Question

1. Find Total Cost, Average Total Cost and Marginal Cost for 2000th, 4000th and 5000th cases

TC= 5000+ 2.75Q - .0005Q^2 + .00000025Q^3

MC= 2.75- .0010Q + .00000075Q^2

2. make a graph for total cost and another for both marginal cost and average total cost

please show all work

Answer 1

1) for 2000th quantity

TC = 5000 + 2.75*(2000) - 0.0005*(2000)^2 + 0.00000025*(2000)^3

= 5000 + 5500 - 2000 + 2000

= 10500

AC = TC/Q

AC = 10500/2000 = 5.25

MC = 2.75 - 0.0010*(2000) + 0.00000075*(2000)^2

= 2.75 - 2 + 3

= 3.75

for 4000 th quantity

TC = 5000 + 2.75*(4000) - 0.0005*(4000)^2 + 0.00000025*(4000)^3

= 5000 + 11000 - 8000 + 16000

= 24000

AC = TC/Q

AC = 24000/4000 = 6

MC = 2.75 - 0.0010*(4000) + 0.00000075*(4000)^2

= 2.75 - 4 + 12

= 10.75

for 5000 case

TC = 5000 + 2.75*(5000) - 0.0005*(5000)^2 + 0.00000025*(5000)^3

= 5000 + 13750 - 12500 + 31250

= 37500

AC = TC/Q

AC = 37500/5000 = 7.5

MC = 2.75 - 0.0010*(5000) + 0.00000075*(5000)^2

= 2.75 - 5 + 18.75

= 16.5

2) to find out the slope/direction of Total cost curve, we can put value of Q= 0, 1,2,and so on

Q = 0, TC = 5000

Q= 1, TC = 5002.7495

Q = 2, TC = 5005.498

so it is a upward sloping curve.

we can find out the direction of the curve by double differentiating the equation of total cost curve. since, they have only asked the graph of the curve, we can perform the above calculations in rough and draw the curve. otherwise, double differentiation would have been the conventional way. (here, we will get double differentiation of TC as a function of Q, which simply means that when we increase the value of Q, value of TC will also increase. hence, upward sloping curve.)

find AC for Q=1,2 and so on (we need to check the values for 1and 2 only, we already got the values for 2000, 4000 and 5000 quantity)

Q = 1, AC = 5002.7495

Q= 2, AC = 2502.749

so here we see that AC is declining initially but at the 2000- 4000 th quantity it is increasing, which means AC follows usual U-shaped curve. to find out the lowest point, we need to differentiate the AC equation. To get the AC equation, divide TC by Q

AC = TC/Q = 5000/Q + 2.75 - 0.0005*Q + 0.00000025*Q^2

differentiate with respect to Q and equate it with the zero

-5000/Q^2 - 0.0005 + 0.00000025*(2)Q = 0

- 0.0005*Q^2 + 0.0000005*Q^3 = 5000

Q = 2544.511528 = 2545 (approx.)    (by using cubic formula/ calculator)

for MC

Q = 0, MC = 2.75

Q = 1, MC = 2.749

hence MC is also declining first, then increasing. to find out it's lowest value, differentiate it's equation with respect to Q and equate it with zero.

-0.001 + 0.00000075*2Q = 0

Q = 666.67 = 667 (approx.)

all these curve follows their respective rules/direction of slope. Following are the diagrams.