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# MICROECONOMIC THEORY ECON 323

Texas A&M

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This 8 page Class Notes was uploaded by Mr. Tito Legros on Wednesday October 21, 2015. The Class Notes belongs to ECON 323 at Texas A&M University taught by T. Turocy in Fall. Since its upload, it has received 12 views. For similar materials see /class/225832/econ-323-texas-a-m-university in Economcs at Texas A&M University.

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Date Created: 10/21/15

ECON 323 Lecture Notes on Individual Choice Based on Lecture of Monday January 26 These notes anticipate the material in Perloff chapters 4 and 5 You should read or review these before reading or reviewing the text chapters 1 A fence problem Suppose you have a xed length of 100m of fencing and you want to enclose the largest rectangle of land possible What shape should you use You may know the answer is a square either from personal experience or because this is a standard example in math courses Let s see how to solve this problem using economic principles To make terminology simple suppose we will lay out our rectangle to match the cardinal directions We can think of this problem as allocating our xed budget of 100m of fencing between two goods the length in the northsouth dimension N and the length in the eastwest dimension E Don t forget that if we allocate one foot in a direction we use two feet of fencing So our lengths N and E will satisfy 2N 2E 100 orNE 50 The area ofthe rectangle is then given by N gtltE In most math courses you ve seen this called the objective function we will call it a utility function Allocating fencing to the N or E directions is costly Remember that the economic concept of cost arises whenever we deal with any resource that is in xed supply that is a scarce resource Recall the mantra that All costs are opportunity costs In our case the cost of allocating a foot towards the N side of our rectangle is allocating a foot towards the E side The price of one foot of N is one foot of E and the price of one foot of E is one foot of N We might start by considering an enclosure that is N gtltE l gtlt 49 for an area of 49 This is horribly suboptimal it s easy to see that a 2 gtlt 48 enclosure would give an area of 96 Let s look at the underlying economics of why 1 gtlt 49 is a poor choice for this problem To do this think about the last foot that was allocated First take away the last foot in the N direction an enclosure that is 0 gtlt 49 has area 0 Therefore the last foot in the N direction is adding 49 7 0 49 to the area Now take away the last foot in the E direction an anclosure that is l gtlt 48 has area 48 so the last foot in the E direction adds 49 7 48 l to the area The last foot in the N direction is adding much more to the area than the last foot in the E direction What this tells you is that you would be better off if you allocated another foot away from the E direction and towards the N direction If you repeat this exercise gradually making the sides of the rectangle closer to equal in length you will nd the same pattern For instance a 15 gtlt 35 rectangle has area 525 The contribution of the last foot in each direction is not equal though A 14 gtlt 35 rectangle has area 490 meaning the last foot in the N direction contributes 525 7 490 35 to the area but a 15 gtlt 34 rectangle has area 510 so the last foot in the E direction adds 525 7 510 15 to the area The size ofthe difference in contributions is smaller than before but they are still not equal so this is not an optimal solution Now consider the 25 gtlt 25 square with area 625 The areas of the 24 gtlt 25 rectangle and the 25 gtlt 24 rectangle are the same at 600 So the last foot in each direction contributes 25 to the area We therefore have the relationship that contribution of last foot in directionN 7 price ofN contribution of last foot in directionE 7 price ofE which can be rearranged to read contribution of last foot in directionN 7 contribution of last foot in directionE price ofN price ofE The intuition behind this equation is simple and is best seen by thinking about what happens if it s not true If the left side is bigger than the right side then we are geting more bang for our buck with the last foot in the N direction than the E direction That means our last foot in direction E is wasteful in that we could get more if we used it in the N direction If the right side is bigger than the left side then the last foot in the E direction is giving us more value than the last foor in the N direction which tells us we should reallocate some length currently used in the N direction and use it in the E direction instead 2 Allocating study time optimally 21 Two courses with equal weight Suppose you have exams coming up in two courses X and Y In both courses you are sure that you can do well enough on the exam to wind up with a B in the course even if you don t do any more studying Your chances of getting an A in the course go up the more you study for the course Suppose you have 10 hours of time available to study which you can allocate between the courses If you spend x hours studying for course X your chances of earning an A are VIZ similarly if you spend y hours studying for course Y your chances of earning an A are The square root function captures the idea that studying a little bit is very helpful to your chances of getting the better grade however as you study more the rate at which more studying helps your chances goes down This could be because studying a lot on one topic results in boredom and fatigue or that there is always some chance there s going to be an exam question that you don t expect and it s unlikely in studying more that you ll happen to prepare for that question Under these assumptions if both courses are weighted equally in your GPA then your expected GPA if you study x hours for course X and y hours for course Y is l x l y uxy351 51 That is the utility function is the expected GPA How should you allocate 10 hours of study time The cost of studying an hour for X is one hour studying for Y and vice versa so the ratio of the prices of x and y is 11 We expect because the problem is symmetric between the courses that x y 5 is the optimal solution To verify this is true observe that u55 37071 to four decimal places The last hour spent on each course has the same marginal value because u54 u4 5 36698 So the last hour spent on each course adds 0373 to your GPA 22 Courses with different credit amounts To make it more interesting suppose instead that course X is a threecredit course and Y only a onecredit course Keeping the same assumptions about how studying affects your chances of getting an A your expected GPA is 3 x 1 y uxy311 11 You d now expect that you d want to spend more time on course X Let s rst look at the allocation x y 5 You d still get an expected GPA of u5 5 3 7071 but the marginals are now different To compute the value of the last hour spent on X we compute u4 5 36511 so the value of the last hour onX is 37071 7 36511 0560 For the value ofthe last hour spent on Y compute u54 36884 and so we get 37071 7 36884 0187 If you studied 5 hours on each course you d be getting a lot more value out of the last hour spent onX than on Y that tells you that your allocation is not optimal and that you should be spending more time on X You can play around with various combinations but it turns out that the optimal allocation is x 9 and y 1 Let s try to verify that First we have that u9 1 37906 To compute the marginal value of studying for X we compute u8 1 37499 and u90 37115 So we get the marginal value of studyingX to be 37906 7 37499 0407 while the marginal value of studying Y is 37906 7 37115 0791 These aren t equal What happened is that we computed these using whole hours In actuality the marginals are the value of the last little bit of studying done If we instead look at fractions of an hour we will get the results we expect For example let s work in units of 1100 of an hour Then the marginal value of studying forX can be estimated by computing u899 1 37902 which gives a marginal value of 0004 for the last 1100 of an hour spent studying on X The marginal value of studying for Y is estimated by computing u9 99 37902 which also gives a marginal value of 0004 So remember that the marginal value is the value of the last little bit In this course we ll generally assume in consumer theory that one can divide consumption into arbitrarily small bits 23 A different time constraint Now let s return to the scenario in Section 21 where both courses have the same weight but let s instead suppose that you promised your friend that you would study together on course Y When you study with your friend you spend about 50 of the total time actually studying and 50 of the time goo ng off So for every hour you spend on Y you actually only get onehalf hour s worth of studying in Therefore your opportunity cost of one hour of studying X is now one halfhour of studying Y Let s search for the optimum allocation Hours onX Hours on Y GPA 50 25 34326 60 20 36109 70 15 36120 80 10 35590 Evidently the optimum allocation is between 6 and 7 hours on X with a little more searching you d nd the optimum is to spend 6 hours onX and 1 hours on Y with 1 hours wasted goo ng 3 off for an expected GPA of 36124 Studying for Y is now more expensive relative to the example in Section 21 therefore you consume less of it Let s now examine the marginal contribution of the last bit of time spent on both courses We have 2 2 u 6717 36124 3 3 2 2 ult6770117 36093 3 3 2 2 u 6717701 36062 3 3 So the value of last hundredth of an hour spent on X is about 36124 7 36093 0031 and the value ofthe last hundredth of an hour spent on Y is about 36124 7 36062 0060 That is the value of the last bit of time spent studying for Y is twice that of X But remember that the price of two units of time studying for Y is one unit of time studying for X So the ratio of the marginal values is the same as the ratio of the prices ECON 323 Lecture Notes on Individual Choice Based on Lecture of Monday January 26 These notes anticipate the material in Perloff chapters 4 and 5 You should read or review these before reading or reviewing the text chapters 1 A fence problem Suppose you have a xed length of 100m of fencing and you want to enclose the largest rectangle of land possible What shape should you use You may know the answer is a square either from personal experience or because this is a standard example in math courses Let s see how to solve this problem using economic principles To make terminology simple suppose we will lay out our rectangle to match the cardinal directions We can think of this problem as allocating our xed budget of 100m of fencing between two goods the length in the northsouth dimension N and the length in the eastwest dimension E Don t forget that if we allocate one foot in a direction we use two feet of fencing So our lengths N and E will satisfy 2N 2E 100 orNE 50 The area ofthe rectangle is then given by N gtltE In most math courses you ve seen this called the objective function we will call it a utility function Allocating fencing to the N or E directions is costly Remember that the economic concept of cost arises whenever we deal with any resource that is in xed supply that is a scarce resource Recall the mantra that All costs are opportunity costs In our case the cost of allocating a foot towards the N side of our rectangle is allocating a foot towards the E side The price of one foot of N is one foot of E and the price of one foot of E is one foot of N We might start by considering an enclosure that is N gtltE l gtlt 49 for an area of 49 This is horribly suboptimal it s easy to see that a 2 gtlt 48 enclosure would give an area of 96 Let s look at the underlying economics of why 1 gtlt 49 is a poor choice for this problem To do this think about the last foot that was allocated First take away the last foot in the N direction an enclosure that is 0 gtlt 49 has area 0 Therefore the last foot in the N direction is adding 49 7 0 49 to the area Now take away the last foot in the E direction an anclosure that is l gtlt 48 has area 48 so the last foot in the E direction adds 49 7 48 l to the area The last foot in the N direction is adding much more to the area than the last foot in the E direction What this tells you is that you would be better off if you allocated another foot away from the E direction and towards the N direction If you repeat this exercise gradually making the sides of the rectangle closer to equal in length you will nd the same pattern For instance a 15 gtlt 35 rectangle has area 525 The contribution of the last foot in each direction is not equal though A 14 gtlt 35 rectangle has area 490 meaning the last foot in the N direction contributes 525 7 490 35 to the area but a 15 gtlt 34 rectangle has area 510 so the last foot in the E direction adds 525 7 510 15 to the area The size ofthe difference in contributions is smaller than before but they are still not equal so this is not an optimal solution Now consider the 25 gtlt 25 square with area 625 The areas of the 24 gtlt 25 rectangle and the 25 gtlt 24 rectangle are the same at 600 So the last foot in each direction contributes 25 to the area We therefore have the relationship that contribution of last foot in directionN 7 price ofN contribution of last foot in directionE 7 price ofE which can be rearranged to read contribution of last foot in directionN 7 contribution of last foot in directionE price ofN price ofE The intuition behind this equation is simple and is best seen by thinking about what happens if it s not true If the left side is bigger than the right side then we are geting more bang for our buck with the last foot in the N direction than the E direction That means our last foot in direction E is wasteful in that we could get more if we used it in the N direction If the right side is bigger than the left side then the last foot in the E direction is giving us more value than the last foor in the N direction which tells us we should reallocate some length currently used in the N direction and use it in the E direction instead 2 Allocating study time optimally 21 Two courses with equal weight Suppose you have exams coming up in two courses X and Y In both courses you are sure that you can do well enough on the exam to wind up with a B in the course even if you don t do any more studying Your chances of getting an A in the course go up the more you study for the course Suppose you have 10 hours of time available to study which you can allocate between the courses If you spend x hours studying for course X your chances of earning an A are VIZ similarly if you spend y hours studying for course Y your chances of earning an A are The square root function captures the idea that studying a little bit is very helpful to your chances of getting the better grade however as you study more the rate at which more studying helps your chances goes down This could be because studying a lot on one topic results in boredom and fatigue or that there is always some chance there s going to be an exam question that you don t expect and it s unlikely in studying more that you ll happen to prepare for that question Under these assumptions if both courses are weighted equally in your GPA then your expected GPA if you study x hours for course X and y hours for course Y is l x l y uxy351 51 That is the utility function is the expected GPA How should you allocate 10 hours of study time The cost of studying an hour for X is one hour studying for Y and vice versa so the ratio of the prices of x and y is 11 We expect because the problem is symmetric between the courses that x y 5 is the optimal solution To verify this is true observe that u55 37071 to four decimal places The last hour spent on each course has the same marginal value because u54 u4 5 36698 So the last hour spent on each course adds 0373 to your GPA 22 Courses with different credit amounts To make it more interesting suppose instead that course X is a threecredit course and Y only a onecredit course Keeping the same assumptions about how studying affects your chances of getting an A your expected GPA is 3 x 1 y uxy311 11 You d now expect that you d want to spend more time on course X Let s rst look at the allocation x y 5 You d still get an expected GPA of u5 5 3 7071 but the marginals are now different To compute the value of the last hour spent on X we compute u4 5 36511 so the value of the last hour onX is 37071 7 36511 0560 For the value ofthe last hour spent on Y compute u54 36884 and so we get 37071 7 36884 0187 If you studied 5 hours on each course you d be getting a lot more value out of the last hour spent onX than on Y that tells you that your allocation is not optimal and that you should be spending more time on X You can play around with various combinations but it turns out that the optimal allocation is x 9 and y 1 Let s try to verify that First we have that u9 1 37906 To compute the marginal value of studying for X we compute u8 1 37499 and u90 37115 So we get the marginal value of studyingX to be 37906 7 37499 0407 while the marginal value of studying Y is 37906 7 37115 0791 These aren t equal What happened is that we computed these using whole hours In actuality the marginals are the value of the last little bit of studying done If we instead look at fractions of an hour we will get the results we expect For example let s work in units of 1100 of an hour Then the marginal value of studying forX can be estimated by computing u899 1 37902 which gives a marginal value of 0004 for the last 1100 of an hour spent studying on X The marginal value of studying for Y is estimated by computing u9 99 37902 which also gives a marginal value of 0004 So remember that the marginal value is the value of the last little bit In this course we ll generally assume in consumer theory that one can divide consumption into arbitrarily small bits 23 A different time constraint Now let s return to the scenario in Section 21 where both courses have the same weight but let s instead suppose that you promised your friend that you would study together on course Y When you study with your friend you spend about 50 of the total time actually studying and 50 of the time goo ng off So for every hour you spend on Y you actually only get onehalf hour s worth of studying in Therefore your opportunity cost of one hour of studying X is now one halfhour of studying Y Let s search for the optimum allocation Hours onX Hours on Y GPA 50 25 34326 60 20 36109 70 15 36120 80 10 35590 Evidently the optimum allocation is between 6 and 7 hours on X with a little more searching you d nd the optimum is to spend 6 hours onX and 1 hours on Y with 1 hours wasted goo ng 3 off for an expected GPA of 36124 Studying for Y is now more expensive relative to the example in Section 21 therefore you consume less of it Let s now examine the marginal contribution of the last bit of time spent on both courses We have 2 2 u 6717 36124 3 3 2 2 ult6770117 36093 3 3 2 2 u 6717701 36062 3 3 So the value of last hundredth of an hour spent on X is about 36124 7 36093 0031 and the value ofthe last hundredth of an hour spent on Y is about 36124 7 36062 0060 That is the value of the last bit of time spent studying for Y is twice that of X But remember that the price of two units of time studying for Y is one unit of time studying for X So the ratio of the marginal values is the same as the ratio of the prices

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