Pat and Kris are roommates. They spend most of their time studying, but they leave some time for their favorite activities: making pizza and brewing root beer. Pat takes 4 hours to brew a gallon of root beer and 2 hours to make a pizza. Kris takes 6 hours to brew a gallon of root beer and 4 hours to make a pizza.
The price of pizza can be expressed in terms of gallons of root beer. What is the highest price at which pizza can be traded that would make both roommates better off?
Hint: think opportunity cost. To make 1 gal of root beer, what does Pat need to give up? what does Kris need to give up. This becomes the price of root beer.
To find the highest price at which pizza can be traded that would make both roommates better off, we need to consider the opportunity cost of making a gallon of root beer for each person.
For Pat: Pat can make 1 pizza in 2 hours, and since it takes 4 hours to brew a gallon of root beer, Pat can make 2 pizzas in the time it takes to make 1 gallon of root beer. So the opportunity cost for Pat to make 1 gallon of root beer is 2 pizzas.
For Kris: Kris can make 1 pizza in 4 hours, and since it takes 6 hours to brew a gallon of root beer, Kris can make 1.5 pizzas in the time it takes to make 1 gallon of root beer. So the opportunity cost for Kris to make 1 gallon of root beer is 1.5 pizzas.
In order for both to be better off, the price of pizza has to fall between the opportunity costs of both roommates. So, for a mutually beneficial trade, the price of pizza should be between 1.5 and 2 gallons of root beer. The highest price at which pizza can be traded that would make both roommates better off is therefore 2 gallons of root beer.
To determine the highest price at which pizza can be traded that would make both roommates better off, we need to understand the opportunity cost for each roommate.
Opportunity cost refers to the value of the next best alternative that is foregone when making a choice. In this case, the opportunity cost for Pat is the time it takes to make a pizza, whereas for Kris, it is the time it takes to brew a gallon of root beer.
Let's start by calculating the opportunity cost for each roommate:
For Pat:
- Pat takes 4 hours to brew a gallon of root beer and 2 hours to make a pizza.
- The opportunity cost of making a pizza is the time it could have been spent brewing root beer, which is 4 hours.
- Therefore, the opportunity cost for Pat to make a pizza is 4 hours per gallon of root beer.
For Kris:
- Kris takes 6 hours to brew a gallon of root beer and 4 hours to make a pizza.
- The opportunity cost of making a pizza is the time it could have been spent brewing root beer, which is 6 hours.
- Therefore, the opportunity cost for Kris to make a pizza is 6 hours per gallon of root beer.
To find the highest price at which pizza can be traded, we need to determine the amount of root beer each roommate is willing to give up for a pizza.
For Pat:
- Pat's opportunity cost of making a pizza is 4 hours per gallon.
- If the price of root beer is lower than Pat's opportunity cost, it would be more beneficial for Pat to give up the pizza and brew root beer instead.
- Therefore, the lowest price at which Pat is willing to give up a pizza is 4 hours.
For Kris:
- Kris's opportunity cost of making a pizza is 6 hours per gallon.
- If the price of root beer is lower than Kris's opportunity cost, it would be more beneficial for Kris to give up the pizza and brew root beer instead.
- Therefore, the lowest price at which Kris is willing to give up a pizza is 6 hours.
To find the maximum price both roommates would be willing to trade pizza for root beer, we compare their opportunity costs. Since Kris's opportunity cost of 6 hours per gallon is higher than Pat's opportunity cost of 4 hours per gallon, the maximum price at which pizza can be traded and make both roommates better off is based on Pat's opportunity cost of 4 hours.
Therefore, the highest price at which pizza can be traded that would make both roommates better off is 4 hours per gallon of root beer.