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# Introduction to Mathematical Economics ECON 4808

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This 14 page Study Guide was uploaded by Bridie Batz on Friday October 30, 2015. The Study Guide belongs to ECON 4808 at University of Colorado at Boulder taught by William Mertens in Fall. Since its upload, it has received 24 views. For similar materials see /class/232137/econ-4808-university-of-colorado-at-boulder in Economcs at University of Colorado at Boulder.

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Date Created: 10/30/15

Econ 4808 Review Set 4 Answers ECONOMIC APPLICATIONS OF CONSTRAINED OPTIMIZATION 1 Assume that an individual s preferences can be described by the following utility function i u uxlx2am1 1190 where a and b are exogenous parameters Let y denote the consumer s income and let p1 and p2 denote the prices of goods I and 2 where y 131 and 132 are exogenous variables Assume that more is always preferred to less A What does the assumption that quotmore is always preferred to lessquot imply about the parameters a and b B Explain how you would go about deriving the individual s demand functions for the two goods Don t solve the problem rather explain in steps how you would solve the problem C Now try and derive the individual s demand function for good 2 D If you get a solution in part C discuss whether or not you are sure that you have found the demand function E What happens to the demand for good 2 if the price of good 1 increases Does it increase or decrease and by how much Answer A First the assumption that more is preferred to less implies that if the consumer is rational she will spend her entire income y on goods I and 2 Second more is preferred to less requires that the marginal utility of each good is always positive That is 811061962 0 am a gt and 811061962 1 fl 7b 2 0 8x2 2 2 gt So a and I must both be positive B The individual s demand functions for goods I and 2 are the solutions to the cone sumer s problem of maximizing utility subject to budget constraint l maxu x1 x2 1901 bx 1112 subject to y 131901 P2962 C First turn the consumer s constrained maximization problem into an unconstrained maximization problem in one variable7 mg by solving the budget constraint for 1 and substituting the result into the objective function 7 l maxu m2 a bx 2 P1 Look for critical points Bu 962 pg 1 fl if 5 2 8x2 p1 2 2 Set this equal to zero and solve for 2 The solution is 2 7 E72 2 7 2bp1 1132 2 71 27 2 lt 5131 i 292 m2 5231 i 1 1ltbp1gt2 m 7 if 2 221g 2 4 P2 17171 D This is possibly the demand function To see if utility is in fact maximized at this level of 27 check if 8211 m2 1 E T I 3 2 2 75 is negative When m Lil Since ii i 1071 gt lt 0 for all 962 gt 07 it is negative am 0172 96d BE 2 2 4 1132 E To determine What happens to 96 when 131 increases7 we have to look at the sign of at 30 so the demand function is the partial derivative of mg With respect to 131 a d 1 b 1 2 L2ltgtltgt 3P1 4 a P2 2 k a 2 That is demand for 2 increases by pig 131 when 131 marginally increases This means that goods 1 and 2 are substitutes 2 Assume that an individual s preferences can be described by the following utility function u x1 x2 x3 1301 17962 can where a b and c are exogenous parameters Let y denote the consumer s income and let 131132 and p3 denote the prices of goods 1 2 and 3 where y 131132 and p3 are exogenous variables Also assume that more is always preferred to less A What does this assumption imply about a b and c B Explain how you would go about deriving the individual s demand functions for the three goods Don t solve the problem rather explain in steps how you would solve the problem C Now solve the consumer s problem D If you get a solution in part C discuss whether or not you are sure that you have found the demand functions E What happens to the demand for good 1 if the price of good 3 increases Answer A More is preferred to less requires that the marginal utility of each good is always positive That is Bu m1 x2 x3 gt0 8961 a 8ux1x2m3 1 fl 7b 2 0 8mg 2 2 gt 8ux1x2m3 1 7 7 gt0 8mg 20mg So a b and 0 must all be positive B The individual s demand functions for the three goods are the solutions to the cone sumer s problem of maximizing utility subject to the budget constraint 1 l maxu m1 x2 x3 1301 bx cm 1112 subject to y 131961 132962 P3963 C Solve the budget constraint for one of the endogenous variables say 301 Solution is y P2962 Pawa P1 m1 Substitue for 1 in the utility function to get maxu m2 m3 1 1213 9 P2902 P3963 P1 Notice that we reduced the constrained optimization problem in three endogenus variables A l gt 1196 cm m1 m2 and 3 to an unconstrained optimization problem in only two endogenous variables m1 and 962 Look for critical points 811062963 7 a 7132 1 7 8x2 7 811062963 a 7133 1 7 8X3 Set each of these partials to zero and solve for 2 and 3 With this simple utility function each can be solved separately P2 1 71 a 7 4796 2 0 lt P1 gt 2 2 a lb P1 2 P2 2922 1 5231 m2 1 m Similarly7 the solution for 3 is 2 f 0P1 3 211P3 D These are possibly the demand functions for goods 2 and 3 To see if these critical values in fact maximize utility u x2 x3 check if the second derivatives are negative 8211 x2 x3 1 1 E 7 7 7 b 2 0 8x5 2 2 2 lt 8211 x2 x3 1 1 7 7 0 mg 2lt 2 3 lt They are negative and so the critical values as and m in fact maximize 11062963 and so they are the demand functions for goods 2 and 3 md 5P1 2 2 2ap2 0P1 2 d 7 x3 7 ZIP And by substituting with m and m in the budget constraint we obtain the demand function for good 1 d y P2962 P3963 m1 P1 7 1701 2 am 2 9 P2 2am P3 21173 P1 2 21 i 22 if y P2 2a 223 2a E To determine what happens to 9631 when p3 increases we have to look at the sign of the partial derivative of 31 with respect to p3 396 P1 0 2 72 i813 13 g 1P3 gt 0 That is demand for 3 increases by 3 ifpgg when 133 marginally increases This means that goods 1 and 3 are substitutes 3 Assume that an individual s preferences can be described by the following utility function U 9617 9627 x3 961962 963 Assume that the consumer s income is y and it is an exogenous variable Let y denote the consumer s income and let 131132 and p3 denote the prices of goods 1 2 and 3 where y 131132 and p3 are exogenous variables Derive the demand functions for the three goods Don t worry about checking the second order conditions for a maximum Answer We want to solve the constrained maximization problem maximize max 1 m1 x2 x3 1902 m3 11121 3 subject to y 131961 p2m2 p3m3 Solve the budget constraint for one of the endogenous variables say 303 I chose 963 because it is the additive term in the utility function Solution is y P1961 P2962 P3 963 Substitue for 3 in the utility function to get 9 P1961 P2962 maxu x1 x2 1962 1112 133 Notice that the objective function is now a function of only two endogenous variables m1 and 302 Look for critical points 811061962 7131 x 7 8961 2 233 811061962 7132 x 7 3962 1 P3 Set each of these partials to zero and solve for 1 and mg respectively With this simple utility function each can be solved separately The solution is 7 P1 722 7 i P3 mi 12 P3 And by substituting for 1 and 2 in the budget constraint we obtain the demand function for good 3 7 12 7 12 2 y m m P2 273 we P3 P3 If we wanted to check the second order conditions we would have 7 37 8211 x1 x2 7 0 8x 7 8211 x1 x2 7 0 8x5 7 So the critical values that we have found do not correspond to a maximum So we have not found the demand functions What is going on is that the maximization problem has a corner solution that is the utility function is maximized when everything is spent on good 3 The demand fuctions are 9610 m2 9037 4 Find the consumer s demand functions for good 1 and good 2 assuming that 11061962 96le Let y denote the consumer s income and let p1 and p2 denote the prices of goods 1 and 2 where y7 p1 and 132 are exogenous variables Answer We want to solve the constrained maximization problem maximize maxu x1 x2 96le 1112 subject to y 131901 P2962 where y denotes the individual s income Solve the budget constraint for one of the endogenous variables7 say 962 and plug the result into the objective function to obtain 9 le1gt maxu m1 m1 7 mm 232 Note that we have turn the constrained optimization in two endogenous variables into an unconstrained optimization problem in one endogenous variable The solution is the consumer s demand function for good 17 xii m1yp1p2 How do we nd the value of x xi 7 that maximizes utility subject to the constraints We nd the critical values for ml by setting the derivative of u 961 with respect to 1 equal to zero7 and solving for 301 erw Bum 8 T l y 7 2131961 0 8X1 8X1 132 The solution is y 96 7 1 2P1 Plugging this into the constraint7 we obtain the solution for x2 90 which is the value of 962 that maximizes the utility function i 2131 These are possibly the demand functions for good 1 and 2 If they are they say that the 1lt 122 individual should spend of his or her income on each good How would you make sure that mi and m are the quantities of 1 and 962 that maximize rather than minimize utility Look at the second derivative 8211 m1 7 2131 396 P2 We want to evaluate this at mi 2 Note that it is the same negative constant71 at every value of as What does this tell us That utility is indeed maximized at mi 2 So the individual s demand functions are d y m 7 1 2131 d y m 7 2 2P2 The individual will spend of his income on each good This should not surprise us given the symmetrical way the two goods enter the demand function 5 Find the consumer s demand functions assuming that the consumer s utility function 11061902 159625 Let y denote the consumer s income and let p1 and p2 denote the prices of goods 1 and 2 where y 131 and 132 are exogenous variables Answer Think about this utility function intuitively and compare it to u x1 x2 1962 from the previous problem Note that 11061902 90159625 m1x2395 is a monotonic transformation of 11061902 1962 and so they describe the same consumer prefernces Recall that the consumer preferences decribe the consumer s ranking of all possible bundles of different amounts of goods 1 and 2 Therefore if a bundle of goods 1 and 2 in amounts x1 x2 is preferred or ranked higher by the consumer with utility function u x1 x2 1962 it will also be preferred by the consumer with utility function u m1 m2 x1m2395 The two utility function yield different values but the same ranking of bundles So the two consumers will have the same demand functions m 1 2131 d y m 7 2 2132 Now let s verify and do the math x1 x2 m1x2395 subject to y 131901 P2962 where y denotes the individual s income First step turn the problem into an unconstrained problem in one variable Do this by solving the budget contraint for 2 and substituting the result into the objective function T y 7 mm 5 maxu m1 m1 7 1112 132 We want to maximize this without respect to 301 Look for a critical values for ml by setting the derivative of 11061 with respect to an equal to zero7 and solving for 1 0 2 5 8 yanrmxl 7 117217111 BMW l m 5 W1 723196 5 y 7 2pm K 272 8x1 8201 I P2 P2 I y11P115 172 The solution is f L 1 2131 This is the same critical value for 961 that we got in the previous problem when u x1 x2 1962 It makes sense The two utility functions represent the same preference ordering In this case7 we don t need to check to make sure utility is indeed maximized at mi 271 We already know that it is Pluging mi 271 into the constraint7 we obtain the demand function for x2 30 y 96 7 2 2P1 6 Assume that an individual s preferences can be described by the following utility function 11061962 1301 bxg where a and b are exogenous parameters Let y denote the consumer s income and let p1 and p2 denote the prices of goods 1 and 2 where y 131 and 132 are exogenous variables Assume more is always preferred to less Derive the individual s demand functions Explain in words what you are doing and the logic of what you are doing Use a graphs to make your argument Answer A More is preferred to less requires that the marginal utility of each good is always positive That is 811061962 a gt 0 8X1 and 811061962 b gt 0 8302 So a and I must both be positive The individual s demand functions for goods 1 and 2 are the solutions to the consumer s problem of maximizing utility subject to budget constraint maxu m1 m2 1301 bxg 1112 subject to y 131901 P2962 where y denotes the individual s income First turn the problem into an unconstrained maximization problem in one variable mg by solving the budget constraint for 1 and substituting the result into the objective function maxu m2 a bxg 2 P1 Then look for critical points W a 22 b 8962 p1 Note that in this case the derivative is not a function of mg and depending on the values of a b 132 and 131 it will always be positive or negative What does this mean It means that maximum utility is either always increaseing or always decreasing in 302 If a 7 17 lt 0 then the individual should consume no 962 and spend all of her income on 301 That is if 10 17273 b lt 0 ii 1 and ii 0 If 17273 b gt 0 then the individual should consume no 961 and spend all of their income on 302 That is if a 7 b gt 0 mi 0 and m 10 Mathematically speaking the individual is at a corner solution spending all of their income on one of the two goods the good that gives the most bang for the buck Think of it this way the individual gets 1 units of utility from each unit of 1 and if it costs 131 per unit the individual gets 10 units of utility for each dollar spent on 1 Likewise he always gets pig units of utility for each dollar spent on 902 He will spend all of his money on the good that gives larger utility Note that this individual s indifference curves are straight lines 2 ldu0 7 and the two goods are perfect substitues 7 Assume the Snerd Corporation produces product as using k and l where x f k l 12515 Further assume that the rm purchases labor and capital at prices w and r A Derive the rm s conditional demand function for labor lg lC x w r Note that the conditional demand function for labor identi es the amount of labor the rm will purchase to minimize the total cost of producing as given w and r You do not have to check the second order conditions for a maximum B Then derive the rm s conditional demand function for capital kg kc x w r C What happens to the conditional demand for capital if the price of labor increases D What is the rms cost function and why Answer A This is the production manager s problem mine l k wl rk subject to m k5l5 Note that there constant returns to scale in production Turn the constraint minimizae tion problem into unconstraint problem by solving the costraint for one of the endogenous variables say k as and substituting the result in the objective function m2 mline l wl r Look for critical points 8 l w 71 magl 2 0 Solution is w rm2l 2 2 E w l ix 1 l w Discard the negative one So the solution is l m r w Check the second order conditions for a minimum 8251 2 73 82 7 71 mo 72H gt 0 It is positive for any positive l including l So 5l is in fact minimized at l mg and therefore l is the conditional deman function T lcxwr x E B The conditional demand function for capital is x2 w kc T W x 7 Note that both of these are linear in output This follows from the fact that we assumed a production function that exhibits constant returns to scale C To determine what happens to the conditional demand for capital when the wage rate increases take the partial derivative aka 1 m 7 7 7 gt 0 8w 2 Mar So if w marginally increases holding m and r constant the rm s demand for capital will 1 2 D Determining the cost function Expenditures on inputs are by de nition the amount increase by spent on labor and the amount spent on capital that is e wl rk Costiminimizing expenditures to produce as given w and r are therefore 12 e wlC rkc r w wmq I i r901 I i w r Z xT w Z xT w Qmrw Cx7r7w 8 This problem has a lot of algebra Assume that a rm produces product as using l and k such that m koallia Where 0 lt 04 lt 1 Assume that the rm buys labor and capital at prices w and r Derive the rm s conditional demand functions for l and k Answer Solve the rm s expenditure minimization problem mine l7 k wl rk subject to m koallia Turn this into an unconstrained minimization problem by solving the constraint for k kw kw x71 i k newly wig and substituting the result in the objective function 1 xii mline l wl mcElT Look for critical values Bl 0471 1 w maul or a w 1 1 1 r arm 04 04 w 7 1 1 7 0t a3 1 a w Verify that e l is infact minimized at l by checking Whether the second derivative is positive 8250 1 0471 17144 W lalTWl 1 1 7 a lt7 farmili gt 0 a a So7 l is the conditional labor demand function 17047 0 Cwivrx 7 a w and by substituting this result in the expression for k interms ofl from the constraint7 we obtain the conditional capital demand function locil kc mglco 17047 an x x 7 a w A ail 17047 0 71 mocch 7 a w ALl liar 0 71 104 a 7 a w 71 17047 a a w

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