ajaz_eco204_2009_chapter_3.1

ajaz_eco204_2009_chapter_3.1 - University of Toronto...

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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain ECO 204 2009 2010 S. Ajaz Hussain (Draft) Chapter 3.1: One Period Consumption & Savings Model 1 Please help improve the course by sending me an e mail about typos or suggestions for improvements In chapter 3 we use the UMP (utility maximization problem) technique to construct two models of consumption and savings. In chapter 3.1 we model a one period decision of allocating (dollar) income between consumption (dollars) and savings (dollars); this model does not contain interest rates. In chapter 3.2, we model a two period (intertemporal) model for decisions of saving and borrowing over time. In contrast to the one period model, the intertemporal model uses interest rates and is the foundation of real interest rates in finance (which is why it’s done extensively in RSM 332). In this chapter, assume that savings are non negative, i.e. either zero or positive. Put simply, the assume that the consumer cannot borrow 2 . 1. One Period Model of Consumption and Savings In this section, we look at two models. In the first model the consumer has one option for savings ‐‐ think of it as the case where the consumer can save in one account 3 . In the second model, the consumer has more than one option for savings ‐‐ think of it as the case when the consumer can save in a regular savings account and/or a tax free savings account. 1.1 One Period Model of Consumption and one Option for Savings In previous chapters, we modeled a consumer’s optimal choice of goods 1 and 2 by the UMP: max ,ொ ܷൌ݂ሺܳ ሻ s.t. ܲ ܳ ൅ ܲ ܳ ൑ܻ 1 Thanks to: Vivian Shi, Betty Wang, Mary Magdaluyo 2 Note to self: for actual book, introduce borrowings, which requires using complex numbers for utilities. See BOOK version. 3 (ECO 209 makes a similar assumption: everyone in the economy lends to and borrows from the “loanable funds”). 1 ECO 204 (Draft) Chapter 3.1
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University of Toronto, Department of Economics, ECO 204 2009 2010 S. Ajaz Hussain Since we’re modeling the allocation of income ܻ (dollars) between consumption (dollars) and savings (dollars), it’s more convenient to use thi model: s max ௌ,஼ ܷൌ݂ሺܵ,ܥሻ s.t. ܵ ൅ ܥ ൌ ܻ Here ܵ is savings in dollars, ܥ is consumption in dollars and ܻ is income in dollars. Notice we don’t say “expenditure should be less than or equal to income” because by definition, dollars spent on consumption plus dollars “saved” must equal income (it’s an accounting identity). Observe also that the “budget line” has slope 1. To solve the consumption and savings UMP we need to indicate preferences: suppose the consumer has Cobb Douglas preferences over consumption and savings. Her UMP is: max ௌ,஼ ܷൌܵ ܥ s.t.
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ajaz_eco204_2009_chapter_3.1 - University of Toronto...

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