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- Wilbur Ramsey, Staff Member
hello everyone i'm sebastian y and this is managerial economics in this video we're going to put supply and demand together to solve for equilibrium let's start out with a quick sketch of a market quantity price demand and supply the market equilibrium occurs at the price at which the quantity supplied and quantity demanded are equal i will denote these as p e and q e if the price is above the equilibrium price then we get quantity supplied greater than quantity demanded which we call a surplus not to be confused with consumer and producer surplus if the price is below the equilibrium price then what we get is quantity demanded bigger than quantity supplied which is a shortage a market experiencing a surplus or shortage will naturally adjust into equilibrium at the equilibrium price we sometimes say that the market clears that is there's no excess demand no excess supply we will use our example beer supply and demand functions from the previous two videos to find the equilibrium of our beer market the way that we're going to go about this is to solve them together as a system of equations we know we need to find the price that makes the two quantities the same so all we're going to do is set these two things equal to each other so what we'll get is negative 600 plus 20 pb equals 900 minus 10 pb now we need to just do a little bit of algebra to solve for this i'm going to add 600 to both sides so on the right hand side we're going to get 1500 and then i'm also going to add 10 pb to both sides that's going to give me 30 p b all that's left is to divide both sides by 30 which is going to give me pb equals 50. i'll throw an e on that to denote that that's our equilibrium for the equilibrium quantity i'm now going to plug the 50 we just got into either of these equations since they're set equal to each other it does not matter which one i use let's use our supply function we get minus 600 plus 20 times 50. 20 times 50 is a thousand so that leaves us with 400. had i used the demand function i would get the same thing 900 minus 10 times 50 10 times 50 is 500 so that's going to leave me with 400 as well let's go ahead and graph our market i won't go through this in as much detail as before since we already showed how to graph these individually in the last two videos for the demand function we use our intercepts of 900 on the quantity side and 90 on the price side draw a straight line between them our supply function has an intercept of 30. we also want to mark out our equilibrium which is at 50 and a quantity of 400 and so we know our supply function is going to look something like this mark those out with supply and demand s d and that's our graph to calculate the consumer surplus that's going to be this triangle right here and our producer surplus is going to be this triangle down here consumer surplus it's a triangle with a base of 400 height of 90 minus 50 that's 40 divided by 2 which is 8 000 our producer surplus also has a base of 400 but a height of 50 minus 30 that's 20. divide by 2 we get 4 000. putting this all together the total economic welfare in the market is measured by the total surplus which is the sum of those two so that's going to come out to 8 000 plus 4 000 which is 12 000. another way we can approach our supply and demand is with the inverse functions that is the inverse demand function and the inverse supply function this just means that we take our regular demand and supply functions and we solve them for price instead of quantity let me show you an example for our demand function our demand function is q b d equals 900 minus 10 pb this is our demand function to get the inverse demand all we need to do is solve this thing for pb so i'm going to add 10 pv to both sides that's going to give me 10 pb plus qbd equals 900 now i'm going to subtract qbd from both sides and give me 10 pb equals 900 minus qbd and now divide both sides by 10 this is going to be our inverse demand function now this is a very common mistake so be careful here when we're dividing both sides by 10 we need to divide every single term by 10. so 900 divided by 10 90. qb divided by 10 as well to do this with our supply our supply function is qbs equals negative 600 plus 20 pv to get the inverse supply i will add 600 to both sides so we get qbs plus 600 equals 20 pb divide both sides by 20. again make sure to be careful about this we get q b s over 20 plus 600 divided by 20 which is 30. and that is our inverse supply just for consistency i'm going to flip this thing around and write pb equals qbs over 20 plus 30. i want to point out here that on the graph these are the exact same equations as before we've just rearranged them the reason for doing this is that sometimes it's more convenient to have things solved for quantity instead of price and several weeks from now we will start to see some examples as to where we might use these we can also use the inverse functions to solve for the equilibrium since both of these are equal to pb we can set them equal to each other so we can have 90 minus qb d over 10 equal qb s over 20 plus 30 and then solve for qb now to do this we're gonna have to add some fractions so for a quick review we can only add fractions when the denominators are the same so i'm going to take my qb d over 10 and multiply both the top and the bottom by two so i'm gonna get minus two q b d over 20 equals qb s over 20 plus 30. since we're finding the equilibrium we know these two quantities are going to be the same so i'm just going to replace the d and s with e for equilibrium and now we are going to add the 2 qb over 20 over to both sides 90 equals 3 qb e over 20 plus 30. subtract 30 from both sides we get 60 over there now multiply both sides by 20 over 3 so qb e will be 60 times 20 over 3 which is 400 just like we got before we can then plug that 400 into either the inverse supply or demand function to get the equilibrium price i'll plug it into the inverse demand function so we get 90 minus 400 over 10. 400 over 10 is 40 so we get 90 minus 40 equals 50 which is exactly what we got before if you want to do it this way do it this way solve for the inverse demands you get the exact same answer if we solve the system of equations because at the end of the day they are the same equations just rearranged our next topic is comparative statics comparative statics is the study of what happens to the equilibrium when some market conditions change that ultimately shifts one of the curves so for example we might think about what would happen to the equilibrium if the price of some inputs increased or the price of a substitute decreased or consumer income increase something like that let's take our example supply and demand functions that we've been using and do a little bit of comparative statics here suppose that the price of inputs decreased which shifts the supply curve 600 units to the right so that's going to add 600 to our supply function the negative 600 positive 600 are going to cancel out and we're just going to have 20 pb to figure out the new equilibrium we'll set all these equal to each other 20 pb is now equal to 900 minus 10 pb add
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- Wilbur Ramsey
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I've studied clinical psychology at Saybrook University in Pasadena and I am an expert in land management. I usually feel flirty. My previous job was casting director I held this position for 19 years, I love talking about mathematics and heliskiing. Huge fan of Andrew Cuomo I practice figure skating and collect disneyana.
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