05-MS&amp;E-241-Demand-Theory-III-Lecture

# 05-MS&amp;amp;E-241-Demand-Theory-III-Lecture - MS&amp;E...

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MS&E 241: ECONOMIC ANALYSIS Thomas A. Weber 5. Demand Theory III Winter 2007 Stanford University Copyright © 2007 T.A. Weber All Rights Reserved -2- MS&E-241-Winter-2007-TAW AGENDA Choice with Time and Budget Constraints Measuring Welfare Changes Demand Aggregation Key Concepts to Remember

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-3- MS&E-241-Winter-2007-TAW In standard demand theory, the set of feasible consumption bundles x is determined solely by income or wealth constraint p.x = w (Walras’ law), where p is the price vector. In the real world, a consumer is also a worker and needs to allocate time to generating an income. In addition, much of consumption not only money but also time ( Æ opportunity cost), Eating Travel Shopping Surfing the Internet The more one works, the less time is available for consumption, including recreation. The tradeoff between allocating to work or consumption implies that there must be a monetary value to the consumer’s time. Hence, the price of consumption goods is augmented by the opportunity cost of the time it takes to consume them. CHOICE WITH TIME AND BUDGET CONSTRAINTS -4- MS&E-241-Winter-2007-TAW Consider a person who has a nonnegative wealth endowment of w 0 and who earns additional income by working L hours at a wage rate of p L > 0. There are N goods, and each good i takes time τ i to consume; let τ be the vector of consumption times. The time available for all work, leisure, or consumption activities is T > 0 . The consumer then solves the following (augmented) utility maximization problem: TIME AND BUDGET CONSTRAINTS: A SIMPLE MODEL (Ignored here, for simplicity) max u(x,L), s.t. p.x # w 0 +p L L, τ .x + L # T, and x,L $0 x,L -5- MS&E-241-Winter-2007-TAW First-Order Necessary Optimality Conditions: D x u(x,L) = 8 p + μ τ = 8 (p + ( μ / 8 ) τ ) D L = - 8 p L + μ 8 (p.x - w 0 -p L L) = 0 ( 8 : Lagrange multiplier associated with budget constraint) μ ( τ .x + L - T) = 0 ( μ : Lagrange multiplier associated with time constraint) Interpret V τ = μ / 8 = p L + ( M u/ M L)/ 8 as the monetary price of time (= opportunity cost) Optimal consumption quantities depend on p + V τ τ (full price of consumption) , w 0 + p L L (total wealth, including earned income), T - L (available leisure time) Either a high price or a high consumption time will reduce the demand for normal goods. Value of time equal to wage rate plus monetized pleasure (or minus displeasure) from working. High-wage-rate people can be expected to have a high value of time. TIME AND BUDGET CONSTRAINTS (Cont’d) -6- MS&E-241-Winter-2007-TAW Example . Consider a consumer who enjoys work, so that M u/ M L > 0. Suppose that s/he earns$50/hour and that the monetized pleasure from working is a constant $20/hour. (1) V τ = μ / 8 = p L + ( M u/ M L)/ 8 =$50 + $20 =$70 $110$70 1 $40 Meal at a Restaurant$65 $14 .2$50 Buy book online $280$280 4 0 Surf the Internet $75$35 .5 $40 Buy book at store$150 $140 2$10 Cook Meal Total Cost Time Cost Hours Price TOTAL COST OF CONSUMPTION: EXAMPLE 1 (1) The monetized pleasure from working will in general depend on exactly which bundle is consumed and is therefore not constant in general. We make the assumption of constancy only to be able to generate the table.

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## 05-MS&amp;amp;E-241-Demand-Theory-III-Lecture - MS&amp;E...

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