I need help with a Calculus question. All explanations and answers will be used to help me learn.
Example 1: The supply and demand equations for a new luxury wristwatch are given below. We will use to denote the price of the wristwatch in dollars and to denote the quantity of watches.
13
100
∶ = ( ) = 1 2 + 1000 5
(a) Verify that the equilibrium quantity is = 100 watches and find the equilibrium (market) price .
(b) Reasoning as in the video on consumer surplus we understand the following integral represents the summing of the “free benefit” to the consumers who are able to purchase the luxury watch as the equilibrium price (even though most were willing to pay MUCH more!). Compute the consumer surplus in this example. You can use a “machine” to compute the definite integral!
∫ ( ( )− )
0
(c) Applying similar logic to the supply side of things we see the producer surplus is given by the following integral. Compute the producer surplus in this example. You can use a machine to compute the integral!
∫ ( − ( )) 0
∶ = ( ) = 13,000−
Bonus: (Just for the nerds…I didn’t even give the answers to the peer leaders…)
Suppose the government decides to impose a $5,600 tax on the sale of these luxury wristwatches. What effect will this have on the economic surplus? (Recall: Economic Surplus is the total free benefit to society…usually the sum of the consumer and producer surplus and perhaps tax revenue…)
Hint: The tax will shift the supply function. The tax should not effect the consumers “willingness” to purchase the item at a given price, but it will certainly effect the suppliers willingness to sell as they will have to send a $5600 check to the government for EACH watch sold!
