Unformatted text preview: s must raise
the same amount of revenues. Holding the income tax constant the excise tax per unit of good 1 must be chosen so that:
⏟ ⏟ ⏟ ⏟ Here,
excise tax per unit and
post excisetax quantity of good 1. Here is an example that shows why the
change in the price is equivalent to the excise tax per unit. Suppose the price of good 1 is initially $10. If the government
imposes a $0.50 quantity tax on good 1 the new price will be:
.
. . Returning to the tax comparison:
⏟ .⏟ .⏟ ⏟ ⏟ . ⏟ Which of these terms is smaller? The LHS is the posttax amount of good 1 and the RHS is the pretax amount of good 1.
Intuitively, the posttax quantity of good 1 must be lower (or the same) as the pretax quantity. That is: This implies that:
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ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. ⏟ ⏟ In sum, an income tax scheme “hurts” consumers less than (or equal to) an excise tax scheme designed to raise equal
amounts of tax revenue. Now that you’ve seen this can you examine whether it’s better to raise revenue by an excise
on good 1 vs. an excise tax on good 2?
We now show how the general UMP can be simplified if we know or assume that the consumer has monotone
preferences.
3. UMP with Monotone Preferences
Earlier we saw this monster UMP:
(, , ).. , ⏟, ⏟ If the utility function is differentiable we’d solve it by:
, , , ⏟( , , ) [ ⏟] [ [ ⏟] This KT problem required checking 8 separate cases. Not wishing to do this, we look a way to simplify the UMP. We now
, prove that if the consumer has monotone preferences (i.e. ) then she will spend her entire income so that the budget set is becomes an equality constraint:
following proof because there’s a high probability it will appear in a test or exam). (you should go through the Consider the FOCs and KT conditions of the general UMP:
, , , (, , [ ) ⏟ ⏟ , , [ 15 ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. , , , , Now, suppose the consumer has monotone preferences. In this case, the first two FOCs can be rearranged to yield: Recall that all pecuniary parameters are positive. From the KT conditions we see that
monotone preferences we have: and so that with That is, with monotone preferences: Now, above, we had these KT conditions:
,
Since we know that , [ the “animal” becomes:
⏟[ The only way for this “animal” to be zero is if: ⏟ [⏟ ⏟ With monotone preferences you must spend all your income (tell that to your parents next time you go clubbing and
blow $500 on bottle service). Why? Suppose you don’t spend all your income: 16
ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. (, 1 ,) Now, if you spent more money (which you have!) you’d be better off because with monotone preferences more of any
good will increase your utility: 2
1 (, ,) Feeling salubrious after spending money what do you do? You spend more money until you hit the budget constraint: 17
ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. 3 2 (, 1 ,) You don’t have any money left but who cares: you’re happier!
Now that we’ve understood why: We examine why: If you perceive all goods as “good” goods then you’ll spend your entire income. In this case, giving you more income lets
you buy more “good” goods which means you must be better off: (,
2 (, , )
,) 1 18
ECO 204 CHAPTER 3 Utility Maximization Problems (this version 20122013) University of Toronto, Department of Economics (STG). ECO 204, S. Ajaz Hussain. Do not distribute. In sum, if you’re given a UMP you should first setup the general UMP:
(, , ).. , ⏟, ⏟ ⏟ Then if the UMP is numerical you’d check if the consumer has monotone preferences and of the UMP is parametric
you’d impose conditions on the utility parameters that guarantee monotone preferences and then setup:
(, , ).....
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 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility, S. Ajaz Hussain, Sayed Ajaz Hussain

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