Question
Question 2Suppose that the demand function is given by q 0.5p 70 and the supply function Is given by q 0.7p 50. Find the Consumer Surplus and Producer Surplus
Question 2 Suppose that the demand function is given by q 0.5p 70 and the supply function Is given by q 0.7p 50. Find the Consumer Surplus and Producer Surplus
Answers
Find the consumer and producer surpluses. $$ \text { Demand Function } \quad \text { Supply Function } $$ $$ p_{1}(x)=50-0.5 x \quad p_{2}(x)=0.125 x $$
This problem. We are given the supply function by s of cure Eagles Cute five house plus two times Q to the 3rd 3 house last 50. And we are also given the fact that supply and the man are in equilibrium that cute 16. So we're guy, we're going to go. Actually, the, uh produces service. So first we had to find a price. The price of a river that is price a week Leave room sequel to IHS of 16. That is the price given by the supply function at cuticle 16 which is the quantity off a equally room. So this see well equal to 16 the five house plus two times 15 16 sir into the three house 50 And that's equal to 1202. With that, we can now calculate the producer selfless So he produces three producers Surplus sequel to the interval from Syria to the quantity of fake Illyria 16 off the price of Hickory Cleary in 1202 minus the supply function. So it is given by the injury from 0 to 16 off, 1200 to minus cute. The five house minus two time skewed to the three house minus 15. And that secret to we can make the certification of the terms. Isn't the injury from 0 to 60 and off 1152 minus Q to the five have minus two times Q 23 Health's and these sequel to 1152 times Q minus Que 25 vowels plus Juan over five house plus one minus two times Q to the three house plus wanted, by the way. Three house plus one. All these expression diluted between Q equals Syria and cuticle 16. So we hit 101,100 feet to Terry Q minus. Here we have been seven house in the denominator. So is tube seventh times Q to the seventh house. Here we have minus two times and and the denominator we have three. Health Plus one is five house toe factories, two fits executed if I helps, and then all that will really between cuticle Syria and Q equal 16. So this is 1152 Q finest to seventh of Cuba to the seventh haves, and here we have 4/5 of Q to the five house between zero and 16 at Syria. We know we're going to have zero, so you're well weighed on the existing. And this is 1100 1152 times 16 honest to seventh of 16 to the 7/2 minus four. 50 16 to 5 lives. And so, uh, he's in a calculator. We find the two separate email people, too. 1 12 7 931.66 So the produces Subra surplus, I said regime illegal too. 12 South in 931 points $66.
While seeing the question looks for the move to the answer. So let's see here that given he is equal to 15 minus 0.5 Q. U. Is equal to 72. Over the value of Q. He is equal to 15 minus 0.5 and two 72. He is equal to 15- Statistics. So P is equal to $14. So prize electricity of demand. He is equal to 15 minus 0.5. Q. 0.5 Q. is equal to 15 B. Elasticity is equal to dick you. Bye. DP into B. Bye you. So it's a girl to minus two and two 14 by 72. So I lost 2 30 is equal to zero 388. So revenue Is equal to p. in two. Q. Is equal to $14 into 72 Is equal to $1,008. So here is the full solution of the question. That given P is equal to 15 minus 0.5. Q. Q. Is equal to 70. To put the value of Q. P. Is equal to 15 minus 0.5 and 2 72. P is equal to 15 minus 30 36. So P. Is equal to $14. So price elasticity of demand, P is equal to 15 minus 0.5. Q. 0.5 Q. Is equal to 15. Dynasty elasticity is equal to D. Q. By DP into P by Q. Is equal to minus two and 2 14 by 72. So the last 2 30 is equal to minus 0.388. Revenue is a call to pain to a QP in Tokyo Is equal to $14 into 72, which is equal to $1,008. So I hope you better understand the full solution. So for any further queries and doubts, I am there for help. So thank you.
This problem we are asked to find the consumer and producer surpluses for a demand function of 300 minus X. They supply function of 100 plus X. So the first thing that we're going to need to do is solve for the X value where we have equilibrium between supply and demand. So we said, we'll see daisy, we said supply 300- X equal to demand or opposite way around, but no difference here significantly. Uh So we can add X to both sides, subtract 100 from both sides, so we'll get to X equals 200 therefore X is going to equal 100. Then we can plug that back into one of our price functions and figure out that the price at 100 The equilibrium price is going to be 200. So now that we have that the consumer surplus, which I'll call sc going to be the integral from zero up to 100 of the demand function minus the price. That's 300 minus x minus 200. Which will be just 100 minus X. D X. So that will be 100 X minus X squared over two, Evaluated from zero up to 100. So that will be uh one second here Will be 100,000 or sorry, 10,000 rather 10,000 -10000 over two. So 10,000 -5000. So the result there should be 5000 and then the producer surplus Is going to be the integral from 0 to 100 Of the price. 100 minus the supply function. There will be 100 minus one. Yeah. Excuse me. One second here. All right. So yeah, that will turn out to be 100 -100 -1. So that will be just negative x dx evaluated from 0 to 100. So that will turn out to be negative X squared over two I evaluated from 0 to 100. So the result there Is going to be negative 5000. Excuse me, That should be 200 minus X. So 200 X. So 20,000 minus 5000. So the result there should be 15,000, I believe. One second. Yeah, no, I take that back. It was only in the producer surplus that I screwed that up. So the actually this should be down here 200 minus X. So that's going to it would be 200 minus 100 minus x. So that will turn out to be 100 minus X there. So the result that we arrive at, when we actually do everything properly here should be the same thing as what we had above. Essentially we should end up having At the end of the day we should end up having 5000 for the producer surplus
Let's begin with the sketch. We have the falling demand curve, which passes through the points 100 on the Q axes and 50 for the price. And they also have the rising supply curve, which goes through 50/3. Here goes up. This one goes down now by inspection. It is not hard to see that equilibrium prices 30 and that makes the Caribbean quantity 40. So let's head, um, in 40 here and 30 here. The consumer surplus is the area of this strangle, which is 1/2 times 20 times 40 which is 400. The producer surplus is here off the strangle. Choose 1/2 times 30 minus 50/3, also times 40. And this is 266 and 2/3.