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Unformatted text preview: ) + 2 log(q2) from q1=0,10 q2=0,10
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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. Impact on due to a small change in minimum quantity of good 1:
⏟ From the KT conditions remember that
cannot be negative and that (with the current consumption set). Recall that
currently
. This result tells us that by raising the lower bound on consumption of good 1 (let’s say the new
constraint is
. ) that (and after optimal readjustment), the consumer’s utility either decreases or stays the same
and can’t go up. Raising the lower bound on consumption can’t make you better off. Here’s the intuition: before raising
the minimum quantity you may have been consuming a positive amount
if that was the case, raising the
minimum quantity by an infinitesimally small amount has no impact on your utility so that ⏟ . On the other hand, you may have been consuming nothing
if that was the case, you were at the optimal quantity of 0 (you
could’ve chosen a positive amount but didn’t) so that raising the minimum quantity by an infinitesimally small amount
will make you worse off so that
Impact on ⏟ . due to a small change in minimum quantity of good 2:
⏟ From the KT conditions remember that
cannot be negative and that (with the current consumption set). Recall that
currently
. This result tells us that by raising the lower bound on consumption of good 2 (let’s say the new
constraint is
. ) that (and after optimal readjustment), the consumer’s utility either decreases or stays the same
and can’t go up. Raising the lower bound on consumption can’t make you better off. Here’s the intuition: before raising
the minimum quantity you may have been consuming a positive amount
if that was the case, raising the
minimum quantity by an infinitesimally small amount has no impact on your utility so that ⏟ . On the other hand, you may have been consuming nothing
if that was the case, you were at the optimal quantity of 0 (you
could’ve chosen a positive amount but didn’t) so that raising the minimum quantity by an infinitesimally small amount
will make you worse off so that ⏟ . Already, without knowing the actual utility function equation or having solving the UMP, we have derived general results
valid for any consumer: more money can’t make you worse off, higher prices can’t make you better off, increasing the
minimum threshold can’t make you better off.
In fact, here’s another general result: suppose the government can raise the same amount of revenue from a lump sum
income tax or an excise tax on a good (when comparing tax schemes we have to make sure that the tax schemes are
“revenue equivalent”, i.e. the two schemes raise the same amount of tax revenues because otherwise we’d be
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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. comparing apples with oranges). Given that taxes will hurt consumers, the government will want to choose that tax
scheme which least hurts consumers. Can we establish a general rule about which tax scheme is better? Yes we can.
, Let’s assume the consumer has monotone preferences. She chooses
, (, ) ..⏟ by solving:
, , We want to compare the impact on utility due to a lump sum income tax versus a tax revenue equivalent excise tax on
(say) good 1. We know how: use the envelope theorem to gauge the impact on optimal utility due to a lump sum tax vs.
an excise tax on good 1:
(, [ ) . . ⏟ (Do you know why the change in ⏟ is the same as the change in ) Now, we want to compare: . ⏟ ⏟ ⏟ . .⏟ ⏟ Because the consumer has monotone preferences we know that
on your own?) which allows us to divide both sides by :
⏟ (we did proved this earlier – can you prove this .⏟ 13
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. What is the change in income? It’s so that we need to compare:
⏟ .⏟ Which side is smaller? There’s no way to tell yet. But wait, we haven’t used the fact that the two tax scheme...
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This note was uploaded on 03/20/2014 for the course ECON 204 taught by Professor Ajazhussain during the Fall '09 term at University of Toronto Toronto.
 Fall '09
 AJAZHUSSAIN
 Microeconomics, Utility

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