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Adjust income to get back household to original IC:

Defining key values of 𝑀:

BAU income, 𝐵𝐶 : 𝑀 Known: original incomePolicy income, 𝐵𝐶 : 𝑀 Known: income after policyCompensated income, 𝐵𝐶 : 𝑀 Need to compute

CV is the change needed to get to 𝑀 from 𝑀 :

𝐶𝑉 = 𝑀 − 𝑀

Last time we set up the CV calculation:

Computing a CV

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𝑄 (𝑀, 𝑃 , 𝑃 )𝑄 (𝑀, 𝑃 , 𝑃 )

Can use demand equations to find bundle:For any given M:1.

𝑈(𝑄 , 𝑄 )Can use utility function to determine the IC

For any given bundle:2.

To compute 𝑀 can use two key ideas:

Guess 𝑀 = 𝑀1.

Put 𝑀 and prices into demand equations to find 𝑄 and 𝑄2.

Compute 𝑈 = 𝑈(𝑄 , 𝑄 )3.

If 𝑈 < 𝑈 , add 1 to 𝑀 and go to step 24.

Could compute the CV iteratively:

Plug the demand equations into the utility function1.

Solve for M2.

Result is the expenditure function: 𝑀(𝑈, 𝑃 , 𝑃 )Gives the minimum M needed to hit the target U

Use expenditure function to compute 𝑀 = 𝑀(𝑈 , 𝑃 , 𝑃 )3.

Much easier and faster to do algebraically:

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Use expenditure function to compute 𝑀 = 𝑀(𝑈 , 𝑃 , 𝑃 )3.

Compute 𝐶𝑉 = 𝑀 − 𝑀4.

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𝑈 = (𝑄 ) 𝑄

𝑄 =𝑎𝑀

𝑃⎯⎯⎯

Utility and demand equations:

Start by deriving the expenditure function for Cobb-Douglas

𝑄 =(1 − 𝑎)𝑀

𝑃⎯⎯⎯⎯⎯⎯⎯⎯

𝑈 =𝑎𝑀

𝑃⎯⎯⎯

(1 − 𝑎)𝑀

𝑃⎯⎯⎯⎯⎯⎯⎯⎯

Step 1: plug demands into utility function

First factor out M:

Use property 1 of exponents: (𝑋𝑌) = 𝑋 𝑌

𝑈 =𝑎

𝑃⎯⎯ 𝑀

1 − 𝑎

𝑃⎯⎯⎯⎯⎯ 𝑀

Reorder multiplication: 𝐴𝐵 = 𝐵𝐴

Step 2: solve for M

Example: Cobb-Douglas Preferences

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𝑈 =𝑎

𝑃⎯⎯

1 − 𝑎

𝑃⎯⎯⎯⎯⎯ 𝑀 𝑀

Use property 2 of exponents: 𝑋 𝑋 = 𝑋

𝑈 =𝑎

𝑃⎯⎯

1 − 𝑎

𝑃⎯⎯⎯⎯⎯ 𝑀

𝑈 =𝑎

𝑃⎯⎯

1 − 𝑎

𝑃⎯⎯⎯⎯⎯ 𝑀

Utility as a function of M and prices rather than Q'sKnown as the indirect utility function:

Now solve for M:

𝑀 =𝑈

𝑎𝑃⎯⎯

1 − 𝑎𝑃

⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

𝑀 =𝑈

𝑎𝑃⎯⎯

1 − 𝑎𝑃

⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Gives M needed to get 𝑈 at given 𝑃 and 𝑃

This is the general CD expenditure function:

Not required but it's convenient to simplify a bit:

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𝑀 =𝑈

𝑎𝑃⎯⎯

1 − 𝑎𝑃

⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⋅

𝑃𝑎⎯⎯

𝑃1 − 𝑎⎯⎯⎯⎯⎯

𝑃𝑎⎯⎯

𝑃1 − 𝑎⎯⎯⎯⎯⎯

⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Denominator is 1 so simplified form is:

𝑀 = 𝑈 ⋅𝑃

𝑎⎯⎯

𝑃

1 − 𝑎⎯⎯⎯⎯⎯

Not required but it's convenient to simplify a bit:

Full set of general Cobb-Douglas equations:

𝑈 = (𝑄 ) 𝑄

Utility:

𝑄 =𝑎𝑀

𝑃⎯⎯⎯, 𝑄 =

(1 − 𝑎)𝑀

𝑃⎯⎯⎯⎯⎯⎯⎯⎯

Demands:

𝑀 = 𝑈 ⋅𝑃

𝑎⎯⎯

𝑃

1 − 𝑎⎯⎯⎯⎯⎯

Expenditure:

Applying to evaluate two policies, A and B

BAU

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𝑀 = $100𝑃 = $10𝑃 = $5

𝑄 = 7.5𝑄 = 5

Suppose preferences are known to be CD. Finding 𝑎:

𝑄 =𝑎𝑀

𝑃⎯⎯⎯

7.5 =𝑎 ∗ 100

10⎯⎯⎯⎯⎯⎯⎯

𝑎 =7.5

10⎯⎯⎯= 0.75

Policy A: $2 tax on 𝑃

Assume supply of X is perfectly elastic so 𝑃 = $10 + $2 = $12

Variable Change Helps or hurts?𝑀 = $100 No change n/a𝑃 = $12 $2 higher -𝑃 = $5 No change n/a

Step 1: Find initial utility 𝑈

𝑈 = 𝑄 . 𝑄.

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𝑈 = (𝑄 ) . 𝑄.

𝑈 = (7.5) . (5) .

𝑈 = 6.777 (avoid rounding U too much: keep 3-4 digits)

Step 2: Find 𝑀 to get 𝑈 at 𝑃 = $12, 𝑃 = $5

𝑀 = 𝑈 ∗𝑃

0.75⎯⎯⎯⎯

.𝑃

0.25⎯⎯⎯⎯

.

𝑀 = 6.777 ∗$12

0.75⎯⎯⎯⎯

.$5

0.25⎯⎯⎯⎯

.

𝑀 = 6.777 ∗ $16.92

𝑀 = $114.67

Step 3: Find the CV

𝐶𝑉 = 𝑀 − 𝑀

𝐶𝑉 = $114.65 − $100 = $14.65

Would require $14.65 to compensate for the policy

CV > 0 indicates household is worse off

Policy B: several changes

Let 𝑃 = $16.92Then 𝑀 = 𝑈 ∗ 𝑃

𝑃 price of utility:Ideal index for household

Aside:

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$5 tax on X1.$1 subsidy on Y (assume perfectly elastic supply)2.$10 lump sum income subsidy3.

Variable Change Helps or hurts?𝑀 = $110 $10 higher +𝑃 = $15 $5 higher -𝑃 = $4 $1 lower +

Impact on household (CV)•Impact on government budget•Overall change in SS (or DWL)•

Want to know:

𝑀 = 6.777 ∗$15

0.75⎯⎯⎯⎯

.$4

0.25⎯⎯⎯⎯

.Find 𝑀 :

𝑀 = 6.777 ∗ $18.91

𝑀 = $128.19

Find the CV:

𝑀 = $100 + $10 = $110

𝐶𝑉 = $128.19 − $110 = $18.19

To find the impact on budget need the new Q's:

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𝑄 =0.75 ∗ $110

$15⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯= 5.5

𝑄 =0.25 ∗ $110

$4⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯= 6.88

Components of revenue change:

Variable Type Rate Q Calculation ValueX: Tax +$5 5.5 +$5*5.5 +$27.50Y: Subsidy -$1 6.88 -$1*6.88 -$6.88M: Subsidy -$10 n/a n/a -$10

Δ𝑅𝑒𝑣 = 27.50 − 6.88 − 10

Δ𝑅𝑒𝑣 = 10.62

Change in SS:

Δ𝑆𝑆 = gain to household + gain to government

Gain to household: −𝐶𝑉 Since CV shows harmGain to government: Δ𝑅𝑒𝑣 As usual

Δ𝑆𝑆 = −𝐶𝑉 + Δ𝑅𝑒𝑣

Δ𝑆𝑆 = −18.19 + 10.62 = −7.57

Policy is inefficient: creates $7.57 of DWL

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Daily exercise on Google Classroom

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