Demand for apartments in a certain town is D(x) = 1052

Demand for apartments in a certain town is D(x) = 1052 - 4x, and the supply is S(x) = 500 + 8x, where x is the number of apartments, in hundreds, and D(x) and S(x) are the rent in dollars per month, per apartment. a) Find the equilibrium point. b) Find the consumer surplus and producer surplus. c) Suppose a maximum rent of $612 per month is imposed by the town council. Find the point (xc, pc). d) Find the new consumer surplus and new producer surplus. e) Find the deadweight loss.

Transcript

00:01 So we have some supply and demand curves here for apartments, saying that our demand is equal to 1052 minus 4 times x, and our supply here is 500 plus 8x.

00:17 Our first step is to find this equilibrium point, which is just where these two things are equal.

00:23 1052 minus 4x equals 500 plus 8x, moving things over to their sides, this becomes 12x, this becomes 552.

00:32 Pull up a calculator, 552 divided by 12 is 46, so our x value is 46, and that's this guy here.

00:48 So, yeah, we've got that supply and demand are each 46, so we've got 46 apartments.

00:54 And to figure out the price, we just need to do 500 plus 8 times x, so 8 times 46 is 868.

01:07 So 868 is our equilibrium point.

01:11 Now, to find the surpluses, we want the areas of each of these triangles here.

01:17 The area of a triangle is in general equal to 1 half times the height times the base.

01:23 And notice both of these triangles have the same base of 46, and then we can just do 1 half of 46 is 23, so 23 times the height of whatever the triangle is.

01:37 To do that, let's see, for the producer surplus down here, which i'm going to color in red, ps is 23 times, what's the height here, what's the difference from 500 is our point here, up to 868, so that'll be 868 minus 500, and that gives us 8464.

02:02 8464, that's great.

02:05 And then our consumer surplus in here will be 23 times, and that same thing, that's 1052, is the height up here, minus 868.

02:18 868, close paren, and we get 4232.

02:24 So the consumer surplus is 4232.

02:29 Great and cool.

02:31 But now we have a maximum rent of $612 per month, so we can ignore those for a minute and set a new maximum rent here, that point, that's the 612.

02:51 And what is the xcpc? so what do we want to do from there? well, what we need is first to find the coordinates of this guy here...

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