Part Incidence: We discussed four principles of incidence. This example problem
ID: 1174242 • Letter: P
Question
Part Incidence: We discussed four principles of incidence. This example problem is meant to illustrate both "inelasticity pays" and "in general, anything can happen." The idea here is to use a sorting problem that is similar to our standard setup in order to show how a policy can have various impacts, and people who are not targeted" by the policy can either benefit or lose from it. There are two neighborhoods B and C. In both neighborhoods, the price of housing is P- N, where N is the number of residents. There are two types of people, Red and Violet. Red and Violet both have utility of the shape U aEQ P, and Reds have and Violets have ?-2. Also, Red people are immobile. They cannot change neighborhoods. Violet people are mobile, and can freely move between neighborhoodsExplanation / Answer
In the question we are given the following information: Utltity of every group is given by the equation:
Utility = a*EQ - P where a is 1 for Red, and 2 for Violet. And P is the total number of people living in a neighbourhood.
(Here I have taken alpha as a)
4. Now we have to caluclate the utility of each group. We will use the utlity function given above. Here EQ is given as 20 and total number of people in each neighbourhood, that is, P = 20.
Therefore:
RedB = a*EQ - P where a = 1 , EQ = 20 and P = 20. Putting these values in the equation, we get:
RedB = 1*20 - 20 = 0
Similarly, for RedC = a*EQ - P
= 1*20 - 20 = 0.
VioletB = a*EQ - P where a is 2, EQ = 20 and P = 20. Putting these values in equation, we get
VioletB = 2*20 - 20
= 40 - 20 = 20
Similarly VioletC = 2*20 - 20 = 20
Therefore respective utltities of the following groups are as follows:
RedB = RedC = 0 and VioletB = VioletC = 20
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5. Now the government has improved neighbourhood C increasing its EQ to 25. Intuitively, we can think of it as: At first both RedC and VioletC will have increased utility. Violet people being mobile will start shifting to neighbourhood C. Doing so will reduce B's population and will affect Red group too. Therefore RedB utlity will also increase. However in city C, the population will start increasing and the utlity will decrease. Violet people will continue to shift until their they are indifferent between staying in B and C. That is they will continue to migrate until their utlity from staying in B = utlity from staying in C.
We will use this fact to solve the number of people in B and C.
Utility of violet people from staying in B = utlity of violet people from staying in C.
Utility of violet people from staying in B = a*EQ - P
=> UB = 2*20 - (Number of RedB + Number of VioletB)
Since RedB are immobile, their total number will be 10
Therefore:
=> UB = 40 - (10 + VioletB) = 30 - VioletB
Similarly utlity of violet people from staying in C = a*EQ - P where EQ now is 25
= UC = 2*25 - (10 + VioletC) = 50 - 10 - VioletC
=> UC = 40 - VioletC
Since UB = UC =>
=> 40 - VioletC = 30 - VioletB
=> 40 - 30 = VioletC - VioletB
=> VioletC - VioletB = 10 --------------------> (1)
Also it is given that total number of Violet people = 20
=> VioletC + VioletB = 20 -----------------------> (2)
Adding equations 1 and 2, we get:
2*VioletC = 30
Therefore VioletC = 30/2 = 15. And VioletB = 20-15 = 5
This gives population in city B = RedB+ VioletB = 10 + 5 = 15
And population in City C = RedC + VioletC = 10 + 15 = 25
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6. We have to calculate the change in utilities of each of the group. This will be given by:
Utility after change - Utility before change.
But first we have to calculate the utilitites after change. We will use the formula again where
utility a*EQ - P but now EQ = 20 for city B and 25 for city C.
Utility of RedB = a*EQ - P where a = 1 EQ = 20 and P = 5. Putting these values we get:
U of RedB = 1*20 - 5 = 15
Similarly Utility of VioletB = a*EQ - P where a= 2 =, EQ = 20 and P = 5
= U of VioletB= 2*20 - 5 = 35
Utility of RedC = a*EQ - P but now a = 1 EQ = 25 and P = 15
U of RedC = 1*25 - 15 = 10
Similalrly, U of VioletC = 2*25 - 15 = 35
Now we will calculate the change:
Change in Utility of RedB = 15 - 0 = 15 which is postive
Change in Utility of VioletB = 35 - 20 = 15 which is positive
Chnage in Utility of RedC = 10 - 0 = 10 which is positive
Chnage in utility of VioletC = 35 - 20 = 15 which is positive.
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7. Observing the change in utilites in previous answer, both VioletB and VioletC experience the same gain in utiity. That is because both the groups have equal utilities in both the conditions. Since they are mobile, if there is scope of greater utility in one neighbourhood than other, this group can migrate until it equalises. Therefore they experience the same incidence.
However this is not the case for RedB and RedC. This is because they are immobile, and completely dependant on their condiiton of neighbourhood and the population. Because neighbourhood C is now better, Violet group has migrated. This overcrowded city C more than city C, causing reduced gain for RedB members than RedC.
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8. In the earlier answers we found that the utilities were as follows:
Utility of RedB = RedC = 0
And Utility of VioletB = Utility of VioletC = 20
Now government imposes a tax of $2 on VioletB. Violet memebers living in city B have reduced utility tha Violet members in City C. Therefore VioletB members will shift to city C until the utlity they get from staying in City B equals utility from staying in City C.
That is Utility in staying in B = utlity from staying in C
=> UB = a*EQ - P - Tax
=> UB = 2*20 - (RedB + VioletB) - 2
=> UB = 38 - (10 + VioletB) since red in each city are 10.
=> UB = 28 - VioletB
Similarly utility for violet people in city C =
=> UC = a*EQ - P
=> UC = 2*20 - (10 + VioletC) =
=> UC = 30 - VioletC
Since UB= UC
=> 28 - VioletB = 30- VioletC
Shuffling terms around we get:
VioletC - VioletB = 2 ----------------------(3)
Also VioletB + VioletC = 20 --------------------(4)
Adding equations 3 and 4, we get
2*VioletC = 22
Theerfore VioletC = 11 and VioletB = 9
New utilities after taxation:
RedB = a*EQ - P
=> RedB = 1*20 - (10 + 9)
=> New RedB = 1
New VioletB = a*EQ - P - tax
=> 2*20 - (10+9) - 2
= New VioletB = 19
Similarly for people in city C:
New RedC = a*EQ - P
=> New RedC = 1*20 - (10 + 11)
=> New RedC = -1
New VioletC = a*EQ - P
=> 2*20 - 21
=> New VioletC = 19
Therefore incidence = change in gain/loss = New gain - old gain
For RedB = 1- 0 = 1 , positive therefore gain
VioletB = 19 - 20 = -1 negative therefore loss
RedC = -1 - 0 = -1 negative therefore loss
VioletC = 19 - 20 = -1 negative therfore loss
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9. Now Tax is applied only to Red people. Since there is no tax on Violet people, their utility is unaffected and there will be no migration.
So now Utility for RedB = a*EQ - P - tax
= U of Red = 1*20 - 20 - 2
= New utility if RedB = -2
Everything else, or everybody's else's utility will be same for all other groups, since RedB is the only group taxed.
Therefore incidence of tax =
For RedB = -2 - 0 = -2
VioletB = 0
RedC = VioletC = 0.
Therefore only RedB incurs a loss of -2 from tax and rest everyone has zero gain/loss.
This difference in incidence is due to inelasticity of Red people. Because they are completely immobile, they bear the full effect of tax. However, when Violet group is taxed, they are elastic. Therefore when they are taxed, they migrate and in doing so, they shift some of the burden of their tax onto other groups.