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FA2023-无代写
时间：2024-03-12
Urban Economics FA2023
Problem Set #1
Due, Tuesday, Sep 26, 11pm EST, ONLINE ONLY!
1) No late problem sets
2) You can only upload one single file. No multiple file submissions.
3) We accept only WORD, EXCEL and PDF formats, no JPEG etc.
Maximum Points: 50
Question1: 7pts
Question 2: 7pts
Question 3: 8pts
Question 4: 8pts
Question 5: 10 pts
Question 6: 10pts
(1) (1) Von-Thünen Model I
A pizzeria is located at M (where x=0) and charges marginal cost prices for its pizza
(i.e., c=p). Customers walk to the pizzeria to eat a pizza and, aside from the purchase
price, incur travel cost of (tx2), measured from the shop (note, this is a squared
function). The shop’s market reach is 2 km and t=2.
If customers’ WTP for pizza is $12.99 what is the price of a pizza in equilibrium? WTP = p+ tx2 (note, c=p) 12.99 = p + 2*22
à p= 12.99-8=4.99
(2) Von-Thünen Model II
Farmers that grow wheat need to sell their product at the market M (where x=0).
Marginal cost of growing equals c=$22, marginal travel cost is t=3 (per mile and ton)
and the market’s WTP for wheat is $79 per ton.
a) what is the maximum distance x* at which farmers would grow wheat?
b) For each distance point x=2, x=6, and x=10, say whether anybody would grow
wheat or not and what the resulting land rent would be. Note, rents cannot be
negative.
a) First calculate x*
WTP = c + tx
79 = 22 + 3*x è x*=19
b) The rent equals rent = WTP-c-tx and cannot be negative.
for x=2, yes, people would grow wheat, rent = 79-22-3*2 = 51
for x=10, yes, people would grow wheat, rent = 79-22-3*6 = 39
for x=20, yes people would grow wheat, rent = 79-22-3*10 = 27
(3) Von-Thünen Model III
Refer to the von-Thünen model where a city’s market reaches to the left and to the right.
The firm is competitive and its price equals marginal cost. The firm’s marginal cost is
c=2; the market reach is x=24 to the left and x=24 to the right. Marginal transportation
cost t equal t=6.
(a) What is customers’ WTP per unit of the good in question? (5 pts)
(b) What is the rent for a person located at x=12? (3 pts)
Answers (needs to be adjusted)
(a) WTP=c+tx à WTP=2+24*6 à WTP=146
(b) at x=12 total cost incurred is c+tx=2+6*12= 74
the rent is WTP-p-tx= 146-74 =72
(4) Von-Thünen Model with multiple cities
There are two cities, A and B, where B is located 11 miles east of A. Both produce an
identical good and consumers’ WTP=42 is the same everywhere. Marginal transportation
cost are and tA=2 and tB=4. Each city prices its good at marginal cost (p=c). We know
that cB=6; cA is unknown.
(a) What is the maximum market reach of city B to the east? (2 pts)
(b) Now assume A wants to reach markets (and sell to at least one customer) to the east
of B. Calculate the maximum cA for this to be true. (6 pts)
Answer
a) calculate the market reach of city B to the east based on WTP=c+tx.
42=6+4x à x=9 (4 pts)
b) since A is 11 miles away from B, and B reaches 9 miles to its east,
A must reach customers that are 20 miles away.
First, calculate the maximum c for this to be true.
WTP=c+tAx and set x=20
42= c + 2*10 à c=2
In order to undercut B, A’s marginasl cost (price) must be below 2.
The maximum price is 1.99
(NOTE: cA=2 is incorrect, it must be cA<2)
(5) Locations of Transfer-Oriented Firms The distance between the resource (R ) and the market (M) is 10 miles. A firm’s procurement cost is given by PC=4x2, where x is the distance from the resource measured in miles. The firm’s distribution cost is given by DC=(10-2x)2. Again, x measures the distance from the resource. (a) What is the Total Freight Cost (TC) at each milestone. This question assumes there are only 11 possible discrete locations, i.e., x=0, 1, 2, 3, …10. Locations between the milestones are not possible. Where should the firm locate? (You can do this “by hand” or in a spreadsheet) Locate at x=2 or at x=3
x PC DC TC
0 0 100 100
A B
cA=pA
cB=pB
1 4 64 68
2 16 36 52
3 36 16 52
4 64 4 68
5 100 0 100
6 144 4 148
7 196 16 212
8 256 36 292
9 324 64 388
10 400 100 500
(b) Now assume, the firm can locate between the milestones, for instance at x=5.145. Provide a mathematically exact solution for the optimal location (with 2 decimals) using calculus. What is the optimal x? Is it different from your solution from (a) above. At this location, what is PC, DC and TC (two decimals). Show your work! Best use calculus since TC = PC + DC = 4! + (10 − 2)! = 4! + (4! + 100 − 40)
050
100150
200250
300350
400
0 1 2 3 4 5 6 7 8 9 10PC DC TC
TC
DC
PC
= 8! − 40 + 100 Build the derivative and set and set equal to zero = 16 − 40 = 0 solve for x ∗ ∗ = 2.50 The best location is x*=2.50 Here, PC=25, DC=25, and TC=50 (c) How does your answer to all parts under (b) change if the distribution cost D where DC= (10-x), i.e., without the squared term. PC remains at PC=4x2. (we also accept if you use DC=10-2x) That is, there are two acceptable answers For DC=(10-x) DC=(10-2x) Optimal location x*=0.125 x*=0.250 PC 0.06 0.25 DC 9.88 9.50 TC 9.94 9.75
(6) A pizzeria delivers pizzas to each customer and incurs a cost of $1 per unit and mile.
Each unit needs to be delivered separately. The locations and the number of customers
are provided in the Table below.
Part 1:
(a) Where is the median customer located?
(b) Where is the delivery cost minimizing location and what is its delivery cost?
Location A B C D E F
Distance from A (miles) 0 10 8 2 6 18
Number of customers 30 140 10 20 30 40
(a) note: to find the median location, you first need to sort the locations according to their distance from A
Location A D E C B F
Distance
from A 0 2 6 8 10 18
Number of
customers 30 20 30 10 140 40
There are 270 customers. The median customer is located inside of B; somewhere inside of B there are 135 customers to the right and 135 to the left). (b) Cost is minimized at CB and amounts to A (10*30=300), D (8*20=160), E(4*30=120), C(2*10=20), B(0), F(8*40=320)
SUM= 920
Part 2:
Where is the median location if the number of customers at B decreased from 140 to 40?
What is the delivery cost from this location?
Location A D E C B F
Distance
from A 0 2 6 8 10 18
Number of
customers 30 20 30 10 40 40
à We now have 170 customers and the median location has 85 customers to its left and 85 to its right. That is, the median location will be at BC. The cost will fall to 72 A (8*30=240), D (6*20=120), E(2*30=60),
C(0), B(2*40=80) F(10*40=400)
SUM=900
Problem Set #1
Due, Tuesday, Sep 26, 11pm EST, ONLINE ONLY!
1) No late problem sets
2) You can only upload one single file. No multiple file submissions.
3) We accept only WORD, EXCEL and PDF formats, no JPEG etc.
Maximum Points: 50
Question1: 7pts
Question 2: 7pts
Question 3: 8pts
Question 4: 8pts
Question 5: 10 pts
Question 6: 10pts
(1) (1) Von-Thünen Model I
A pizzeria is located at M (where x=0) and charges marginal cost prices for its pizza
(i.e., c=p). Customers walk to the pizzeria to eat a pizza and, aside from the purchase
price, incur travel cost of (tx2), measured from the shop (note, this is a squared
function). The shop’s market reach is 2 km and t=2.
If customers’ WTP for pizza is $12.99 what is the price of a pizza in equilibrium? WTP = p+ tx2 (note, c=p) 12.99 = p + 2*22
à p= 12.99-8=4.99
(2) Von-Thünen Model II
Farmers that grow wheat need to sell their product at the market M (where x=0).
Marginal cost of growing equals c=$22, marginal travel cost is t=3 (per mile and ton)
and the market’s WTP for wheat is $79 per ton.
a) what is the maximum distance x* at which farmers would grow wheat?
b) For each distance point x=2, x=6, and x=10, say whether anybody would grow
wheat or not and what the resulting land rent would be. Note, rents cannot be
negative.
a) First calculate x*
WTP = c + tx
79 = 22 + 3*x è x*=19
b) The rent equals rent = WTP-c-tx and cannot be negative.
for x=2, yes, people would grow wheat, rent = 79-22-3*2 = 51
for x=10, yes, people would grow wheat, rent = 79-22-3*6 = 39
for x=20, yes people would grow wheat, rent = 79-22-3*10 = 27
(3) Von-Thünen Model III
Refer to the von-Thünen model where a city’s market reaches to the left and to the right.
The firm is competitive and its price equals marginal cost. The firm’s marginal cost is
c=2; the market reach is x=24 to the left and x=24 to the right. Marginal transportation
cost t equal t=6.
(a) What is customers’ WTP per unit of the good in question? (5 pts)
(b) What is the rent for a person located at x=12? (3 pts)
Answers (needs to be adjusted)
(a) WTP=c+tx à WTP=2+24*6 à WTP=146
(b) at x=12 total cost incurred is c+tx=2+6*12= 74
the rent is WTP-p-tx= 146-74 =72
(4) Von-Thünen Model with multiple cities
There are two cities, A and B, where B is located 11 miles east of A. Both produce an
identical good and consumers’ WTP=42 is the same everywhere. Marginal transportation
cost are and tA=2 and tB=4. Each city prices its good at marginal cost (p=c). We know
that cB=6; cA is unknown.
(a) What is the maximum market reach of city B to the east? (2 pts)
(b) Now assume A wants to reach markets (and sell to at least one customer) to the east
of B. Calculate the maximum cA for this to be true. (6 pts)
Answer
a) calculate the market reach of city B to the east based on WTP=c+tx.
42=6+4x à x=9 (4 pts)
b) since A is 11 miles away from B, and B reaches 9 miles to its east,
A must reach customers that are 20 miles away.
First, calculate the maximum c for this to be true.
WTP=c+tAx and set x=20
42= c + 2*10 à c=2
In order to undercut B, A’s marginasl cost (price) must be below 2.
The maximum price is 1.99
(NOTE: cA=2 is incorrect, it must be cA<2)
(5) Locations of Transfer-Oriented Firms The distance between the resource (R ) and the market (M) is 10 miles. A firm’s procurement cost is given by PC=4x2, where x is the distance from the resource measured in miles. The firm’s distribution cost is given by DC=(10-2x)2. Again, x measures the distance from the resource. (a) What is the Total Freight Cost (TC) at each milestone. This question assumes there are only 11 possible discrete locations, i.e., x=0, 1, 2, 3, …10. Locations between the milestones are not possible. Where should the firm locate? (You can do this “by hand” or in a spreadsheet) Locate at x=2 or at x=3
x PC DC TC
0 0 100 100
A B
cA=pA
cB=pB
1 4 64 68
2 16 36 52
3 36 16 52
4 64 4 68
5 100 0 100
6 144 4 148
7 196 16 212
8 256 36 292
9 324 64 388
10 400 100 500
(b) Now assume, the firm can locate between the milestones, for instance at x=5.145. Provide a mathematically exact solution for the optimal location (with 2 decimals) using calculus. What is the optimal x? Is it different from your solution from (a) above. At this location, what is PC, DC and TC (two decimals). Show your work! Best use calculus since TC = PC + DC = 4! + (10 − 2)! = 4! + (4! + 100 − 40)
050
100150
200250
300350
400
0 1 2 3 4 5 6 7 8 9 10PC DC TC
TC
DC
PC
= 8! − 40 + 100 Build the derivative and set and set equal to zero = 16 − 40 = 0 solve for x ∗ ∗ = 2.50 The best location is x*=2.50 Here, PC=25, DC=25, and TC=50 (c) How does your answer to all parts under (b) change if the distribution cost D where DC= (10-x), i.e., without the squared term. PC remains at PC=4x2. (we also accept if you use DC=10-2x) That is, there are two acceptable answers For DC=(10-x) DC=(10-2x) Optimal location x*=0.125 x*=0.250 PC 0.06 0.25 DC 9.88 9.50 TC 9.94 9.75
(6) A pizzeria delivers pizzas to each customer and incurs a cost of $1 per unit and mile.
Each unit needs to be delivered separately. The locations and the number of customers
are provided in the Table below.
Part 1:
(a) Where is the median customer located?
(b) Where is the delivery cost minimizing location and what is its delivery cost?
Location A B C D E F
Distance from A (miles) 0 10 8 2 6 18
Number of customers 30 140 10 20 30 40
(a) note: to find the median location, you first need to sort the locations according to their distance from A
Location A D E C B F
Distance
from A 0 2 6 8 10 18
Number of
customers 30 20 30 10 140 40
There are 270 customers. The median customer is located inside of B; somewhere inside of B there are 135 customers to the right and 135 to the left). (b) Cost is minimized at CB and amounts to A (10*30=300), D (8*20=160), E(4*30=120), C(2*10=20), B(0), F(8*40=320)
SUM= 920
Part 2:
Where is the median location if the number of customers at B decreased from 140 to 40?
What is the delivery cost from this location?
Location A D E C B F
Distance
from A 0 2 6 8 10 18
Number of
customers 30 20 30 10 40 40
à We now have 170 customers and the median location has 85 customers to its left and 85 to its right. That is, the median location will be at BC. The cost will fall to 72 A (8*30=240), D (6*20=120), E(2*30=60),
C(0), B(2*40=80) F(10*40=400)
SUM=900
