Microeconomics principles and analysis phần 9 ppsx

506 APPENDIX A. MATHEMATICS BACKGROUND Figure A.9: A strictly concave-contoured (strictly quasiconcave) function  There are functions for which the contours look like those of a concave function but which are not themselves concave. An example here would be ' (f(x)) where f is a concave function and is an arbitrary monotonic transformation. These remarks lead us to the de…nition: De…nition A.23 A function f is (strictly) concave-contoured if all the sets B(y 0 ) in (A.31) are (strictly) convex. A synonym for (strictly) concave-contoured is (strictly) quasiconcave. Try not to let this (unfortunately necessary) jargon confuse you. Take, for example, a “conventional”looking utility function such as U(x) = x 1 x 2 : (A.32) According to de…nition A.23 this function is strictly quasiconcave: if you draw the set of points B() := f(x 1 ; x 2 ) : x 1 x 2  g you will get a strictly convex set. Furthermore, although U in (A.32) is not a concave function, it is a simple transformation of the strictly concave function ^ U(x) = log x 1 + log x 2 ; (A.33) and has the same shape of contour map as ^ U. But when we draw those contours on a diagram with the usual axes we would colloquially describe their shape A.7. MAXIMISATION 507 as being “convex to the origin”! There is nothing seriously wrong he re: the de…nition, the terminology and our intuitive view are all correct; it is ju st a matter of the way in which we visualise the function. Finally, the following complementary property is sometimes useful: De…nition A.24 A function f is (strictly) quasiconvex if f is (strictly) qua- siconcave. A.6.6 The Hessian property Consider a twice-di¤erentiable function f from D  R n to R. Let f ij (x) denote @ 2 f(x) @x i @xj . The symmetric matrix 2 6 6 4 f 11 (x) f 12 (x) ::: f 1n (x) f 21 (x) f 22 (x) ::: f 2n (x) ::: ::: ::: ::: f n1 (x) f n2 (x) ::: f nn (x) 3 7 7 5 is known as the Hessian matrix of f. De…nition A.25 The Hessian matrix of f at x is negative semide…nite if, for any vector w 2 R n , it is true that n X i=1 n X j=1 w i w j f ij (x)  0: A twice-di¤erentiable func tion f from D to R is concave if and only if f is negative semi-de…nite for all x 2 D. De…nition A.26 The Hessian matrix of f at x is negative de…nite if, for any vector w 2 R n ,w 6= 0, it is true that n X i=1 n X j=1 w i w j f ij (x) < 0: A twice-di¤erentiable function f from D to R is strictly concave if f is negative de…nite for all x 2 D; but the reverse is not true – a strictly concave function f may have a negative semi-de…nite Hessian. If the Hessian of f is negative de…nite for all x 2 D we will say that f has the Hessian property. A.7 Maximisation Because a lot of economics is concerned with optimisation we will brie‡y overview the main techniques and results. However this only touches the edge of a very big subject: you should consult the references in section A.9 for more details. 508 APPENDIX A. MATHEMATICS BACKGROUND A.7.1 The basic technique The problem of maximising a function of n variables max x2X f(x) (A.34) X  R n is straightforward if the function f is di¤erentiable and the domain X is unbounded. We adopt the usual …rst-order condition (FOC) @f(x) @x i = 0; i = 1; 2; :::; n (A.35) and then solve for the values of (x 1 ; x 2 ; :::; x n ) that satisfy (A.35). However the FOC is, at best, a necessary condition for a maximum of f. The problem is that the FOC is essentially a simple hill-climbing rule: “if I’m really at the top of the hill then the ground must be ‡at just where I’m standing.” There are a number of di¢ culties with this:  The rule only picks out “stationary points” of the function f. As Figure A.10 illustrates, this condition is satis…ed by a minimum (point C) as well as a maximum (point A), or by a point of in‡ection (E). To eliminate points such as C and E we may look at the sec ond-orde r conditions which essentially require that at the top of the hill (a point such as A) the slope must be (locally) decreasing in every direction.  Even if we eliminate minima and points of in‡ection the FOC may pick out multiple “local” maxima. In Figure A.10 points A and D are each local maxima, but obviously A is the point that we really want. we may be able to eliminate. This problem may be sidestepped by introducing a priori restrictions on the nature of the function f that eliminate the possibility of multiple stationary points –for example by requiring that f be strictly concave.  If we have been careless in spec ifying the problem then the hill-climbing rule may be completely misleading. We have assumed that each x-component can range freely from 1 to +1. But suppos e – as if often in the case in economics –that the de…nition of the variable is such that only non- negative values make sense. Then it is clear from Figure A.10 that A is an irrelevant point and the maximum is at B. In climbing the hill we have reached a logical “wall”and we can climb no higher.  Likewise if we have overlooked the requirement that the function f be everywhere di¤erentiable the hill-climbing rule represented by the FOC may be misleading. If we draw the function f(x) = 8 < : x x  1 2 x x > 1 A.7. MAXIMISATION 509 Figure A.10: Di¤erent types of stationary point it is clear that it is continuous and has a maximum at x = 1. But the FOC as stated in (A.35) is useless because the di¤erential of f is unde…ned exactly at x = 1. If we can sweep these di¢ culties aside then we can use the solution to the system of equations provided by the FOC in a powerful way. To see what is usually done, slightly rewrite the maximisation problem (A.34) as max x2R n f(x; p) (A.36) where p represents a vector of parameters, a set of numbers that are …xed for the particular maximisation problem in hand but which can be used to characterise the di¤erent members of a whole class of maximisation problems and their solutions. For example p might represent prices (outside the control of a small …rm and therefore taken as given) and might x represent the list of quantities of inputs and outputs that the …rm cho oses in its production process; pro…ts depend on both the parameters and the choice variables. We can then treat the FOC (A.35) as a system of n equations in n unknowns (the components of x).Without further regularity conditions such a system is not guaranteed to have a solution nor, if it has a solution, will it necessarily be unique. However, if it does then we can write it as a function of the given 510 APPENDIX A. MATHEMATICS BACKGROUND parameters p: x  1 = x  1 (p) x  2 = x  2 (p) ::: x  n = x  n (p) 9 > > = > > ; (A.37) We may refer to the functions x  1 () in (A.37) as the response functions in that they indicate how the optimal values of the choice variables (x  ) would change in response to changes in values of the given parameters p. A.7.2 Constrained maximisation By itself the b asic technique in section A.7.1 is of limited value in economics: optimisation is usually subject to some side constraints which have not yet been intro d uce d. We now move on to a simple case of constrained optimisation that, although restricted in its immediate applicability to economic problems, forms the basis of other useful techniques. We consider the problem of maximising a di¤erentiable function of n variables max x2R n f(x; p) (A.38) subject to the m equality constraints G 1 (x; p) = 0 G 2 (x; p) = 0 ::: G m (x; p) = 0 9 > > = > > ; (A.39) There is a standard technique for solving this kind of problem: this is to in- corporate the constraint in a new maximand. To do this introduce the Lagrange multipliers  1 ; :::;  m , a set of non-n egative variables, one for each constraint. The constrained maximisation problem in the n variables x 1 ; :::; x n , is equivalent to the following (unconstrained) maximisation problem in the n + m variables x 1 ; :::; x n ;  1 ; :::;  m , L(x; ; p) := f(x; p)  m X j=1  j G j (x; p) (A.40) , where L is the Lagrangean function. By introducing the Lagrange multipliers we have transformed the constrained optimisation problem into one that is of the same format as in section A.7.1, namely max x; L(x; ; p) (A.41) A.7. MAXIMISATION 511 The FOC for solving (A.41) are foun d by di¤erentiating (A.40) with respect to each of the n + m variables and setting each to zero. @L(x  ;   ; p) @x i = 0; i = 1; :::; n (A.42) @L(x  ;   ; p) @ j = 0; j = 1; :::; m (A.43) where the “  ”means that the di¤erential is being evaluated at a solution point (x  ;   ). So the FOC consist of the n equations @f(x  ; p) @x i = m X j=1   j @G j (x  ; p) @x i ; i = 1; :::; n (A.44) plus the m constraint equations (A.39) evaluated at x  . We therefore have a system of n + m equations (A.44,A.39) in n + m variables. As in section A.7.1, if the system of equations does have a unique solution (x  ;   ), then this can be written as a function of the parameters p: x  1 = x  1 (p) x  2 = x  2 (p) ::: x  n = x  n (p) 9 > > = > > ; (A.45)   1 =   1 (p)   2 =   2 (p) :::   m =   m (p) 9 > > = > > ; (A.46) Once again the functions x  1 () in (A.45) are the response functions and have the same interpretation. The Lagrange multipliers in (A.46) also have an interesting interpretation which is handled in A.7.4 below. If the equations (A.44,A.39) yield more than one solution, but f in (A.38) is quasiconcave and the set of x satisfying (A.39) is convex then we can appeal to the commonsense result in Theorem A.12. A.7.3 More on constrained maximisation Now modify the problem in section A.7.2 in two ways that are especially relevant to economic problems  Instead of allowing each component x i to range freely from 1 to +1.we restrict to some interval of the real line. So we will now write the domain restriction x 2 X where we will take X to be the non-negative orthant of R n . The results below can be adapted to other speci…cations of X. 512 APPENDIX A. MATHEMATICS BACKGROUND  We replace the equality constraints in (A.39) by the corresponding in- equality constraints G 1 (x; p)  0 G 2 (x; p)  0 ::: G m (x; p)  0 9 > > = > > ; (A.47) This is reasonable in economic applications of optimisation. For example the appropriate way of stating a budget constraint is “expenditure must not exceed income”rather than “ must equal ”. So the problem is now max x2X f(x; p) subject to (A.47). The solution to this modi…ed problem is similar to that for the standard Lagrangean – see Intriligator (1971), pages 49-60. Again we transform the problem by forming a Lagrangean (as in A.40): max x2X;>0 L(x; ; p) (A.48) However, instead of (A.42, A.43)we now have the following FOCs: @L(x  ;   ; p) @x i  0; i = 1; :::; n (A.49) x  i @L(x  ;   ; p) @x i = 0; i = 1; :::; n (A.50) and @L(x  ;   ; p) @ j  0; j = 1; :::; m (A.51)   j @L(x  ;   ; p) @ j = 0; j = 1; :::; m (A.52) This set of equations and inequalities is conventionally known as the Kuhn- Tucker conditions. They have important implications relating the values of the variables and the Lagrange multipliers at the optimum. Applying this result we …nd @f(x  ; p) @x i  m X j=1   j @G j (x  ; p) @x i ; i = 1; :::; n (A.53) with (A.44) if x  i > 0. Note that if, for some i, x  i = 0 we could have strict inequality in (A.53). Figure A.11 illustrates this possibility for a case where the objective function is strictly concave: note that the conventional condition of “slope=0” (A.42) (which would appear to be satis…ed at point A) is irrelevant here since a point such as A would violate the constraint x i  0; at the optimum (point B) the Lagrangean has a strictly decreasing slope. Similar interpretations will apply to the Lagrange multipliers: A.7. MAXIMISATION 513 Figure A.11: A case where x  i = 0 at the optimum 1. If the Lagrange multiplier associated with constraint j is strictly positive at the optimum (  j > 0), then it must be binding (G j (x  ; p) = 0). 2. Conversely one could have an optimum where one or more Lagrange mul- tiplier (  j = 0) is zero in which case the constraint may be slack – i.e. not binding –(G j (x  ; p) < 0). So, for each j at the optimum, there is at most one inequality condition: if there is a strict inequality on the Lagrange multiplier then the corresponding constraint must be satis…ed with equality (case 1); if there is a strict inequality on the constraint then the corresponding Lagrange multiplier must be equal to zero (case 2). These facts are conventionally known as the complementary slackness condition. However, note that one can have cases where both the Lagrange multiplier (  j = 0) and the constraint is binding (G j (x  ; p) = 0). Again if the system (A.53,A.47) yields a unique solution it can be written as a function of the parameters p which in turn determines the response functions; but if it yields more than one solution, but f in (A.38) is quasiconcave and the set of x satisfying (A.47) is convex then we can use the following. Theorem A.12 If f : R n 7! R is quasiconcave and A  R n is convex then the set of values x  that solve the problem max f (x) subject to x 2 A is convex. 514 APPENDIX A. MATHEMATICS BACKGROUND A.7.4 Envelope theorem We now examine how the solution, conditional on the given set of parameter values p changes when the values p are changed. Let v(p) = max x2X f(x; p) subject to (A.39). Using the response functions in (A.37) we obviously have v(p) = f(x  (p); p) (A.54) The maximum-value function v has an important property: Theorem A.13 If the objective function f and the constraint functions G j are all di¤erentiable then, for any k: @v(p) @p k = @f(x  ; p) @p k  m X j=1  j @G j (x  ; p) @p k Proof. Evaluating the constraints (A.39) at x = x  (p) we have G j (x  (p); p) = 0 (A.55) and di¤erentiating (A.55) with respect to p k and rearranging gives: n X i=1 @G j (x  ; p) @x i @x  i (p) @p k =  @G j (x  (p); p) @p k (A.56) Di¤erentiate (A.54) with respect to p k @v(p) @p k = @f(x  (p); p) @p k + n X i=1 @f(x  (p); p) @x i @x  i (p) @p k (A.57) Using (A.44) evaluated at x = x  (p) (A.57) becomes @v(p) @p k = @f(x  (p); p) @p k + m X j=1  j n X i=1 @G j (x  (p); p) @x i @x  i (p) @p k : (A.58) Using (A.56) in (A.58) gives the result. The envelope theorem has some nice economic corollaries. One of the most important of these concerns the interpretation of the Lagrange multiplier(s). Suppose we modify any one of the constraints (A.39) to read G j (x; p) =  j (A.59) where  j could have any given value. This does not really make the problem any more general because we could have rede…ned the parameter list as p := (p;  j ) and used a modi…ed form of the jth constraint  G j de…ned by  G j (x; p) := G j (x; p)  j = 0: (A.60) In e¤ect we can just treat  as an extra parameter which does not enter the function f. Then A.8. PROBABILITY 515 Corollary A.3 @v(p) @ j =  j The result follows immediately from Theorem A.13 using the de…nition of  j in (A.60) and the fact that @f (x  ;p) @ = 0. So  j is the “value”that one would put on a marginal change in the jth constraint, (represented as a small displacement of  j ). A similar result is available f or the case where the relevant constraints are inequality constraints –as in section A.7.3 rather than section A.7.2. In partic- ular, notice the nice intuition if constraint j is slack at the optimum. We know then that the associated Lagrange multiplier is zero (see page 513), and the im- plication of Corollary A.3 is that the marginal value placed on the jth constraint is zero: you would not pay anything to relax an already-slack constraint. A.7.5 A point on notation For some maximisation problems in microeconomics it is convenient to use a special notation. Consider the problem of choosing s from a set S in order to maximise a function '. To characterise the set of values that do the job of maximisation one uses: arg max s ' (s) := fs 2 S : ' (s)  '(s 0 ) ; s 0 2 Sg where the function ' may, of course, incorporate side constraints. A.8 Probability For the basic de…nition of a random variable and the me aning of probability see, for example, Spanos (1999). We will assu me that the random variable X is a scalar. This is not essential to most of the discussion that follows, but it makes the exposition easier. The case where the random variable is a vector is discussed in standard books on probability and statistics – see section A.9 below. The support of a random variable is de…ned to be the smallest closed set whose complement has probability zero. For the applications in this book we can take the support to be either an interval on the real line or a …nite set of real numb ers. For the exposition that follows we take the support of X to be the interval S := [x; x]. A convenient general way of characterising the distribution of a random variable is the distribution function F of X. This is a non-decreasing function F (x) := Pr (X  x) (A.61) where 0  F(x)  1 for all x and F (x) = 1; the symbol Pr stands for “proba- bility.”In words F(x) in (A.61) gives the probability that the random variable [...]... ( 199 9) On optimisation in economics see Dixit ( 199 0) and Sundaram (2002) For more on applications of convexity and …xed-point theorems see Green and Heller ( 198 1) and (for the mathematically inclined) the very thorough treatment by Border ( 198 5) A useful introduction to the elements of probability theory for economists is given in Spanos ( 199 9); for a more advanced treatment see Ho¤man-Jørgensen ( 199 4)... parameters and B (a; b) := 0 xa [1 The corresponding distribution function is found by integration of f A .9 b x] dx Reading notes For an overall review of concepts and methods there are several suitable books on mathematics designed for economists such as Chiang ( 198 4), de la Fuente 520 APPENDIX A MATHEMATICS BACKGROUND ( 199 9), Ostaszewski ( 199 3), Simon and Blume ( 199 4) or Sydsæter and Hammond ( 199 5) A... density of 1 0 on the value x1 x < x1 x 0 x1 9 > > > > = > > > > ; on the value x0 and a probability A .9 READING NOTES 5 19 Rectangular distribution The density is assumed to be uniform over the interval [x0 ; x1 ] and zero elsewhere: 8 9 if x0 x x1 = < x1 1 x0 f (x) = : ; 0 elsewhere F (x) = 8 > > > > < > > > > : 0 if x x0 x1 x0 if 1 x < x0 if x0 x < x1 x x1 9 > > > > = > > > > ; Normal distribution This... bliss point then he would buy this bundle and leave the rest of the income unspent 8 All the results go through except that the optimal commodity demands x may no longer be well-de…ned functions of p and (or of p and y): at certain sets of prices there may be multiple solutions, and we have demand correspondences which will, however, be upper semi-continuous 9 Indi¤erence curves with the direction of... simply been a price fall then the cost must have fallen, and so the expression is positive, the same sign as the welfare change 26 Use equation (4.23) and apply reasoning similar to the answer to question 19 27 Compare equations (4.37) and (4.40) By de…nition of the optimal commodity demands and the cost function the denominators on the right-hand side must be equal; but by de…nition of the cost function... price of good j on the demand for good i and vice versa are: i i Dj (p; y) = Hj (p; ) i xj Dy (p; y) (B.27) j j Di (p; y) = Hi (p; ) j xi Dy (p; y) (B.28) Although the substitution term (…rst term on the right-hand side) has to be equal in equations (B.27) and (B.28), the income e¤ects could be very di¤erent So it is possible for the left-hand side to be negative in one case and positive 534 APPENDIX... curves are nonhomothetic then as income expands x will move along a facet and eventually may switch between facets When such a switch occurs one input j is no longer purchased and another input is substituted In this case as income increases the demand for good j at …rst increases and then, when, the switch occurs, a further increase in income causes the demand for j to fall to zero 12 Apply an induction... (B. 29) , yields: n X i pi Hj (p; ) = 0 (B.32) i=1 Now, because the own-price substitution e¤ect must be negative, equation (B.32) implies that at least one of the cross-price substitution e¤ects must be positive In other words at least one pair of goods i, j must be net substitutes 18 In Figure B .9 the income e¤ect is from x to and the substitution e¤ect from to x 19 The …rst term on the right-hand... (2.13) and (B.8) imply 0= m X j wj Hi (w; q) : (B .9) j=1 From this we immediately see that the last term in (B.7) must be zero and the result follows See also the remarks on the envelope theorem in section A.7.4 on page 514 14 Let there be increasing returns to scale over the output levels q to tq where t > 1, and let z be cost-minimising for q at input prices w Now consider the input vector ^ := tz, and. .. purchased, and xt must lie in A 9 Every input is always essential so such any such change is bound to shift the cost of any given output bundle 10 If R3 > 0 the household’ budget y increases and the frontier moves s outwards at all points: consumption of goods 1 and 2 increases 11 Given the linear technology in equation (5.20) it is clear that if the person’ income increases then the attainable set expands . economists such as Chiang ( 198 4), de la Fuente 520 APPENDIX A. MATHEMATICS BACKGROUND ( 199 9), Ostaszewski ( 199 3), Simon and Blume ( 199 4) or Sydsæter and Ham- mond ( 199 5). A useful summary of results. Sydsæter et al. ( 199 9). On optimisation in economics see Dixit ( 199 0) and Sundaram (2002). For more on applications of convexity and …xed-point theorems see Green and Heller ( 198 1) and (for the mathematically. Probability For the basic de…nition of a random variable and the me aning of probability see, for example, Spanos ( 199 9). We will assu me that the random variable X is a scalar. This is not essential