My textbook had the following question: The town of Mario is occupied by familie

Question

My textbook had the following question:

The town of Mario is occupied by families and single individuals. These groups have separate demand formulas for their residence:

* for families it is
Rf = 9000 - 3Q

* for single individuals it is
Rf = 6,600 - 6Q

where R is the rent in dollars and Q is the number of apartments sought.

There are 3,000 families and 1,100 single individuals living in the town.

The supply of apartments in the town is:

V = 12,000 + 5Q

where V is the value of the apartment in dollars and Q is the number of apartments available.

The connection between the rental price of an apartment and the value of an apartment is

p = 0.1

Assume that all the apartments in the town are identical and that the demand for an apartment is not affected by where in the town the apartment is located.

At equilibrium, what'll be the price to rent an apartment in Mario, and how many apartments will be rented by families and individuals? The answer given in the book is:

* The rental price will be $2600. A total of 2,800 apartments will be rented out:
* families will rent 2,133.33 apartments
* single individuals will rent 666.66 apartments

Please explain how they arrived at this answer. Thank you.

Explanation / Answer

Start by rewriting our equations: p=0.1V, so rewrite the equation V=12000+5Q as 10p=12000+5Q, or p=1200+0.5Q That is our first equation. Now that is supply curve, representing total quantity, Q. Demand curves each represent partial quantities, Q1 could be quantity of apartments for families, and Q2 could be for individuals. Note Q=Q1+Q2. Now rewrite the demand equations: p=9000-3*Q1 and p=6600-6*Q2. If we multiply the first equation by 2, we can add the equations together and combine Q1 and Q2 as follows: 2p=18000-6*Q1 and p=6600-6*Q2 Adding the equations we get 3p=24,600-6(Q1+Q2) = 24,600-6Q. Use 3p=24,600-6Q (Demand) and p=1200 + 0.5Q (supply) to solve for Q. Dividing the first equation by 3, p=8200-2Q=1200+0.5Q. So 7000=2.5Q, so Q=2800. Now find p. p=1200 + 0.5Q = 1200 + 0.5(2800) = 1200+1400 = 2600 Now find Q1 and Q2 p=9000-3Q1. 2600=9000-3Q1. 3Q1=6400. Q1 =2133.3333 p=6600-6Q2. 2600=6600-6Q2. 6Q2=4000. Q2=666.6666666 Thus we arrive at our answers. Hope this helps!!!

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