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In a purely competitive market, the price of a good is naturally driven to the value where the quantity demanded by consumers matches the quantity made by producers, and the market is said to be in $ equilibrium $. These values are the coordinates of the point of intersection of the supply and demand curves.

(a) Given the demand curve $ p = 50 - \frac{1}{20} x $ and the supply curve $ p = 20 + \frac{1}{10} x $ for a good, at what quantity and price is the market for the good in equilibrium?

(b) Find the consumer surplus and the producer surplus when the market is in equilibrium. Illustrate by sketching the supply and demand curves and identifying the surpluses as areas.

a) Equilibrium occurs when $x=200$ and $p=\$ 40$

b) consumer surplus $=\$ 1000 \quad$ producer surplus $=\$ 2000$

Applications of Integration

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Okay, so the question gives us a supply curve of P is equal to 20 plus one over 10 x, and the demand curve of P is equal to 50 minus 11 over 20 x, and it ask first for part A to find equilibrium, which it defines for us as the intersection of the supply and demand curve. And then it asks us to identify and find producer and consumer surplus. So for part egg, the equilibrium point it is defined as the point where these two lines intersect, which I've labeled with the big dot and having dash lines going to the P and ex axes. So to get it, we simply set the supply and demand curve equal. So that's going to be 20 plus 1 10th X is equal to 50 minus 1/20 X answer now adding 1/20 to both sides and subtracting 20 from both sides. We get that three over 20 X is equal to 30 and then we want to multiply by two by 20 and then divide by three on both sides and then on this side those will cancel and we left with X as equal two 200. So that is our equilibrium Quantity Now to get the equilibrium price since the curves are equal for exes 200 you can plug in tow either one you want. I've gone ahead and plugged into the supply curve. So to do that, we just simply say 20 plus one over 10 times. 200. So that's 20 and then 200 over tennis. 20 again. So 20 plus 20 is 40. Okay, so now that we have that part, a excess 20 on P is $40. No, for part B, it asked us to for find producer and consumer surplus going to the graph. This region is consumer, sir Plus and this region is producer surplus now, since the price of 40 is just a constant line and then these two functions for supply and demand or linear these two regions will actually be triangles. So it makes it much easier to solve for them because we can say that consumer surplus is equal to 1/2 times the base of the tribal, which in this case is 50 minus 40 times the height which we solved for earlier as 200 which gives us 1/2 of 200 is 100 times 10 consumer surplus of $1000. And then for producer surplus we have This is 1/2 times its base, which is going to be 40 minus 20. And then the height begin is 200 Um, 1/2 of 200 is 101 100 times 20. There's 2000 So in the end we have producer surplus equal a 2000 and consumers who surplus equal to 1000 for a part being.