dollars per gallon This is an increasing function since the producers will supply more like as the price increases.

Transcription

1 Qustion 2: What ar consumrs and producrs surplus? An intrsting application of th ara btwn curvs is consumrs and producrs surplus. To undrstand ths concpts, w nd to racquaint ourslvs with dmand and supply functions. W ll do this through spcific functions. A dmand function P D( Q) rlats th quantity of som product Q to th pric P that consumrs ar willing to pay for it. Suppos w hav th dmand function for milk in som rgion, DQ ( ).5Q 7.75 dollars pr gallon whr th quantity Q is in thousands of gallons. Sinc this function has a ngativ slop, as th quantity incrass th pric must dcras. For th consumr, whn th pric is chapr thy ar willing to purchas mor milk. Th supply function for milk is S Q Q dollars pr gallon This is an incrasing function sinc th producrs will supply mor lik as th pric incrass. Th dmand and supply functions intrsct at th quilibrium point Q, P P, th consumr dmands Q units and th producr is willing to supply. At a pric Q units. Th subscript rfrs to quilibrium on ach lttr. W can find th valus algbraically by stting th dmand and supply function qual. W may also stimat th valus by xamining a graph. 9

2 S Q Q DQ ( ).5Q7.75 Figur 4 Th dmand and supply function for milk in som markt. For th dmand and supply functions for milk, th markt is in quilibrium at a pric of $ pr gallon. At this pric, th consumrs ar willing to purchas thousand gallons of milk and th producrs ar willing to sll thousand gallons. This would rsult in rvnu for th producrs of dollars Rvnu thousand gallons gallon 285 thousand dollars This is th ara of th rctangl outlind by th dashd lins and axs. On this graph, ara dscribs rvnu. W could also dscrib this ara using a dfinit intgral, Q PdQ Q dq St P qual to Apply th Fundamntal Thorm of Calculus to valuat th dfinit intgral 285 This amount assums ach gallon is sold at a constant pric of $ pr gallon. 1

3 For lowr quantitis th consumrs would b willing to pay mor than $ pr gallon sinc th dmand function DQ is highr than $ dollars pr gallon. If w comput th ara undr th dmand function, w find th total amount of mony consumrs would b willing to pay for milk. Th dfinit intgral for this ara is Q DQ ( ) dq.5q7.75 dq Substitut.5Q 7.75 for DQ ( ) 2 Q Q Apply th Fundamntal Thorm of Calculus to valuat th dfinit intgral Th consumrs would b willing to spnd thousand dollars on milk. Although consumr s would b willing to pay $51,625 for milk, thy only pay $285, by paying th quilibrium pric. Th amount savd by paying th quilibrium pric is calld th consumrs surplus Consumrs' Surplus thousand dollars This amount corrsponds to th ara btwn th dmand function DQ ( ) and th constant P from Q to Q Q. 11

4 Figur 5 Th consumrs surplus is th ara btwn th dmand function and th quilibrium pric from Q Q. Q to To hlp illustrat th maning of this ara, w hav calculat th ara blow th dmand function, blow th quilibrium pric, and subtractd th rsults. W can also find th consumrs surplus by subtracting th functions first and thn computing a dfinit intgral. Consumrs Surplus Consumrs' Surplus DQ ( ) P dq Q Exampl 4 Comput th Consumrs Surplus Us th formula abov to calculat th consumrs surplus for th milk dmand function DQ ( ).5Q 7.75 dollars pr gallon whr Q is th quantity of milk in thousands of gallons. Assum an quilibrium quantity of thousand and an quilibrium pric of $ pr gallon. 12

5 Solution Substitut th dmand function and quilibrium point into th formula, Consumrs' Surplus.5Q7.75 dq.5q4.75 dq Simplify th intgrand 2 Q Q Apply th Fundamntal Thorm of Calculus by finding th antidrivativ Th supply function dscribs th prics at which a producr would b willing to supply a product. Th ara undr th supply function corrsponds to th amount of rvnu a producr would b willing to accpt for a product. Th ara from Q to Q is 2 Q QdQ This mans that th producrs would b willing to accpt thousand dollars for thousand gallons of milk. Howvr, if consumrs pay th quilibrium pric for all of this milk thy will pay a total of 285 thousand dollars. Th xcss rvnu th producrs rciv is calld th producrs surplus. Th producrs surplus, Producrs' Surplus thousand dollars corrsponds th ara btwn th constant P and th supply function SQ. 1

6 Figur 6 - Th producrs; surplus is th ara btwn th quilibrium pric and th supply function from Q Q. Q to Producrs Surplus Producrs' Surplus Q P S Q dq Exampl 5 Comput th Producrs Surplus Us th formula abov to calculat th producrs surplus for th milk supply function SQ ( ) Q dollars pr gallon whr Q is th quantity of milk in thousands of gallons. Assum an quilibrium quantity of thousand and an quilibrium pric of $ pr gallon. Solution Substitut th supply function and quilibrium point into th formula, 14

7 Producrs' Surplus QdQ 2 Q Q Apply th Fundamntal Thorm of Calculus by finding th antidrivativ 15