PUBLIC ECONOMICS ECO 479
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This 5 page Class Notes was uploaded by Miss Annamarie Kovacek on Friday October 23, 2015. The Class Notes belongs to ECO 479 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/228265/eco-479-university-of-kentucky in Economcs at University of Kentucky.
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Date Created: 10/23/15
Notes from ECO 479 Lecture 2 8282007 amp 8302007 Preferences Indifference Curves Calculus Review Marginal Utility Marginal Rate of Substitution Budget Constraints 7 shifting and rotating Utility Maximization Demand Curves 1 Preferences and Indifference Curves note see the attached powerpoint slides from last semester for pictures etc a b 9 We use theoretical tools understand economic decision making They are primarily graphical and mathematical Constrained utility maximization means all decisions are made in order to maximize the utility of the individual subject to his or her available resources This involves preferences and budget constraints As illustrated in Figure 1 7 with movies and CD s 7 different bundles of goods might be available for the person to consume Bundle C in the figure 7 which gives at least as many movies and CD s ad Bundles A or B 7 must be preferred to those bundles because of non satiation more is preferred to less i Almost all goods satisfy this property Perhaps you can drink so much beer or eat so much pizza however that you do feel satiated 7 meaning you are worse off if you consume more of the good A utility function is a mathematical representation that translates the consumption of goods into utility i Note that the utility number itself simply ranks the different bundles 7 it doesn t say how much better one bundle is than another There are ways for representing the same preferences with what looks like different utility functions For example UXY and U10XY give the same rankings for all bundles This is called a monotonic transformation iii We derive indifference curves from utility functions Simply hold utility constant at some amount and the different combinations of goods that give this utility emerges For example with the following utility function 1 1 U QEJQCZD We could write the indifference curve as 2 QCD U 2QM3 For any fixed level of utility I7 If you can t figure out how this indifference curve was derived we simply moved QCD to the other side If you are comfortable with exponents this isn t hard 2 Calculus Review Specific Examples General Case y x y x dy dy 2x nxn71 dx dx y 2101c3 y kx dy dy 302 knx 1 dx dx Partial Derivative y x2z4 y xaz Qz42x iz 6azc H amp amp g4z3x2 We often use partial derivatives in our micro problems Here we are computing marginal utility l l 3 2 U Q M QCD i 1 MUM QCZ39D EJQMS 3 Marginal Utility and Marginal Rate of Substitution 9 Marginal Utility is the additional utility from consuming an infinitesimal amount more of some good We take the partial derivative of the utility function to compute it If the marginal utility of movies was equal to 5 then if we consume two additional movies our total utility goes up by 10 U Figure 3 in the powerpoints shows diminishing marginal utility 7 as we consume more of a good the additional happiness we get falls Note that utility continues to rise but just at a slower rate C Q 539 The Marginal Rate of Substitution MRS is the slope of the indifference curve We can derive a relationship between the MRS and Marginal Utility as illustrated in Figure 6 of the powerpoints In general AU MUMAQM MUCDAQCD The first term is the change in utility level The second term is the change in utility due to changing movie consumption And the third term is the change in utility due to changing CD consumption That is MU MA QM is multiplying the change in quantity by the utility change per unit Going back to the above example if the marginal utility of movies was equal to 5 and we consume two additional movies our total utility goes up by 10 Along an indi erence curve A U 0 by definition So consider small changes in movies and CD s that keep us on the same indifference curve Then we have 0 AU MUMAQM MUCDAQCD Ain Z A Q M M U CD The term on the left side is the slope 7 the MRS 7 and is related to the ratio of the marginal utilities 1 1 For the utility function U Q3 QgD the marginal utility of movies is 1 1 1 MU M QCZDEEJ Q M3 and the marginal utility of CD s is 1 1 1 MU CD QC The MRS is therefore wlm 1 1 1 2 7 dCD QCDE3 QM MRS7 1 dM 3 1 7 QM E QCD Which simplifies to 2 MRS amp 3QM quot7 Exponents Review In order to simply the above expression the following rules for manipulating exponents are helpful 4 Budget Constraints 7 shifting and rotating The budget constraint is represented by Y PM Q M PCDQCD where the first term is total income the second term is expenditure on movies and the third term is expenditure on CD s Figure 7 in the powerpoint slides shows the mechanics of the budget constraint 5 Utility Maximization We put preferences and budget constraints together for utility maximization and deriving demand curves 2 conditions must be met for utility maximization first the indifference curve is tangent to the budget constraint meaning MRSprice ratio and second that we spend all of our money In class we reviewed a bang per buck argument If MU M lt MU CD does that tell us that we should reallocate from movies to CD s No because we don t know how much movies cost and how much CD s cost What we need to do is normalize by price If MU M lt M U CD PM PCD Then reallocating 1 from movies to CD s will raise utility and is feasible given our budget Because of diminishing marginal utility as we spend more on CD s its marginal utility will fall while the marginal utility of movies will rise as we spend less on it When MUM MUCD PM PCD There is no possible reallocation of spending from movies to CD s or vice versa that can raise utility Note that the expression above is simply another way of writing the MRS equal to the price ratio 6 Demand Curves 1 1 Assume that U Q3 QCZD and that the price of movies is 1 the price of CD s is 7 and income is 1000 MU 2 P 1 First note that MRS 7M E 7M or 14QCD 3QM MUCD 3QM PCD In addition the budget constraint is QM 7QCD 1000 3000 35 14 By substituting we get QCD 7QCD 1000 or QCD Solving for movies is also straightforward
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