am2x2 + ¢ ¢ ¢ + amnxn; There are three sorts of elements here: The constants: fyi : i = 1; : : : ; mg, The unknowns: fxj : j = 1; : : : ; ng, The coe±cients: faij : i = 1; : : : ; m; j = 1; : : : ; ng; and they can be gathered into three arrays: 2 3 2 3 2 3 y1 a11 ¢ ¢ ¢ a1n x1 6 7 6 7 6 7 6 y2 7 6 a21 ¢ ¢ ¢ a2n 7 6 x2 7 y = 6 . 7 ; A = 6 . 7 ; x = 6 . 7 4 . 5 4 . .. 5 4 . 5 ym am1 ¢ ¢ ¢ amn xm The arrays y and x are column vectors of order m and n, respectively whilst the array A is a matrix of order m £ n, which is to say that it has m rows and n columns. In matrix terms the above system can be represented as follows: www.allonlinefree.comy = Ax (2.2) There are two objects on our initial agenda. The ¯rst is to show, in detail, how the summary matrix representation corresponds to the explicit form in (2.1). For this purpose we need to de¯ne, at least, the operation of matrix multiplication. The second object is to describe a method for ¯nding the values of the unknown elements. Each of the m equations is a statement about a linear relationship amongst the n unknowns. The unknowns can be determined if and only if there can be found, amongst the m equations, a subset of n equations which are mutually independent in the sense that none of the corresponding statements can be deduced from the others.
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