Consumers' and Producers' Surplus Suppose the supply function for a certain item is given by S(q

Consumers' and Producers' Surplus Suppose the supply...

Key Concepts

Consumer Surplus

Consumer surplus represents the difference between the maximum price consumers are willing to pay for a good and the actual market price they pay. It is illustrated as the area between the demand curve and the market price, up to the equilibrium quantity. Consumer surplus is an important measure of the economic benefit that consumers receive from participating in the market.

Producer Surplus

Producer surplus is the difference between the market price that producers receive and the minimum price at which they are willing to supply a good. It is shown graphically as the area between the supply curve and the market price, up to the equilibrium quantity. Producer surplus quantifies the benefit that producers gain from selling a good at a price higher than their minimum acceptable price.

Supply and Demand Curves

Supply and demand curves are graphical representations of the relationship between the quantity of a good that producers are willing to supply at various prices and the quantity that consumers are willing to purchase at those prices. The supply curve typically slopes upward, reflecting the fact that higher prices incentivize increased production, while the demand curve generally slopes downward, indicating that higher prices reduce the quantity that consumers are willing to buy.

Market Equilibrium

Market equilibrium is the point at which the quantity supplied by producers equals the quantity demanded by consumers. At this point, the market clears, meaning there is neither excess supply nor excess demand. The equilibrium price and quantity are determined by the intersection of the supply and demand curves, and they serve as key indicators of the overall market condition.

Transcript

00:01 We know that the supply function for a certain item is given by s of q equals q plus one square and the demand function is d of q equals 1000 over q plus one so we are going to graph both functions in the same axis and we have that here we can see that the supply function which is the blue curve has its typical increase in behavior in this case is slowly increase in behavior and then we have the red line which is the manned function with its typical decrease in behavior and then they both meet in a point which is the point of equilibrium which we get to calculate in par p so to find the point of equilibrium we got to solve the equation s of q equals d of q.

01:13 And this equation is, in this case, q plus 1 square equals to 1 ,000 over q plus 1.

01:30 So passing the term q plus 1 to the left, multiplying q plus 1 square, we get q plus 1.

01:38 Cube equals 1 ,000 and we can take cubic root both sides of the equation there is only one solution because the power is an odd power power 3 and then we get q plus 1 equals the cubic root of 1000 is 10 and so we get finally that q is equal to 9 quantity is q star equals 9.

02:28 Now we can calculate the equilibrium price, equilibrium price, using no matter what of the supply or demand function we use.

02:52 Because this is an intersection or point of meeting of the two curves, they both have the same value at this q star here.

03:02 So, the equilibrium price is p star equal, let's say s of q star, that's equal to 2 star plus 1 square, that is 9 plus 1 square, that is 100.

03:31 Then we can say that the equilibrium point, that is the point where supply and demand are equal, is p star, q star, equals to 1009.

03:59 So it's the part p and part c, we are going now to calculate the consumer surplus.

04:18 And we know that the consumer surplus, which we are going to call cs, is equal to the integral from 0 to the quantity of equilibrium, that is, 9, of the demand function, d of q, minus the price of equilibrium 100.

04:43 So this is equal to the interval of 0 to 9 of 1 ,000 over q plus 1 minus 100.

04:55 And that's equal to 1 ,000 times the integral from 0 to 9 of the q over q plus 1, 1, a 1 over q plus 1, differential of q minus 100 times q evaluated between.

05:15 Q equals 0 and q equals, sorry, and q equals 9.

05:28 And here i can see i made a mistake, so it's 9.

05:36 And that's equal to 1 ,000 times.

05:41 And we recognize here the natural algorithm of the absolute value of q plus 1, because the derivative of q plus 1 is 1, which is in the numerator.

05:51 So this is the natural...

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