Question #85922

b) The demand for a commodity is given by p = 400 – q. The average total cost of producing the commodity is given by 1000/q+100-5q+q2

where p is the price in shillings and q is the quantity in kilograms.

Required

i) What does in the ATC equation represent economically?

ii) Determine the output that leads to maximum profit and the profit at the level of output.

where p is the price in shillings and q is the quantity in kilograms.

Required

i) What does in the ATC equation represent economically?

ii) Determine the output that leads to maximum profit and the profit at the level of output.

Expert's answer

b) p = 400 – q, ATC = 1000/q + 100 - 5q + q2.

i) Average total cost is the per unit cost that includes all fixed costs and all variable costs. Knowing average total cost is critical in making pricing decisions, as any price below average total cost will result in a financial loss.

ii) The output that leads to maximum profit is:

MR = MC,

MR = TR' = (p*q)' = 400 - 2q,

MC = TC' = (ATC*q)' = 100 - 10q + 3q^{2},

100 - 10q + 3q^{2} = 400 - 2q,

3q^{2 }- 8q - 300 = 0,

q = (8 + 60.53)/(2*3) = 11.42 kilograms.

p = 400 - 11.42 = 388.58.

The profit at the level of output is:

TP = (p - ATC)*q = (388.58 - (1000/11.42 + 100 - 5*11.42 + 11.42^{2}))*11.42 = 1458.31.

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