A producer is a price-taker with cost function C(y)=4y3−32y2+96y. Determine the supply function.
Antwoord 1 correct
Correct
Antwoord 2 optie
y(p)={0if p<4223+124√48p−512if p≥4
Antwoord 2 correct
Fout
Antwoord 3 optie
y(p)={0if p<48+18√16p−512if p≥4
Antwoord 3 correct
Fout
Antwoord 4 optie
y(p)={0if p<324+18√16p−512if p≥32
Antwoord 4 correct
Fout
Antwoord 1 optie
y(p)={0if p<32223+124√48p−512if p≥32
Antwoord 1 feedback
Correct: AC(y)=4y2−32y+96 Then AC′(y)=8y−32 gives y=4. Since AC″(y)=8 it holds that AC″(4)=8>0. Hence, according to the second-order condition for an extremum AC(4)=32 is the minimum of AC(y).
The profit function of the producer is given by π(y)=py−(4y3−32y2+96y). which gives
π′(y)=0⇔p−(12y2−64y+96)=0⇔−12y2+64y−96+p=0.
Hence,
y=−64−√642−4⋅(−12)⋅(−96+p)−24=223+124√48p−512
and
y=−64+√642−4⋅(−12)⋅(−96+p)−24=223−124√48p−512.
Since π′(y) is a quadratic function with a<0 we find the maximum profit for an output quantity of y=223+124√48p−512, whenever p≥32. We conclude that the supply function is defined by y(p)={0if p<32223+124√48p−512if p≥32
Go on.
The profit function of the producer is given by π(y)=py−(4y3−32y2+96y). which gives
π′(y)=0⇔p−(12y2−64y+96)=0⇔−12y2+64y−96+p=0.
Hence,
y=−64−√642−4⋅(−12)⋅(−96+p)−24=223+124√48p−512
and
y=−64+√642−4⋅(−12)⋅(−96+p)−24=223−124√48p−512.
Since π′(y) is a quadratic function with a<0 we find the maximum profit for an output quantity of y=223+124√48p−512, whenever p≥32. We conclude that the supply function is defined by y(p)={0if p<32223+124√48p−512if p≥32
Go on.
Antwoord 2 feedback
Wrong: y=4 is the minimum location of the average cost function, not the minimum itself.
See Minimum/maximum.
See Minimum/maximum.
Antwoord 3 feedback
Wrong: y=4 is the minimum location of the average cost function, not the minimum itself.
See Minimum/maximum.
See Minimum/maximum.
Antwoord 4 feedback