Info Jnp
'Applicable only after the first-order necessar condition has been satistied. 'Applicable only after the first-order necessar condition has been satistied. configuration hill or valley, as the case may be not only in the two basic directions east-west and north-south , but in all other possible directions such as northeast-southwest as well. The above result, together with the first-order condition 11.5 , enables us to construct Table 11.1. It should be understood that all the second partial...
Solow Growth Model
The growth model of Professor Solow is purported to show, among other things, that the razor's-edge growth path of the Domar model is primarily a result of the Robert M. Solow, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, February, 1956, pp. 65-94. particular production-function assumption adopted therein and that, under alternative circumstances, the need for delicate balancing may not arise. In the Domar model, output is explicitly stated as a function of...
Exercise 107
1 Find the instantaneous rate of growth a v 3r c y ab' e y t 3' 2 If population grows according to the function H Hi 2 ' and consumption by the Miction C C e find the rates of growth of population, of consumption, and of per capita consumption by using natural log. 3 If y is related to x by y xk, how will the rates of growth rv and be related 4 Prove that if y u v, where u f t and v g t , then the rate of growth of y will be r ru - rr, as shown in 10.25 . 5 Prove the rate-of-growth rule 10.27 ....
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Although Fig. 11.10 specifically illustrates the one-variable case, the definitions of S and S in 11.27 and 11.28 are not limited to functions of a single variable. They are equally valid if we interpret x to be a vector, i.e., let x x,, , xn . In that case, however, 11.27 and 11.28 will define convex sets in the -space instead. It is important to remember that while a convex function implies 11.27 , and a concave function implies 11.28 , the converse is not true for 11.27 can also be satisfied...
Exercise 116
1 If the competitive firm of Example 1 has the cost function C 2Q 2Q instead, then a Will the production of the two goods still be technically related b What will be the new optimal levels of Q and Q c What is the value of nr 2 What does this imply economically 2 A two-product firm faces the demand and cost functions below 0, 40 - 2P, - P2 Q2 35 - Pt - P2 C Qj 2Q 10 a Find the output levels that satisfy the first-order condition for maximum profit. Use fractions. b Check the second-order...
Exercise 86
1 Let the equilibrium condition for national income be S Y T Y I Y G0 S',T',r gt 0 S' V gt ' where S, Y, T, , and G stand for saving, national income, taxes, investment, and government expenditure, respectively. All derivatives are continuous. a Interpret the economic meanings of the derivatives S', T', and '. b Check whether the conditions of the implicit-function theorem are satisfied. If so, write the equilibrium identity, c Find dY dG,, and discuss its economic implications. 2 Let the...
The Supply Function Of A Certain Commodity Is Q A Bp2 R1 2
3 The supply function of a certain commodity is Q a bP2 B 2 a lt 0, gt gt 0 R rainfall Find the price elasticity of supply eqP, and the rainfall elasticity of supply EqR. 4 How do the two partial elasticities in the last problem vary with P and R1 In a monotonic fashion assuming positive P and R 5 The foreign demand for our exports X depends on the foreign income Yf and our price level P X 1 2 P 2. Find the partial elasticity of foreign demand for our exports with respect to our price level.
Info Wxb
1 z 0 minimum 3 z - 11 40 minimum 5 z 2 - e minimum , attained at 5c, y, w 0,0,1 7 a rt 2 r2 4 f6 4- 6 1 a Strictly convex. c Strictly convex. 2 a Strictly concave. c Neither. 1 a Convex combination, with 6 0.5. b Convex combination, with 9 Exercise 11.6 1 a No. b Q Pl l 4 and Q2 P20 4 3 1 ,1 If ed2 Is U 5 a tt P0Q a, b 1 i0y2 - aPa0 - bPh0 1 da dPaQ P0Qhhe' J lt 0 db dPM -P0Qahe- J lt 0 2 a Four. A 3a 3 gt 0 QhQah - QaQhh P0 1 i0 2 J gt 0 c as a 0 - e efcft - e eflfc gt 02 i 0 -3 I I lt o...
Inflation And Unemployment In Discrete Time
The interaction of inflation and unemployment, discussed earlier in the continuous-time framework, can also be couched in discrete time. Using essentially the same economic assumptions, we shall illustrate in this section how that model can be reformulated as a difference-equation model. The earlier continuous-time formulation equations 15.33 p a T PU htr Sec. 15.5 consisted of three differential adaptive expectations monetary policy Three endogenous variables are present p actual rate of...
Info Xiq
a Is AB defined Calculate AB. Can you calculate BA1 Why b Is BC defined Calculate BC. Is CB defined If so, calculate CB. Is it true that BC CB1 3 On the basis of the matrices given in Example 9, is the product BA defined If so, calculate the product. In this case do we have AB BA1 4 Find the product matrices in the following in each case, append beneath every matrix a dimension indicator 5 Expand the following summation expressions 5 4 6 Rewrite the following in L notation a - 1 2x2 x2 - 1 3x3...
Exercise 41
1 Rewrite the equation system 3.1 in the format of 4.1 , and show that, if the three variables are arranged in the order Qd, Qs, and P, the coefficient matrix will be How would you write the vector of constants 2 Rewrite the equation system 3.12 in the format of 4.1 with the variables arranged in the following order Qd , QsU Qd2, Qs2, Px, P2. Write out the coefficient matrix, the variable vector, and the constant vector.
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domain. Now each point on the said line segment is in the nature of a weighted average of u and v. Thus we can denote this line segment by 8u 1 8 v. where 8 the Greek letter theta unlike u and v is a variable scalar with the range of values 0 lt 8 lt 1. By the same token, line segment MN, representing the set of all weighted averages of f u and f v , can be expressed by 8 u 1 - 8 v , with 8 again varying from 0 to 1. What about arc MN along the surface Since that arc shows the values of the...
Partial Derivatives
where the variables x, i 1,2, , n are all independent of one another, so that each can vary by itself without affecting the others. If the variable x, undergoes a change Ax, while x2, , x all remain fixed, there will be a corresponding change inj gt , namely, Ay. The difference quotient in this case can be expressed as If we take the limit of A y Ax, as Ax, - gt 0, that limit will constitute a derivative. We call it the partial derivative of y with respect to xx, to indicate that all the other...
Info Uhp
More generally, if we have n differentiable functions in n variables, not necessarily y ' i,x2, ,x 7.26 y2 f2 xux2, ,xn where the symbol denotes the nth function and not the function raised to the nth power , we can derive a total of n2 partial derivatives. Together, they will give rise to the Jacobian A Jacobian test for the existence of functional dependence among a set of n functions is provided by the following theorem The Jacobian J defined in 7.27 will be identically zero for all values...
Logarithmic Functions
When a variable is expressed as a function of the logarithm of another variable, the function is referred to as a logarithmic function. We have already seen two versions of this type of function in 10.12 and 10.13 , namely, t logft v and t logt, v In gt which differ from each other only in regard to the base of the logarithm. Log Functions and Exponential Functions As we stated earlier, log functions are inverse functions of certain exponential functions. An examination of the above two log...
Exercise 23
1 Write the following in set notation a The set of all real numbers greater than 27. b The set of all real numbers greater than 8 but less than 73. 2 Given the sets S, 2,4,6 , S2 7,2, 6 , S, 4, 2,6 , and S4 2,4 , which of the following statements are true a 5, S2 d 3 S2' g 5, d S4 b Sl R e 4 iS4 h 0 c S2 c 5 e S2 S4 c R , S 1,2 3 Referring to the four sets given in the preceding problem, find a s,us2 t S nS, gt s4 n s2 n s, fe S, u s, d S2nS4 S, u s, u S4 4 Which of the following statements are...
Secondorder Conditions
The introduction of a Lagrange multiplier as an additional variable makes it possible to apply to the constrained-extremum problem the same first-order condition used in the free-extremum problem. It is tempting to go a step further and borrow the second-order necessary and sufficient conditions as well. This, however, should not be done. For even though Z is indeed a standard type of extremum with respect to the choice variables, it is not so with respect to the Lagrange multiplier....
R Pq
the AR curve can also be regarded as a curve relating price P to output Q P f Q . Viewed in this light, the AR curve is simply the inverse of the demand curve for the product of the firm, i.e., the demand curve plotted after the P and Q axes are reversed. Under pure competition, the AR curve is a horizontal straight line, so that ' gt 0 and, from 7.7' , MR - AR 0 for all possible values of Q. Thus the MR curve and the AR curve must coincide. Under imperfect competition, on the other hand, the...
Info Szp
2 Use the matrices given in the preceding problem to verify that a A B ' A' B' b AC ' C'A' 3 Generalize the result 4.11 to the case of a product of three matrices by proving that, for any conformable matrices A, B, and C, the equation ABC ' C'B'A' holds. 4 Given the following four matrices, test whether any one of them is the inverse of another 5 Generalize the result 4.14 by proving that, for any conformable nonsingular matrices A, B, and C, the equation ABC 1 C B 'A '. a Must A be square Must...
The Difference Quotient
Since the notion of change figures prominently in the present context, a special symbol is needed to represent it. When the variable x changes from the value x0 to a new value xt, the change is measured by the difference x, x0. Hence, using the symbol A the Greek capital delta, for difference to denote the change, we write Ax x, - x0. Also needed is a way of denoting the value of the function x at various values of x. The standard practice is to use the notation x to represent the value of f x...
Market Model
First let us consider again the simple one-commodity market model of 3.1 . That model can be written in the form of two equations Q a - bP a, b gt 0 demand These solutions will be referred to as being in the reduced form the two endogenous variables have been reduced to explicit expressions of the four mutually independent parameters a, b, c, and d. To find how an infinitesimal change in one of the parameters will affect the value of P, one has only to differentiate 7.14 partially with respect...
Exercise 32
find P and Q by a elimination of variables and b using formulas 3.4 and 3.5 . Use fractions rather than decimals. 2 Let the demand and supply functions be as follows a Qd 51 - 3P b Qd 30-2P find P and Q by elimination of variables. Use fractions rather than decimals. 3 According to 3.5 , for Q to be positive, it is necessary that the expression ad be have the same algebraic sign as b d . Verify that this condition is indeed satisfied in the models of the preceding two problems. 4 If b d 0 in...
Note On Jacobian Determinants
The study of partial derivatives above was motivated solely by comparative-static considerations. But partial derivatives also provide a means of testing whether there exists functional linear or nonlinear dependence among a set of n functions in n variables. This is related to the notion of Jacobian determinants named after Jacobi . If we get all the four partial derivatives dy, and arrange them into a square matrix in a prescribed order, called a Jacobian matrix and denoted by J, and then...
Finding The Inverse Matrix
If the matrix A in the linear-equation system Ax d is nonsingular, then A ' exists, and the solution of the system will be x A 'd. We have learned to test the nonsingularity of A by the criterion A 0. The next question is How can we find the inverse A 1 if A does pass that test Expansion of a Determinant by Alien Cofactors Before answering this query, let us discuss another important property of determinants. Property VI The expansion of a determinant by alien cofactors the cofactors of a wrong...
B
the marginal falls short of the average in numerical value thus the function is inelastic at point A. The exact opposite is true in diagram b. Sometimes, we are interested in locating a point of unitary elasticity on a given curve. This can now be done easily. If the curve is negatively sloped, as in Fig. 8.3a, we should find a point C such that the line OC and the tangent BC will make the same-sized angle with the x axis, though in the opposite direction. In the case of a positively sloped...
Set Notation
A set is simply a collection of distinct objects. These objects may be a group of distinct numbers, or something else. Thus, all the students enrolled in a particular economics course can be considered a set, just as the three integers 2, 3, and 4 can form a set. The objects in a set are called the elements of the set. There are two alternative ways of writing a set by enumeration and by description. If we let 5 represent the set of three numbers 2, 3, and 4, we can write, by enumeration of the...
Comparativestatic Analysis Of Generalfunction Models
The study of partial derivatives has enabled us, in the preceding chapter, to handle the simpler type of comparative-static problems, in which the equilibrium solution of the model can be explicitly stated in the reduced form. In that case, partial differentiation of the solution will directly yield the desired comparative-static information. You will recall that the definition of the partial derivative requires the absence of any functional relationship among the independent variables say, x,...
Info Ivt
Property II The interchange of any two rows or any two columns will alter the sign, but not the numerical value, of the determinant. ad be, but the interchange of the two rows yields cb ad ad be
Limit Theorems
Our interest in rates of change led us to the consideration of the concept of derivative, which, being in the nature of the limit of a difference quotient, in turn prompted us to study questions of the existence and evaluation of a limit. The basic process of limit evaluation, as illustrated in Sec. 6.4, involves letting the variable v approach a particular number say, N and observing the value which q approaches. When actually evaluating the limit of a function, however, we may draw upon...
Digression On Inequalities And Absolute Values
We have encountered inequality signs many times before. In the discussion of the last section, we also applied mathematical operations to inequalities. In transforming 6.7' into 6.7 , for example, we subtracted 1 from each side of the inequality. What rules of operations apply to inequalities as opposed to equations To begin with, let us state an important property of inequalities inequalities are transitive. This means that, if a gt b and if b gt c, then a gt c. Since equalities equations are...
Info Qli
Here, even the descriptive statement is symbolically expressed. A set with a finite number of elements, exemplified by set S above, is called a finite set. Set I and set each with an infinite number of elements, are, on the other hand, examples of an infinite set. Finite sets are always denumerable or countable , i.e., their elements can be counted one by one in the sequence 1,2,3, . Infinite sets may, however, be either denumerable set I above , or nondenumerable set J above . In the latter...
Mathematical Economics Versus Econometrics
The term mathematical economics is sometimes confused with a related term, econometrics. As the metric part of the latter term implies, econometrics is concerned mainly with the measurement of economic data. Hence it deals with the study of empirical observations using statistical methods of estimation and hypothesis testing. Mathematical economics, on the other hand, refers to the application of mathematics to the purely theoretical aspects of economic analysis, with little or no concern about...
Economic Models
As mentioned before, any economic theory is necessarily an abstraction from the real world. For one thing, the immense complexity of the real economy makes it impossible for us to understand all the interrelationships at once nor, for that matter, are all these interrelationships of equal importance for the understanding of the particular economic phenomenon under study. The sensible procedure is, therefore, to pick out what appeal to our reason to be the primary factors and relationships...