Demand for apartments in a certain town is dx680 x and the...

Demand for apartments in a certain town is D(x) = 680 - x and the supply is S(x) = 500 + 9x, 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 $554 per month is imposed by the town council. Find the point (x_C, P_C). d) Find the new consumer surplus and new producer surplus. e) Find the deadweight loss.

Transcript

00:01 Hi here for the given question.

00:03 We are given that the value of demand function s of x is equal to supply function.

00:09 So as of x equals to demand function will give us equilibrium point if both the values are equal.

00:14 So here we have 680 minus x is equal to 500 plus 9x.

00:19 So simplifying further we have 10x is equal to 180 which gives the value of x is equal to 18.

00:26 So here using the value of x is equal to 18.

00:29 We can say that we can find our equilibrium point.

00:32 So here now further this is the point we have equilibrium at x equals to 80.

00:38 Now, we need to find consumer surplus.

00:40 So consumer surplus is equal to integration over 0 to 18 and we need to do for supply function.

00:46 So here we have 680 minus x dx.

00:50 So this is equal to after integrating 680 x minus x square upon 2 and the limit is from 0 to 18.

00:56 So substituting the value and limit and simplifying this we can say that here in our case, we got the value of consumer surplus is equal to $12 ,078.

01:06 So this is the solution of the second part.

01:09 Now here in the next part, we need to calculate the value of procedure producer surplus.

01:15 So producer surplus can be calculated as integration over 0 to 18 and here we have demand function.

01:21 So demand function is 500 plus 9x dx.

01:24 So integrating this we have producer surplus is equal to 500x plus 9x square upon 2 and the value of integration is 0 to 18.

01:34 So here by substituting the value of limit and simplifying this we can say that here in our case producer surplus is equal to $10 ,458.

01:44 So this is the value of producer surplus.

01:47 Now here for the next part, we need to calculate the equilibrium point if the maximum rent is maximum rent of $554 per month is imposed per month is imposed.

02:05 So here we set our equation as 680 minus x is equal to 554.

02:10 So solving this we have x is equal to minus 126 minus x.

02:15 This further implies x is equal to 126 units.

02:18 So here in our case further using the value of this x we can say that our new equilibrium point will be new equilibrium point will be equal to 126 ,554...

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