I need help with math for socialsciences

Question

Firm’s total cost function is of the form of TC=2Q + 14 (i.e., a firm’s fixed costs are 14 and the variable costs are 2 per unit). The corresponding profit function is of the form 𝜋 = AQ² + 𝛽Q + F . The firm makes a loss of 6 when Q=1 and a profit of 4 when Q=6. a)

(10 p.) Find A, 𝛽 and F.

b) (10 p.) Find the break-even points and hence find the maximum profit.

c) (10 p.) Find an expression for total revenue and deduce the demand function.

Answer 1

Let's see what we know. TC = 2Q + 14 . Fixed cost is 14 and variable cost is 2Q. this means marginal cost = average variable cost which = 2. finally Profit = aQ^2 +bQ + F. When Q = 0 Profit = F, Total revenue =0 ( you aren't selling anything and total cost = 14 hence profit is minus 14 ( your fixed costs) When q = 1 Profit = -6. Your total cost is 2Q + 14 = 16. If your costs are 16 and you lost only 6 dollars you must have made $10 in revenue. Total revenue =10. Since you only sold 1 unit, the price must also be 10Now, Let increase Q to 6. (change in Q = 5) Profit rises by 10 from -6 to plus 4. When Q = 6, TC = 14 + 2*6 = 26. If total cost = 26 and profit = 4, then toal revenue = 30. Average revenue is total revenue/Q which is sometimes known as Price/ Selling 6 units for $5 a piece will get you 30 dollars. We now have 2 points on the Demand curve (1,10) and (6,5) we saw sales rise by 5 when price falls by 5 which tells us the demand curve has a slope of -1. We will show this in a bit. If you are really smart you will see the demand curve is given as p=11-Q. We will prove this in a bit. Your teacher want to play with you for a bit first.

Now, lets solve for a and b. we Know aQ^2 +bQ -14 = =6 when q=1 and +4 when q=6. This will give us 2 equations in 2 unknowns. namely,

a(1^2) +b(1) -14 = -6 (i) and a(6^2) +b(6) -14 =4 (ii)

we can expand (ii) so 36a+6b =18 or 6a +b =3 (iia)

simplifying (i) we have a+b=8 or b= 8-a (ia)

Substitute the right hand side of (ia) into the left hand side of (iia) and get

(6*(a) +(8-a) =3 or 60 or 5a=5 or a = 1. when a = 1 b= 8-a or b=9.The profit function can be written

Can now be written as Profit = -Q^2 +9Q-14 which is parabolic and DOES NOT GROW WITHOUT BOUND. There will be 2 levels of output where profit is zero. This is determined by evaluating the quadratic:

-Q^2 +9Q-14= 0 a=-1 b=9 c =-14 the two roots are Q** = (-9 +-(81-4*(-1*(-14)).)^(1/2)/(-2)=

= 9/2 +- [(81-56)^(1/2)] = 9/2 +- (25^(1/2))/2 = (9/2)+- 5/2 so the break evens are 4/2 and 14/2. That is, q less than 2 or greater than 7 will incur losses

Recognize that Profit = TR -TC and TC = 2Q -14. Soooo, TR is simply Profit + Total Cost so -Q^2 +9Q -14 +2Q +14 = 11Q-Q^2 =TR.. Now, Average Revenue = TR/Q = 11-Q=P. The Marginal Revenue curve will be twice as sttep as the demand curve so MR = 11-2Q. Since marginal cost is constant MC=2 Profits are maximized where MC=MR or 2=11-2Q or 9=2Q or q=4.5. When Q = 4.5 price will equal 11-4.5=6.5.

This is not a Giffen good. This is the profit function of a monopolist facing a downward sloping demand curve The concept of a Giffen good is an academic curiosity. The idea is that if you have an inferior good and ONLY THE PRICE OF THAT GOOD goes up it is theoretically possible that the quantity demanded might increase. The during the Irish Famine INCOMES FELL by 25 to 50 % the demand for inferior goods SHIFTED TO THE RIGHT. THIS EXAMPLE SHOULD BE STRICKEN FROM ALL IN INTRODUCTORY TEXTBOOKS!!!!

Answer 2

a) F=-14. IF Q=0, Revenue = 0 but fixed costs are 14

Profit = Revenue - Cost = R - C = 0 -14

A^2Q + BQ -14 =4 when Q = 6

36Q + 6B = 18

A^2Q +BQ -14 = -6 when Q =4

16A + 4B = 8

4A +B = 2

24A +6B =12 subtract from 36Q+6B = 18 to get

12A =6

A = 1/2

B=0

Profit = Q^2/2 -14

b) Break even is when Q^2/2 -14 = 0, Q^2 =28, Q =sqr28 = 2sqr6 = 5.3

since Profit = -6 at Q=4 and =4 at Q=6, the breakeven point has to be between 4 and 6.

Maximum profit is infinite. The higher Q, the higher the profit

c) Revenue = R = Profit plus Cost = Q^2/2 -14 + 2Q + 14 = Q^2/2 +2Q = Q(Q+2)= QP = Q times Price

R = PQ = Q^2+2Q. P=Q+2 which is an upward sloping demand curve, a rarity, a "Giffen" good

illustrated by the Irish potato famine, where as price of potatoes rose, people bought more, as

they were so poor, they switched out of any luxuries and substituted the most basic necessities,

even buying more at higher prices.