Write down mathematical formulae
Mathematics

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In this question write down a simple, formal, mathematical model. We wish to analyze the consumer's choice of housing. In particular, we want to compare the impact on utility of a change in the price of housing for a renter and a homeowner. You may assume that all individuals consume two things  housing (H), and a second consumption good (C).
1. (10 points) Write down the renter's problem. Write down the homeowner's problem.
Thanks and best of luck
The problem is we have two people  a renter and homeowner  and there are two goods to be consumed  general consumption and the cost of housing through rent or through homeownership.
So from the renters point of view the lower the rent that they have to pay, the more money they have available for additional consumption and therefore their total value of their consumption has increased so they are maximizing their utility by minimizing the rent and maximizing additional consumption
Don't let the complexity of the problem getting away of understanding the basics but this is really as simple as in straightforward as it sounds: renters want pay less money so they had more money to buy other stuff and this is the way in which they maximize their utility.
The homeowner is different in the sense that they have to lay out a huge amount of capital upfront to purchase the home and then they can rent it out to a renter at hopefully a very high rate of rent. So the greater the cost of the home to the homeowner, the more they are going to charge rent for that home but it also means the home owner has less money to purchase additional goods and services  additional consumption  because they paid so much for the house upfront.
1)
This is gonna be important for which is how when we get to how the renter any homeowner are directly or indirectly affected by price changes. So finally the answer to the first question is the homeowners problem is going to be that the higher the price of housing the lower the utility of living in that house will be because they are able to charge a higher rent to a renter. So higher cost of housing implies a high potential rent possibility for the homeowner and therefore a decreased utility of living at home also keep in mind for that a higher cost of housing the homeowner can purchase fewer other consumable goods which is see in this problem so they want to charge as much rent as possible so they can use that went to pay for housing as well as other consumption. The renters problem is the Niger the cost of rent, the lower the utility of renting the house since they have less money for other consumption.
Now the answer for the quick second part depends upon the direct or indirect impact of changes in pricing for housing so since the renter does not own the house a change in housing in directly impact the renters utility because they are renting it from someone else and have no control over the price they pay they basically taking or paying whatever that homeowner demands in terms of rent. So when the price of housing changes the renters utility is going to change in directly with that.
Please let me know if you need any clarification. I'm always happy to answer your questions.
MERCI
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