Lecture2_Detailed_Example

Lecture2_Detailed_Example - This is a (painfully) detailed...

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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 Lets 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 todays consumption so little that youre 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 todays 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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Lecture2_Detailed_Example - This is a (painfully) detailed...

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