<p>With regard to situation with 25 apartments: The reservation prices of renters is not known in advance.  The only price signals a landlord will get happen at the extremes.  So if a landlord lists a price of 501, he finds that nobody will rent.  So he rents below that. And if he rents at 200, he has excess demand, so he will rent above that.  So the market finds a way to end up at say 201 to 499 inclusive (ignore partial dollars for now).  If all the landlords rent at 300, there is no excess demand and no shortfall in demand, so everything is rented.  No reason to do swaps. Yes, the landlords COULD have charged 499, but they don't know that since at 300 there was no excess demand.</p>
<p>With regard to situation with 26 apartments: Varian should have specified whether or not there was a 27th person here.  If you have 26 people in the world and 26 apartments, they will rent out at 0&lt;=rent&lt;=200.  Varian's answer which is $200 implies a 27th person in the mix.</p>

CC BY-SA 4.0
2021-08-25T20:39:40.913Z
2021-08-25T20:54:57.300Z
2021-08-25T20:54:57.300Z
HFM
<p>I have quickly skimmed the chapter and there does not seem to be much about bargaining models so as the question says you have to solve this using supply and demand.</p>
<p>We know that an equilibrium price on the market will occur when supply intersect demand. Given that the supply is fixed at <span class="math-container">$24$</span> and <span class="math-container">$25$</span> apartments respectively it will be just straight vertical line in each case. Now when it comes to demand, if we assume each person wants only one single apartment the demand will be given in step-wise fashion, with two steps given by reservation prices <span class="math-container">${\\\$}500$</span> and <span class="math-container">${\\\$}200$</span> respectively. So this situation will look like shown on the tikz picture I made below (supply is blue, demand is red):</p>
<p><a href="https://i.sstatic.net/pXzgu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pXzgu.png" alt="enter image description here" /></a></p>
<p>At point where supply is equal to <span class="math-container">$24$</span> the equilibrium price given by intersection of supply and demand (<span class="math-container">$S=D$</span>) is clearly <span class="math-container">${\\\$}500$</span>. However, when supply is equal to <span class="math-container">$25$</span>, supply and demand will intersect at a whole range of different prices <span class="math-container">$[200,500]$</span>.</p>
<p>Consequently, in this case using this simple supply and demand model we cannot say anything more that price will be something between <span class="math-container">${\\\$}200$</span> and <span class="math-container">${\\\$}500$</span>. There are more advanced ways how we could model this situation which might yield unique price instead of range (maybe some models with explicitly modeled bargaining process or some matching models), but here the author of the book gives you hints by asking how demand would look like etc., that you are supposed to use this solution concept (also the first chapter uses only simple supply and demand models so that is another hint).</p>

CC BY-SA 4.0
2020-10-24T11:26:45.267Z
Thank you for your answer. By your logic, though, shouldn't the S = 26 graph intersect with the demand curve at every price from [\$0, \$200]? Why then does Varian say that the equilibrium price for 26 apartments is a fixed \$200?
Ray Bradbury
CC BY-SA 4.0
2020-10-24T11:28:18.630Z
@ijm no because there is no excess demand after Q=26, so in that case the price will be given just by the maximum reservation price.
1muflon1
CC BY-SA 4.0
2020-10-24T11:54:18.377Z
I don't really understand why the excess demand at S = 25 (the sole tenant with a reservation price of \$200) should affect the price, given that 25 apartments can be leased to the 25 people with the highest reservation price, i.e., \$500. The argument involving the intersection of the demand-supply graphs makes sense, but where can I acquire the economic intuition behind this idea?
Ray Bradbury
CC BY-SA 4.0
2020-10-24T11:57:07.480Z
@ijm Since in your Q you only wanted to know how price changes between Q=24 and Q=25, I did not put that in the picture but at the boundaries (e.g even if case if there are only 24 consumers with all reservation prices 500 and supply =24) the price will be given by the reservation prices. However, within the boundaries you should look for intersections of supply and demand
1muflon1
CC BY-SA 4.0
2020-10-24T12:07:57.360Z
I get that within boundaries we should look at the intersections of supply and demand, but that still seems a quirk of the model we've set up, and does not seem to say anything about the behaviour of the actors involved. For example, as I wrote in the Q, "if the equilibrium price is anything lower than \$500, landlords would have an incentive to arbitrarily increase the rent to the highest the market can absorb, i.e., \$500." I was asking about the economic intuition for that not happening.
Ray Bradbury
CC BY-SA 4.0
2020-10-24T12:21:39.853Z
@ijm the intuition here is that since landlords can’t price discriminate  here and can’t directly observe the reservation prices at S=24 there is only single price at which D=24 which is 500, at S=25 there is a whole range of prices  that will make supply and demand equal because we do not know what is the mechanism of competition between the buyer with 500 and 200 reservation price so any price there is possible, and at S=26 there again will be one price that is guaranteed to produce D=26 and that is 200
1muflon1
CC BY-SA 4.0
2020-10-24T12:33:11.093Z
Thanks a lot. This seems like an interesting topic in its own right. Where can I learn more about "mechanisms of competition between buyers"?
Ray Bradbury
CC BY-SA 4.0
2020-10-24T13:18:44.213Z
@ijm these things are covered mainly by market/mechanism design and auction theory. I have older version of the textbook so I am not sure if chapters are correct but the Varian textbook ch 17 should cover auctions which is an intro to auction theory. Also the book will make the concepts for standard supply and demand more rigorous in later chapters. Other than that this is also covered in some length in MWG microeconomic theory and in Tadelis game theory (game theory is actually recommended prerequisite in these fields).
1muflon1
CC BY-SA 4.0
2020-10-24T13:21:43.360Z
There are for sure texts that focus only on auction theory and market design but I can’t give more narrow recommendations there as I do not specialize in these two subfields, but the books I listed above have introductory treatments to those topics and they contain references to further papers/books. Alternatively, you can do your own additional search using the names of the field as key words, or you can make separate question as a reference request for such texts (but prior that you should do at least a bit of search yourself since users here don’t like Q that don’t show any effort).
1muflon1
CC BY-SA 4.0
2020-10-24T09:19:05.923Z
2020-10-24T09:19:05.923Z
1muflon1
<p>From the Review Questions to Chapter 1: The Market in Hal Varian's <em>Intermediate Microeconomics with Calculus, 9th ed.</em>:</p>
<blockquote>
<ol>
<li>Suppose that there are 25 people who had a reservation price of <span class="math-container">$\\\$500$</span>, and the 26th person had a reservation price of <span class="math-container">$\\\$200$</span>. What would the demand curve look like?</li>
<li>In the above example, what would be the equilibrium price if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?</li>
</ol>
</blockquote>
<p>Please note that in this chapter, the supply of apartments is taken to be fixed.</p>
<p>I have drawn the demand curve, and attempted to solve the first two cases in the second question in the following way:</p>
<p>If there are 24 apartments, they go to the 24 people with the highest reservation price. This happens to be <span class="math-container">$\\\$500$</span>, so that is the equilibrium price in this case.</p>
<p>When there are 26 apartments, we can imagine 25 landlords successfully leasing their property for <span class="math-container">$\\\$500$</span>, and one landlord who is unable to net the <span class="math-container">$\\\$500$</span> lowering his price until he reaches <span class="math-container">$\\\$200$</span>. This prompts a surge of potential tenants at his doorstep, but the apartment does not necessarily go to the tenant who had a reservation of price of <span class="math-container">$\\\$200$</span>; it goes to whomever spends most time bargaining with the landlord, has the best connections, etc. Since this apartment generally goes to someone who had a reservation price of <span class="math-container">$\\\$500$</span>, the new tenant vacates an apartment in her wake, which is then leased for <span class="math-container">$\\\$200$</span>, and so on until everybody is living in a <span class="math-container">$\\\$200$</span> apartment.</p>
<p>The third case is confusing. The answer at the end of the book says that the equilibrium price would be &quot;any price between <span class="math-container">$\\\$200$</span> and <span class="math-container">$\\\$500$</span>&quot; when there are 25 apartments. Why is this true? It seems to me that since the supply of apartments is fixed, the price of one apartment would depend only on the reservation price of tenants. If the equilibrium price is anything lower than <span class="math-container">$\\\$500$</span>, landlords would have an incentive to arbitrarily increase the rent to the highest the market can absorb, i.e., <span class="math-container">$\\\$500$</span>. What's keeping the equilibrium price below <span class="math-container">$\\\$500$</span>?</p>
<p>Please also tell me if my reasoning about the other two cases is flawed, or there are other, better ways to reason about them.</p>

<microeconomics><equilibrium-price>
What is the equilibrium price in this case?
Ray Bradbury
# What is the equilibrium price in this case?

(Asked by: username_0 on 2020-10-24)

From the Review Questions to Chapter 1: The Market in Hal Varian's *Intermediate Microeconomics with Calculus, 9th ed.*:

> 1. Suppose that there are 25 people who had a reservation price of $\\\$500$, and the 26th person had a reservation price of $\\\$200$. What would the demand curve look like?
> 2. In the above example, what would be the equilibrium price if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?

Please note that in this chapter, the supply of apartments is taken to be fixed.

I have drawn the demand curve, and attempted to solve the first two cases in the second question in the following way:

If there are 24 apartments, they go to the 24 people with the highest reservation price. This happens to be $\\\$500$, so that is the equilibrium price in this case.

When there are 26 apartments, we can imagine 25 landlords successfully leasing their property for $\\\$500$, and one landlord who is unable to net the $\\\$500$ lowering his price until he reaches $\\\$200$. This prompts a surge of potential tenants at his doorstep, but the apartment does not necessarily go to the tenant who had a reservation of price of $\\\$200$; it goes to whomever spends most time bargaining with the landlord, has the best connections, etc. Since this apartment generally goes to someone who had a reservation price of $\\\$500$, the new tenant vacates an apartment in her wake, which is then leased for $\\\$200$, and so on until everybody is living in a $\\\$200$ apartment.

The third case is confusing. The answer at the end of the book says that the equilibrium price would be "any price between $\\\$200$ and $\\\$500$" when there are 25 apartments. Why is this true? It seems to me that since the supply of apartments is fixed, the price of one apartment would depend only on the reservation price of tenants. If the equilibrium price is anything lower than $\\\$500$, landlords would have an incentive to arbitrarily increase the rent to the highest the market can absorb, i.e., $\\\$500$. What's keeping the equilibrium price below $\\\$500$?

Please also tell me if my reasoning about the other two cases is flawed, or there are other, better ways to reason about them.

**Tags:** microeconomics, equilibrium-price


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## 2 answers

### Answer by username_2 on 2020-10-24. Score: 3 | Accepted answer

I have quickly skimmed the chapter and there does not seem to be much about bargaining models so as the question says you have to solve this using supply and demand.

We know that an equilibrium price on the market will occur when supply intersect demand. Given that the supply is fixed at $24$ and $25$ apartments respectively it will be just straight vertical line in each case. Now when it comes to demand, if we assume each person wants only one single apartment the demand will be given in step-wise fashion, with two steps given by reservation prices ${\\\$}500$ and ${\\\$}200$ respectively. So this situation will look like shown on the tikz picture I made below (supply is blue, demand is red):

[![enter image description here](https://i.sstatic.net/pXzgu.png)](https://i.sstatic.net/pXzgu.png)

At point where supply is equal to $24$ the equilibrium price given by intersection of supply and demand ($S=D$) is clearly ${\\\$}500$. However, when supply is equal to $25$, supply and demand will intersect at a whole range of different prices $[200,500]$.

Consequently, in this case using this simple supply and demand model we cannot say anything more that price will be something between ${\\\$}200$ and ${\\\$}500$. There are more advanced ways how we could model this situation which might yield unique price instead of range (maybe some models with explicitly modeled bargaining process or some matching models), but here the author of the book gives you hints by asking how demand would look like etc., that you are supposed to use this solution concept (also the first chapter uses only simple supply and demand models so that is another hint).

> **Comments:**
> **username_0**: Thank you for your answer. By your logic, though, shouldn't the S = 26 graph intersect with the demand curve at every price from [\$0, \$200]? Why then does Varian say that the equilibrium price for 26 apartments is a fixed \$200?  
> **username_2**: @ijm no because there is no excess demand after Q=26, so in that case the price will be given just by the maximum reservation price.  
> **username_0**: I don't really understand why the excess demand at S = 25 (the sole tenant with a reservation price of \$200) should affect the price, given that 25 apartments can be leased to the 25 people with the highest reservation price, i.e., \$500. The argument involving the intersection of the demand-supply graphs makes sense, but where can I acquire the economic intuition behind this idea?  
> **username_2**: @ijm Since in your Q you only wanted to know how price changes between Q=24 and Q=25, I did not put that in the picture but at the boundaries (e.g even if case if there are only 24 consumers with all reservation prices 500 and supply =24) the price will be given by the reservation prices. However, within the boundaries you should look for intersections of supply and demand  
> **username_0**: I get that within boundaries we should look at the intersections of supply and demand, but that still seems a quirk of the model we've set up, and does not seem to say anything about the behaviour of the actors involved. For example, as I wrote in the Q, "if the equilibrium price is anything lower than \$500, landlords would have an incentive to arbitrarily increase the rent to the highest the market can absorb, i.e., \$500." I was asking about the economic intuition for that not happening.  
> **username_2**: @ijm the intuition here is that since landlords can’t price discriminate  here and can’t directly observe the reservation prices at S=24 there is only single price at which D=24 which is 500, at S=25 there is a whole range of prices  that will make supply and demand equal because we do not know what is the mechanism of competition between the buyer with 500 and 200 reservation price so any price there is possible, and at S=26 there again will be one price that is guaranteed to produce D=26 and that is 200  
> **username_0**: Thanks a lot. This seems like an interesting topic in its own right. Where can I learn more about "mechanisms of competition between buyers"?  
> **username_2**: @ijm these things are covered mainly by market/mechanism design and auction theory. I have older version of the textbook so I am not sure if chapters are correct but the Varian textbook ch 17 should cover auctions which is an intro to auction theory. Also the book will make the concepts for standard supply and demand more rigorous in later chapters. Other than that this is also covered in some length in MWG microeconomic theory and in Tadelis game theory (game theory is actually recommended prerequisite in these fields).  
> **username_2**: There are for sure texts that focus only on auction theory and market design but I can’t give more narrow recommendations there as I do not specialize in these two subfields, but the books I listed above have introductory treatments to those topics and they contain references to further papers/books. Alternatively, you can do your own additional search using the names of the field as key words, or you can make separate question as a reference request for such texts (but prior that you should do at least a bit of search yourself since users here don’t like Q that don’t show any effort).  

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### Answer by username_1 on 2021-08-25. Score: 0

With regard to situation with 25 apartments: The reservation prices of renters is not known in advance. The only price signals a landlord will get happen at the extremes. So if a landlord lists a price of 501, he finds that nobody will rent. So he rents below that. And if he rents at 200, he has excess demand, so he will rent above that. So the market finds a way to end up at say 201 to 499 inclusive (ignore partial dollars for now). If all the landlords rent at 300, there is no excess demand and no shortfall in demand, so everything is rented. No reason to do swaps. Yes, the landlords COULD have charged 499, but they don't know that since at 300 there was no excess demand.

With regard to situation with 26 apartments: Varian should have specified whether or not there was a 27th person here. If you have 26 people in the world and 26 apartments, they will rent out at 0<=rent<=200. Varian's answer which is $200 implies a 27th person in the mix.


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