There is a beach that runs east-west. Ice cream vendors can choose any of nine locations along the beach. Location 1 is at the west end, location 2 is just to the east of 1, 3 is just to the east of 2, and so on. There are 10 consumers at each location. Consumers follow these rules:
If they buy ice cream, they will buy from the closest vendor. If there are two vendors the same distance away, half will buy from one and a half from the other.
Each consumer will buy one ice cream cone as long as they do not need to walk more than two locations. So, for example, customers at 4 will buy from a vendor at 2, 3, 4, 5, or 6 but they will not buy from a vendor at 1, 7, 8, or 9.
If there is no vendor within two locations, a customer will not buy ice cream.
Vendor A and vendor B will choose their locations simultaneously. Each wants to maximize the number of ice cream cones it sells. They could choose the same location.
Find the Nash equilibrium (or Nash equilibria if you believe there is more than one) of this game.
5. 1. Player 1 and Player 2 will play the following game. Player 1 will choose x1 and Player 2 will choose x2 simultaneously. Their payoffs Ï1 and Ï2 are
Ï1 = 90x1 x12 x1x2
Ï2 = 90x2 x22 x1x2
(a) Find each player's best reply.
(b) Show that x1 = x2 = 30 is the Nash equilibrium of this game.
There is a beach that runs east-west. Ice cream vendors can choose any of nine locations along the beach. Location 1 is at the west end, location 2 is just to the east of 1, 3 is just to the east of 2, and so on. There are 10 consumers at each location. Consumers follow these rules:
If they buy ice cream, they will buy from the closest vendor. If there are two vendors the same distance away, half will buy from one and a half from the other.
Each consumer will buy one ice cream cone as long as they do not need to walk more than two locations. So, for example, customers at 4 will buy from a vendor at 2, 3, 4, 5, or 6 but they will not buy from a vendor at 1, 7, 8, or 9.
If there is no vendor within two locations, a customer will not buy ice cream.
Vendor A and vendor B will choose their locations simultaneously. Each wants to maximize the number of ice cream cones it sells. They could choose the same location.
Find the Nash equilibrium (or Nash equilibria if you believe there is more than one) of this game.
5. 1. Player 1 and Player 2 will play the following game. Player 1 will choose x1 and Player 2 will choose x2 simultaneously. Their payoffs Ï1 and Ï2 are
Ï1 = 90x1 x12 x1x2
Ï2 = 90x2 x22 x1x2
(a) Find each player's best reply.
(b) Show that x1 = x2 = 30 is the Nash equilibrium of this game.
Related textbook solutions
Related questions
1. Use the table below for the following question. Suppose you have a budget of $6 and that brownies and Twizzlers cost $1 each. What is the consumer optimum?
|
Number of Brownies |
Total Utility of Brownies |
Marginal Utility of Brownies |
Number of Twizzlers |
Total Utility of Twizzlers |
Marginal Utility of Twizzlers |
|
0 |
0 |
- |
0 |
0 |
- |
|
1 |
10 |
10 |
1 |
6 |
8 |
|
2 |
19 |
9 |
2 |
13 |
7 |
|
3 |
26 |
7 |
3 |
19 |
6 |
|
4 |
30 |
4 |
4 |
23 |
4 |
|
5 |
30 |
0 |
5 |
25 |
2 |
|
6 |
29 |
-1 |
6 |
25 |
0 |
| 6 brownies |
| 3 brownies and 3 Twizzlers |
| 4 brownies and 2 Twizzlers |
| any combination of 6 because they're both $1 |
| 8 |
2. Use the table below for the following question. What is the consumer equilibrium for a $6 budget if the price of brownies rises to $1.50 and the price of Twizzlers remains at $1.00?
|
Number of Brownies |
Total Utility of Brownies |
Marginal Utility of Brownies |
Number of Twizzlers |
Total Utility of Twizzlers |
Marginal Utility of Twizzlers |
|
0 |
0 |
- |
0 |
0 |
- |
|
1 |
10 |
10 |
1 |
6 |
8 |
|
2 |
19 |
9 |
2 |
13 |
7 |
|
3 |
26 |
7 |
3 |
19 |
6 |
|
4 |
30 |
4 |
4 |
23 |
4 |
|
5 |
30 |
0 |
5 |
25 |
2 |
|
6 |
29 |
-1 |
6 |
25 |
0 |
| The consumer will buy all Twizzlers because they're cheaper. |
| 4 brownies |
| 3 Twizzlers and 2 brownies |
| 3 brownies and 3 Twizzlers |
| 7 |
3. At Nice Price for the Ice, an ice cream parlor, customers routinely buy a scoop of ice cream for $3. If consumers purchase one scoop of ice cream at $3, then why don't they keep buying more and more scoops for $3 until the store sells out?
| People would not want more than one scoop of ice cream because then they would have less money to spend on other goods. |
| People do not want to consume all the scoops of ice cream because their total utility is higher the less they eat. |
| At some point, customers do not value an additional scoop at $3, so they will not pay $3 for a scoop after they reach that point. |
4. Suppose you plan on eating 50 potato chips. As you start consuming potato chips, your marginal utility is very high, but it begins to fall slowly until you've eaten 10 chips. After you have eaten 10 chips, your marginal utility decreases even faster with each additional chip. The marginal utility continues to decline until you've eaten 49 chips. The fiftieth chip does not give you any additional utility. After 50 chips, your mouth gets so salty that it's unpleasant to eat anymore, so marginal utility is actually negative for those chips. How many chips should you eat in order to maximize your total utility?
| 1 |
| 10 |
| 49 |
5. As you start consuming potato chips, your marginal utility is very high, but it begins to fall slowly until you've eaten 10 chips. After you have eaten 10 chips, marginal utility decreases even faster with each additional chip. The marginal utility continues to decline until you've eaten 49 chips. The fiftieth chip does not give you any additional utility. After 50 chips, your mouth gets so salty that it's unpleasant to eat anymore, so marginal utility is actually negative for those chips. How many chips should you eat in order to maximize your marginal utility?
| 1 |
| 10 |
| 49 |
| 50 |
6. At current levels of consumption, Alice is spending her entire budget. If Alice gets 3 utils of satisfaction per dollar spent on ice cream and 2 utils per dollar spent on shampoo, then how should she adjust her consumption to get closer to the consumer optimum?
| She is already at the consumer optimum, so no adjustment is necessary. |
| She should buy more ice cream and less shampoo. |
| She should buy more shampoo and less ice cream. |
| She should buy more of both goods. |
7. Suppose you are at a restaurant and your favorite dish costs $20. You can get your next-favorite dish for $17. If your next-favorite dish gives you 100 utils, how many additional utils do you need from your favorite dish to spend the extra $3?
| 15 |
| 17.65 |
| 85 |
| 117.65 |
8. The diamond water paradox is the observation that water is essential to life and inexpensive, while diamonds are not essential and are highly-priced. Which of the other pairs of goods exhibit a pricing structure similar to water and diamonds?
| economy & luxury cars |
| rice & truffles (very expensive mushrooms) |
| paper clips & gasoline |
| paper & textbooks |
9. Rice is a cheap staple food eaten several times a day by many people all over the world. In Trufflelandia, residents also eat expensive mushrooms known as truffles once every year as a birthday celebration. Rice keeps the people alive and truffles are not necessary for sustaining their lives. Why is rice so cheap and truffles so expensive?
| Truffles taste better, so they are worth more money. |
| Rice is easy to cook, so people buy a lot of it. If people are going to buy so much, then it has to be cheap. |
| Truffles are more nutritious, and healthy food is always more expensive than unhealthy food. |
| People eat so much rice that an additional serving of rice has little marginal value, but the marginal value of another serving of truffles is very high. |
