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Unformatted text preview: ◮ This is a (painfully) detailed example demonstrating the concepts we learned in Lecture 2. Note that when you try to replicate there results, your numbers may differ slightly due to rounding ◮ Let’s start with a utility function that specifies that your total happiness is simply the product of what you eat today and what you eat next period: U = C · C 1 ◮ Suppose that your income (endowment) today and next period are 10 and 1, respectively: ( Y , Y 1 ) = (10 , 1) ◮ Your total utility if you simply consume your endowment is therefore U = C · C 1 = 10 · 1 = 10 ◮ This can be rearranged to give us the equation for the indifference curve at your endowment point: C 1 = 10 / C ◮ This indifference curve shows all possible points at which you are equally happy (at which your total utility is 10) ◮ For example, it includes points (10,1), (5,2), (2.5,4), (1,10), etc ◮ We start at the endowment point 1 Lecture 2 Example RSM 332 C C 1 C 1 = 10 / C ; U 1 = 10 1 Lecture 2 Example RSM 332 ◮ The slope of the indifference curve at point 1 tells us how willing you are to give up your consumption today to get some extra consumption tomorrow ◮ The slope is called marginal rate of substitution and is equal to the negative of the ratio of two derivatives which in turn equals (1 + r i ): Slope = ∂ U /∂ C ∂ U /∂ C 1 = (1 + r i ) ◮ r i is your subjective rate of time preference, and you can think of it as ‘return’ earned by giving up some of your consumption today to get more consumption tomorrow ◮ In our case MRS = ∂ U /∂ C ∂ U /∂ C 1 = C 1 C ◮ At the endowment point 1, this slope and the rate of time preference are MRS = 1 10 = . 10, and therefore r i = . 90 = 90% ◮ This means that at point 1 you value your today’s consumption so little that you’re willing to accept a negative return on whatever you give up today ◮ Roughly speaking, you are willing to give up 1 unit of food today to get just 0.10 units of food next period, which is a return of (0 . 10 1) / 1 = 90% ◮ Even though you value today’s consumption so little, if there is no production and if no financial markets exist , you are stuck at point 1 Lecture 2 Example RSM 332 C C 1 C 1 = 10 / C ; U 1 = 10 1 Slope = C 1 / C = . 10; r i = . 90 1: C * = 10 . 0, C * 1 = 1 . 00, U 1 = 10 . 00 Lecture 2 Example RSM 332 ◮ Suppose now that there is a production opportunity such that any amount I you invest today grows to value F next period F = 2 · √ I ◮ You can think of it as building a plant that costs you I dollars today but produces income of 2 · √ I next period ◮ Now, given that you have income of 10 today, if you eat some amount C today that means you have 10 C left to invest ◮ So your consumption next period can be written as the original income of 1 plus income from investing C 1 = 1 + 2 · radicalbig 10 C ◮ This line defines the production opportunity set in our example Lecture 2 Example RSM 332 C C 1 C 1 = 10 / C ; U 1 = 10...
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This note was uploaded on 10/21/2011 for the course RSM 332 taught by Professor Raymondkan during the Spring '08 term at University of Toronto.
 Spring '08
 RAYMONDKAN

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