Advanced Microeconomic Theory
Advanced Microeconomic Theory ECON 204
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This 35 page Class Notes was uploaded by Isidro Stoltenberg on Monday September 7, 2015. The Class Notes belongs to ECON 204 at University of California - Santa Cruz taught by Staff in Fall. Since its upload, it has received 61 views. For similar materials see /class/182320/econ-204-university-of-california-santa-cruz in Economcs at University of California - Santa Cruz.
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Date Created: 09/07/15
1 PROFIT MAXIMIZATION The theory of the firm is first presented in terms of general functional forms Lectures 1 4 and then in Lecture 6 we consider the Cobb Douglas production function For Lectures 1 4 the homework is to redo the previous lecture under the assumption that the production function Q 2 Log K1 2 Log L 1 Also one should be able to reproduce the lecture without looking at your notes Note that Log always means natural log QK L output Q is a function of inputs K 2 capital and L labor 3Q 9Q 3L QLKL QL K QK That is the partial derivatives are denoted by subscripts Assume Hessian of 2nd partials is negative definite this implies concavity of the production function PERFECTLY COMPETITIVE FIRM P w and i are exogenous P 2 price of output w wage i 2 interest rate Objective Function H PQK L Lw Ki Profit 2 revenue minus cost KT conditions FOC KT Conditions HLPQL w50 L20 HL L0 HKPQK iso K20 HKKzo Wage 2 Marginal Revenue Product of Labor Verbal Interpretation Interest Rate 2 Marginal Revenue Product of Capital For an Interior Solution HL 2 Hk 0 Implied Relations assuming L K gt 0 w K i Ratio of Marginal Products 2 Ratio of Payments to Factors ISOQUANT Isocurve QL K Q isoquant d6 QLdL QKdK 0 OR dK g slope of the isoquant dL QK QL W As shown above a profit max1m121ng f1rm sets Q So for a profit K 1 maximizing firm the slope of the isoquant at the profit maximizing point equals d w E i This relationship can be understood Via the following diagram ISOQUANT L The straight line is the budget line A profit maximizing firm will choose the lowest budget line for any isoquant At that point the slopes will be identical HOMOGENIETY From the first order K T conditions we can establish homogeneity QLS QKS wls wh Therefore the First Order Conditions FOC are homogeneous of degree zero in w P and i If double w P and i then K and L remain the same rts and so does Q Therefore the maximum of H PQ K L Lw Ki is homogeneous of degree 1 in P i and w since Q K L rts remain the same and thus doubling P i and w doubles H I have said that profits are homogenous of degree 1 in w i and P This should not be confused with constant returns to scale of the production function Q may or may not be homogeneous of degree 1 in K and L We will discuss returns to scale later IMPLICIT AND EXPLICIT DEMAND FUNCTIONS Note that the first order conditions make the profit maximizing L and K implicit functions of w P and i Since QLL and QKK are negative and QLLQKK gt QLKQLK by assumption in principle L and K can be solved as explicit functions of w P i L Lw P i K Kw P i These derived factor demand functions are the profit maximizing amounts of L and K given w P i An will be used in this course to denote an explicit function of the exogenous variables Characteristics of the implicit function such as homogeniety hold true for the explicit function as well LTw TP Ti Lw P i and KTw TP Ti Kw P i because the first order condions are homogeneous of degree 0 in w P and i SECOND ORDER CONDITIONS Second PQLL PQLK 2 QLL Q LK Ordef PQKL PQKK QKL QKK Conditions H is negative definite since Q is negative definite by assumption H1 lt 0 H2 gt 0 HI1 1 X 1 the upper left term H2 is the 2 X 2 determinant COMPARATIVE STATICS We want to find the effect of a change of an exogenous variable w i or P on an endogenous variable K or L assuming that the firm is maximizing profits That is we want to find the effect of a change in w i or P on the profit maximizing K or L and not on just any possible K or L we could denote the profit maximizing K and L by K and but this would clutter up the notation further We will make use of the implicit function theorem To make things simpler we will assume that both K and L are greater than 0 Both before and after the exogenous change the first order conditions hold That is the marginal profitability of increased K or L is zero More formally dHL HLLdL HLKdK HLde 2 0 W changes dHK HKLdL 11Kde 11Kde 0 dHL PQLLdL PQLKdK dw 0 0 OR lPQLL PQLKl d1 ldWl lPQKL PQKKJ ldkl lol Since there are 2 linear equations linear in terms of dl and dk we can solve using Cramer s rule First we find the effect on L dw P dL QLK 0 PQKK H dL PQKK dL QKK lt 0 by assumption and H 2 H2 gt 0 by assumption Therefore m lt 0 Downward sloping factor demand curve Locally the derived demand curve for the factor is always downward sloping Next we find the effect of w on K PQLL dw dK P d lm 0 Q W H dK PQKL dK 0 gt 0 dw dw lt QKL gt Comparative statics when i changes Kl IL HLLdL HLKdK HLidi 0 dHK HKLdL HKKdK HKidi 0 dHL PQLLdL PQLKdK Odi 0 dHK PQKLdL PQKKdK di 2 o The effect of a change in i on L 0 PQLK dL ldi PQKK IHI PQLKdi Cross Demands dL PQLK dK Always Equal di H dW PQLL d By a similar process one can also show that d1 H Note that we use the implicit function theorem to find dKdi Suppose that we had solved the first order conditions explicitly for K Then the partial of the explicit function K with respect to the partial of i would yield the same answer as the total derivative of the implicit function with respect to the derivative of i dKdi All of this can be illustrated via a simple example Suppose that Q 2 Log L 1 2 Log K 1 Then HPLogL12PLogK1 Lw Ki KT conditions FOC HLzPL1 w50 L20 HLL0 HK2PK1 isOK20 HKK20 We can solve for L as an explicit function of P w and i For L gt 0 LP w i Pw 1 The partial of this explicit function L with respect to the partial of w LW Pw2 Alternatively we can make use of the implicit function theorem dHL HLLdL 11Lde 11Lde 2 0 dHK HKLdL HKKdK 11Kde 2 0 Kl IL PL 12dL 0 dK dw 0 dHK 0d ZPK 12 dK 0 dw 0 dL Hence 2 L 12P dw dL But by the first order conditions L 1 Pw So d Pw2 w Thus the total dervivative of the implicit function L with respect to the derivative w is equivalent to the partial derivitive of the explicit function L with respect to w THE EFFECT OF A CHANGE IN P ON L dHL PQLLdL PQKLdK QLdP 0 PQKLdL 0 QLdP PQKL dL QKdP PQKK tQLQKKP QKQKLPIdP lHl IHI wIH Since QL g and QK l QKKW iQKLJ dL dP IHI QKK lt 0 Therefore QKKW gt 0 If QKL gt 0 then dL gt 0 Note however that dL might be greater than zero even if QKL Le Chatelier Principle lt0 LE CHATELIER PRINCIPLE Long Run Changes in absolute value gt Short Run changes First assume K is fixed What is the effect of a change in w dHL PQLLdL dw 0 dL 1 mmlt0 Now suppose K is not fixed then dL PQKK PQKK lt 0 dW PZQLLQKK PZQLK The inequality holds Since gt 0 and QKK lt 0 S P PQLL QEK QKK PQLL CONSTANT RETURNS TO SCALE Constant QTK TL TQK L 52ng to Q is homogeneous of Degree 1 in K and L If there are constant returns to scale then the hessian of second derivatives is negative semidefinite We first take the derivative of QTK TL 2 TQK L with respect to K and get TQKTK TL 2 TQKK L or TQKTK TL 2 QKK L That is the first derivative of a function homogeneous of degree 1 in K and L is itself homogeneous of degree 0 in K and L This is just a special case of Euler s Theorem Next take derivative of both sides of QTK TL 2 TQK L with respect to T KQKTK TL LQLTKT L KQKK L LQLK L QK L Equivalently PKQK PLQL PQK L But by the first order conditions from profit maximization note that when there are constant returns to scale the first order conditions will give us ratios but not amounts of K and L we have the following PKQK iK and PLQL Lw Therefore Cost 2 iK wL PKQK PLQL PQK L 2 Revenue That is there are zero profits when there are constant returns to scale Note that the above partial notation is somewhat sloppy QK in TQKTK TL really means the derivative of Q with respect to the first argument TK which is denoted by QK not Q1 for mnemonic purposes the derivative of the argument TK with respect to K is then T Please note that Euler39s theorem says that if the function is homogeneous of degree 1 with respect to certain variables then the derivatives of the function are homogeneous of degree 0 with respect to the same variables and vice versa Do not conflate this with the homogeniety discussed earlier Earlier we showed that the first order conditions with respect to K and L were homogeneous of degree 0 with respect to P w and i not with respect to K and L Inspection of the profit equation then showed that the maximum profit equation was homogeneous of degree 1 with respect to P w and i However this homogeniety would not in general be true if the firm were not maximizing profits 2 PROFIT FUNCTIONS AND ENVELOPE THEOREMS Envelope Theorems for profit maximization The effect of an exogenous change on the objective function assuming profit maximization Exogenous change in P dH 2 1391de HLdL HKdK But by the First Order Conditions HLdL0 and HKdK0 Hence at Profit Maximization Point dH H dP P Total Derivative 2 Partial Derivative H PQK L Lw Ki Then 11 13 Up Q Hotellings Lemma Next we consider the effect of an exogneous change in w on profits HI HKdK HLdL dew dH EHW L dH EHi K Sometimes students get confused between comparative statics and the envelope theorem The envelope theorem deals with the objective function while comparative statics deal with the first order condions Furthermore under the envelope theorem the total derivative of the maximized function equals the partial of the maximized function while under the implicit function theorem the partial of the expicit function equals locally the total derivative of the implicit function The envelope theorem gives the effect of an exogenous change on the maximized value of the objective function PROFIT FUNCTION The first order conditions make K and L implicit functions of P w and i Suppose that we solved for these variables explicitly and then plugged them into our original profit equations Then we would have a profit function which was solely a function of the exogenous variables This profit function would give us the maximum profit for any set of w i and P HW i P 11 profit function The profit function contains the same information as the envelope theorems for profit maximization There profit is a function of the endogenous variables K and L but they are constrained to be the profit maximizing amounts by the first order conditions The profit function has the first order condition inherent in the function itself Thus the envelope theorems hold for the profit function since the profit function is the maximized value of the objective function The profit function is increasing in P H Q H is decreas1ng in factor pr1ces HW LvviP Hf KW39LP Another way of viewing this is that in the envelope theorem we take the total derivative of profits with respect to w for example The K T conditions imply that the total derivative equals the marginal derivative For the profit function we immediately find the partial derivative Some students confuse maximizing profits with the profit function One does not find the first order conditions for the profit function since they are already built in Furthermore the profit function is an explicit function of the exogenous variables w i and P while the maximizing profit equation is a function of the endogenous variables K and L The ideas in the previous paragraphs can be solidified by considering a particular example Suppose that Q 2 Log K1 2 Log L 1 then HPlogK1210gL1 Ki wL and the first order conditions for an interior maximum are PK 1 i 0 2PL1 w0 Equivalently K Pi 1 and L ZPw 1 Plugging these explicit functions back into the profit equation we get 11 PlogPi Zlog2Pw iPi w2Pw This function is stating the same thing as H subject to the first order conditions PROPERTIES OF Hw i P Note that 11 is a function of the exogenous variables Properties of Us 1 H is continuous in P and w 1 2 H is non decreas1ng in P non 1ncreas1ng in w 1 3 H is convex in P w and 1 4 11 is homogeneous of degree 1 in P w i 1 Follows from the Theorem of the Maximum QK L is twice differentiable by assumption and K and L are just variables Therefore Q K and L are continuous H PQ Ki Lw Therefore H is continuous in P i and w The Theorem of the Maximum states that if a function is continuous with a compact range and the constraint set is a non empty compact valued continuous coorespondence of A then the maximum of the function is a continuous function of A Here we have shown that continuity requirements for the Theorem of the Maximum are satisfied 2 Follows from the envelope theorem See Hotelling39s lemma and the related derivationsonpage 1311 LwiP s 0 H KwiPs 0 H Q 2 0 3 I will use mathematics to prove convex in w and 1 One should use a 3 by 3 determinant to show that convex for w i and P together but this is a hopeless complex task in this case For the more general argument see the diagrams following the mathematical proof 1 is convex in w and i if and only if the following determinant is positive semi definite HWW I IVV1 iw ii We knowthat n LwiP n KwiP gtk SO ww Hvvi Lw Li H 11 K K 1W 11 W 1 It is readily demonstrated that Liv 19ng Hi gt Htvi HTW LE Ktv Therefore H is convex in w and 1 This can also be illustrated Via the following diagram HP PY WX H is concave in K L but i and w have H inverse relation to K L respectively so H is convex in i w HW Looking at the first diagram start with a given P and maximize profits this is the tangency point If P changes but K L and Q remain the same then profits change in a linear way just look at the equation for profits But 11 is maximal profits and must not be below this straightline of profits Therefore 11 is locally convex 4 11 is homogeneous of degree 1 in P w i We have already shown on page 4 that the first order conditions are homogeneous of degree 0 in P w and i and that H is homogeneous of degree 1 in P w and i We will brie y sketch and review the proof here HPQKL Lw Ki If P w and i multiplied by T then Q L and K remain the same by the first order conditions Therefore HTP Tw Ti TPQ LTw KTi THP w i Note well that the profit function HP W i is a function of the exogenous variables P w and i while HK L is a function of K and L OVERVIEW Starting with a production function with a negative definite determinant of second order partials we have shown that there exists an associated profit function with 4 charactersitics We could have reversed the process Starting with a profit function that has these 4 characterstics we could have derived an associated production function that had a negative definite determinant of second order partials PROPERTIES OF Lw i P Kw i P Qw i P These are obviously closely related to the properties of H I present them here because this is the approach used by Mas Colell et al 1 L K and Q are homogeneous of degree 0 2 The matrix of derivatives M LW Liquot LP Kwquot K1quot Kpquot Qw Qi Q1 is symmetric and positive semidefinite That is quantities respond in the same direction as price price of output increases supply increases price of input increases demand decreases 3 Mw i P 0 That is the matrix of derivatives times the price vector 2 0 PROOF 1 Since H is homogeneous of degree 1 its derivatives are homogenous of degree 0 Proof If H is homogeneous of degree 1 then by definition T1Hw i P HTWT i TP Taking the derivative of this equation with respect to w we get T Hmw i P T nwrwr i TP Equivalently Tonmw i P HtVTwT i TP We derived these results earlier via a different method by appealing to the first order conditions of profit maximization see the section on implicit and explicit demand functions We then went on to derive properties of the profit function Now given the properties of the profit function we derive properties of its derivatives including homogeniety of degreee 0 for L and K So we have come full circle 2 M is the matrix of second derivatives of H By Young39s theorem the matrix is symmetric By convexity of H the matrix of second derivatives is positive semidefinite 3 This is just a special case of Euler39s theorem From 1 we have T0 Htvw i P HtVTwT i TP Equivalently Lw i P LTwT i TP Differentiating both sides with respect to T and evaluating at T 1 we get 0 w Liv i LiPLP HOMEWORK 1 In order for WAIBFC to be a profit function what restrictions are there on A and B and C W Wage I 2 Interest Rate P 2 Price of Good Find LP iw Show that the profit function is continuous Hint If a function is differentiable it is continuous but not necessarily vice versa 2 Suppose QK L K L and the firm is competitive in input and ouput markets A Derive Profit maximizing relations B What are second order conditions C Find the slope of the isoquant D Show that maximal H is homogeneous of degree 1 in P i and w 3 A monopolist has the following demand curve P A BQ A B gt 0 1 2 The production function is QK L K314 The exogenous variables are A B C w and i A Find FOC SOC B Find the effect of an increase in w on Q C Find the effect of an increase in w on H D Write out the profit function H explicitly Remember that a profit function is a function of the exogenous variables only 4 A monopolist has demand curve P DQ D Q lt 0 The production function is QK L where Q is strictly concave A Find FOC SOC Are there additional restrictions on D that insure that the SOC hold Only answer this yes or no C Find the effect of an increase in w on H 3 COST MINIMIZATION Minimize C Ki Lw subject to QK L 2 a The solution can be characterized by introducing the following Lagrange multiplier form Objective quotMinquot CK LL Ki Lw AQK L 6 Function Note that I use the word quotMinquot in quotes We are really not finding a minimum but rather a saddle point Min just reminds us that we started with a minimization problem Kuhn Tucker Conditions KT EKi AQK20 K20 Conditions As always it is useful to interpret these first order condition and put them into words First take the derivative of N with respect to a and get A marginal cost This technique will be further developed when we get to the envelope theorem Now for the perfectly competitive firm price equals marginal cost so A price and AQK the marginal revenue product of capital Hence the first constraint states that the interest rate is greater than or equal to the marginal revenue product of capital And if it is strictly greater than then no capital will be used Now this statement is sometimes bothersome to students because in their undergraduate classes the marginal revenue product of capital was less than the interest rate and the firm increased its use of capital until the marginal revenue product equaled the interest rate Here as in all programming the relationship is reversed because the lagrange multiplier is the shadow value or opportunity cost and the opportunity cost is the highest valued use We will now take a brief diversion in order to gain an appreciation and understanding of the Lagrange multiplier technique From the Kuhn Tucker conditions we know that EA A A6 Q K L 0 Looking back at quotMinquot CK Lt Ki Lw AQK L 6 all that we are doing is adding a zero term to the original minimization problem Minimize C Ki Lw While the constraint QK L 2 from the original minimization problem is now found in the Kuhn Tucker conditions Now back to the main event Once again more relationships can be teased out by considering ratios Implif d If K Lgt 0 and i w gt 0 then Q K L Relations QL W Again the first order conditions imply homogeniety FOC are homogeneous of degree 0 in i w and A If double all 3 variables Homogeneity nothing is changed In particular K L and Q remain the same C is homogeneous of degree 1 in i w and A But can drop A as A QK L 3 0 From FOC K and L are implicit functions of 3 i and w Implicit Functions Their explicit solution is denoted by Li w Q and Ki w Q SUFFICIENT CONDITIONS We next find the sufficient conditions for a minimum when there are constraints There are 2 different methods of dealing with SUFFICIENT conditions when there are constraints A Look at objective function and constraints separately CK L Ki Lw is convex in K L CKi CLw CKchKchLLzo CKK CKL 390 0 Hence objective function positive ClK CLL 0 0 semidefinite and thus C is convex QK L QKK QKL Assumed to be negative definite QLK QLL therefore concave Since 3 gt 0 there is an interior point and Arrow Enthoven conditions satisfied Thus the Kuhn Tucker conditions are necessary and sufficient for a global minimum See the Mathematical Background section for a general presentation of these conditions This approach is much simpler than the bordered Hessian method which involves 3 by 3 matrices However the bordered Hessian method is often used in establishing the second order conditions and in comparative statics analysis There is a very close relationship between these two methods Wittman Lemma If the objective function is positive semidefinite and the constraints are negative semidefinite with one or the other being definite and not just semidefinite and the first derivatives of the constraints are not all zero then the Bordered Hessian condition will be satisfied The proof relies on the property that the addition of positive sernidefinite determinants is positive sernidefinite If everything were sernidefinite then the determinant would be zero B Bordered Hessian approach to showing that second order conditions are satisfied Do not confuse this with the test of quasi concavity which puts a border of first partials Here the border is composed of the constraints although in this example they will look similar 3M 0 CKK quotQKK 0 CK CAL CKA CKK CKL Cm CLK CLL 0 QK 2 QK AQKK QL AQLK AK QK LL QLL 0 QK QK QKK QL QLK QL AQKL AQLL CAL QL CKL quotQKL QL 39QKL 39QLL If this 3 X 3 determinant is negative then a constrained minimum Note that each border flips sign That is unconstrained 3x3 minimum would be positive but here we have 1 constraint plus two variables so reversed If we had two constraints plus two variables then 4 X 4 would be positive Note that with 0 in the upper corner the 1x1 is always 0 and the 2x2 never gt 0 Note also that a strict negative is only a sufficient condition but not necessary We divide the second and third columns by A and then multiply the first row by and then to keep things unchanged we multiply the determinant by A 0 QK QL L QK QKK QKL tH QL QLK QLL We know that Q is concave Therefore Q is also quasiconcave and thus the determinant is positive making J H s 0 It is useful to demonstrate that J H s 0 via a different method This method is really derivative of the previous method but some people approach this way Multiplying out we get 0 QK QL 0 QK AH QK AQKK AQKL QK AQKK QL AQLK AQLL QL AQLK QKQL QKL QKQLAQLK AQLQLQKK AQKQKQLL 39MZQKQLQKL 39 QLQLQKK 39QKQKQLL Note that Q lt 0 Therefore Q gt 0 We will show that the above expression is negative by demonstrating that the expression in brackets is positive The following term will be shown to resemble but not exactly so the expression in brackets I QLQKK39 5 QK QLL 5 r gt 0 since we have a squared term Note that we are finding the square root of Qii which is a positive term 2 2QKQL p KKQ LL 39 QLQLQKK 39 QKQKQLLl By assumption the Hessian of Q is negative definite Therefore QKKQLL gt Q L and 39 2 IQKKQLL lt 39 IZQKLI Therefore 0 lt ZQKQL KKQLL QLQLQKK QKQKQLL lt ZQKQLQKL 39 QLQLQKK 39 QKQKQLL This method of squares is often used to sign a determinant Comparative COMPARATIVE STATICS Statics Both before and after exogenous change firm is in cost minimizing equilibrium 6 deK NEALdL de 5de 0 dNK KKdK KLdL mdm Kde 0 dNL 6LKdK LLdL 6mm 6Lwdw0 d A QKdK QLdL 0d oaw 0 d5 K gt QKKdK AQKLdL Qde 0dw o dN L gtQLKdK AQLLdL Qde dw 0 Using Cramer s Rule we find the effect of a change in w on K dK dw 0 0 QK 0 QL dw QL AQKL AQLL dW QL QK Conditional factor demand If two factors must be substitutes
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