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- NOW THAT WE'VE DETERMINED THE EQUILIBRIUM POINT
GIVEN THE SUPPLY AND DEMAND FUNCTIONS,
WE WANT TO DETERMINE THE CONSUMER SURPLUS AT THIS POINT.
THE FORMULA FOR CONSUMER SURPLUS IS GIVEN HERE,
BUT LET'S REVIEW WHAT THIS ACTUALLY MEANS
BEFORE WE CALCULATE THIS.
WHEN A CONSUMER RECEIVES X ITEMS OF A PRODUCT,
A CERTAIN AMOUNT OF PLEASURE OR UTILITY IS DERIVED.
AND CONSUMER SURPLUS IS THE UTILITY OR PLEASURE RECEIVED
FOR WHICH THERE IS NO PAYMENT.
IF WE TAKE A LOOK AT THIS GRAPHICALLY,
IF THIS WAS THE EQUILIBRIUM POINT,
WHERE THE X COORDINATE IS THE QUANTITY
AND THE Y COORDINATE IS THE PRICE,
IF WE INTEGRATED THE DEMAND FUNCTION
ON THE INTERVAL FROM ZERO TO Q,
THIS WOULD GIVE US THE AREA HERE IN BLUE,
WHICH WOULD REPRESENT THE TOTAL PLEASURE OR UTILITY RECEIVED
FROM THE QUANTITY Q ITEMS.
BUT SOME OF THAT UTILITY IS PAID FOR,
AND THAT'S REPRESENTED BY THE PRICE x THE QUANTITY,
OR THE AREA OF THIS RECTANGLE HERE.
SO IF WE INTEGRATE THE DEMAND FUNCTION FROM ZERO TO Q
TO GET THE TOTAL UTILITY,
AND THEN WE SUBTRACT OUT THE PRICE PAID FOR THAT UTILITY,
THIS REGION HERE,
THIS AREA HERE REPRESENTS THE CONSUMER SURPLUS.
THIS IS THE UTILITY OR PLEASURE RECEIVED
THAT WAS NOT ACTUALLY PAID FOR.
ANOTHER WAY TO THINK OF CONSUMER SURPLUS
IS LET'S SAY YOU'RE WILLING TO PAY $100
TO PURCHASE A NEW MP3 PLAYER,
BUT THEN WHEN YOU GO TO THE STORE IT'S ON SALE FOR $80.
WELL, BECAUSE YOU WERE WILLING TO PAY $100
AND IT ONLY COST YOU $80, OF COURSE, YOU'RE HAPPY,
AND THAT $20 THAT YOU SAVED
COULD BE VIEWED AS CONSUMER SURPLUS.
NOW LET'S GO BACK AND TAKE A LOOK AT OUR EXAMPLE.
THE CONSUMER SURPLUS IS GOING TO BE EQUAL TO THE DEF INTEGRAL
ON THE INTERVAL FROM ZERO TO Q,
SO THE INTERVAL WOULD BE FROM ZERO TO 1600.
AND OUR INTEGRAND IS GOING TO BE THE DEMAND FUNCTION HERE,
SO WE'LL WRITE THIS AS 3,200X TO THE -1/2DX
MINUS Q x P, WHICH WILL BE 1,600 x $80.
SO HERE WE'LL APPLY THE POWER RULE.
WE'LL HAVE 3,200 x THIS'LL BE X TO THE -1/2 + 1.
THAT'S GOING TO BE X TO THE 1/2.
BUT THEN INSTEAD OF DIVIDING BY 1/2,
WE'RE GOING TO MULTIPLY BY THE RECIPROCAL,
SO WE'LL MULTIPLY BY 2.
AND WE'LL HAVE TO EVALUATE THIS AT 1,600 AND 0 - 1,600 x 80.
SO HERE WE'LL HAVE 6,400, AND THEN WE'LL HAVE--
THIS IS THE SQUARE ROOT OF X,
SO WE'LL HAVE THE SQUARE ROOT OF 1,600
MINUS THE SQUARE ROOT OF 0 - 1,600 x 80.
WELL, THE SQUARE ROOT OF 1,600 WOULD BE 40,
SO WE'LL HAVE 6,400 x 40, THAT'S 256,000 - 1,600 x 80 IS 128,000.
SO THE CONSUMER SURPLUS WOULD BE THIS DIFFERENCE,
WHICH IS $128,000.
SO THIS REPRESENTS THE AMOUNT OF UTILITY OR PLEASURE RECEIVED
THAT WAS NOT ACTUALLY PAID FOR.
IN THE NEXT VIDEO WE'LL TAKE A LOOK AT THE LAST QUESTION
TO DETERMINE THE PRODUCER SURPLUS
FROM THIS SAME SITUATION.