Suppose that a firm faces the demand curve, P = 100 - 3Q, where P denotes price

Question

Suppose that a firm faces the demand curve, P = 100 - 3Q, where P denotes price in dollars and Q denotes total unit sales. The cost equation is TC = 200 + 22Q.

a. Determine the firm’s profit-maximizing output and price.   

b. Suppose that there is a change in the production process so that the cost equation becomes TC = 80 + 12Q + Q2.   Determine the resulting effect on the firm’s output:

c. Using the two different cost structures from part a and b, compute Total Cost and Marginal Cost at the quantity value of 12.   

Cost structure a: TC =

                           MC =

Cost structure b: TC =

                           MC =

d. Do the values computed in part c support the difference you found in the quantity values (compared output in part a and part b)?

e. Suppose that the firm sells in a competitive market and faces the fixed price:    P = $56. State the Total Revenue (TR) functions, and using the cost function in part b, find the firm’s new profit maximizing (optimal) quantity.

Please provide step by step ( i am trying to figure out how to do it)

Explanation / Answer

a.Given demand curve is P= 100- 3Q

We multiply it by Q to get total revenue (TR)

TR = P.Q = Q(100-3Q) = 100Q- 3Q2

Now differentiating the TR equation wrt Q to get marginal revenue (MR)

So,

MR = 100 -6Q

Similarly on cost side given that TC = 200 +22Q

Differentiating TC wrt Q gives us the marginal cost (MC)

So,

MC = 22

Now for profit maximization condition is MR = MC. Putting these equations together we get,

100- 6Q = 22,

Solving we get,

Q = 13

And to get profit maximizing price we put this value of Q in the demand function.

i.e. P = 100 – 3*13 = 100 – 39 = 61

So,

P = 61

-------------------

b.If TC now changes to TC = 80 +12Q +Q2

Differentiating TC wrt Q to get MC will give

MC = 12 + 2Q

And we have found in part a that MR = 100 – 6Q

Equating MR = MC

100 – 6Q = 12+ 2 Q

Solving we get,

Q = 11

Putting this value of Q in demand function we get

P= 100-3*11 = 100 – 33 = 67

So,

P = 67

---------------------

c.In part a MC and TC equations were

MC = 22 and TC = 200 + 22Q

So at Q = 12,

MCa = 22

TC a = 464

In part b MC and TC equations were

MC = 12 +2Q and TC = 80 +12Q +Q2

So at Q = 12

MCb = 36

TCb = 368

-------------------

d.Yes, the values computed in part c support the difference we found in the quantity values as compared in both parts a and b.

--------------------

e.The total revenue function will remain the same whereby TR = P*Q

i.e. TR = (100- 3Q)*Q = 100Q – 3Q2

in part b we have found out MC equation as,

MC =12 +2Q

Now in given condition when the firm is operating under perfect condition and the price is given, the condition for profit maximization is P = MC

Given that P = $56 as given equating these equations gives us,

12 + 2Q = 56

Solving we get,

Q = 22

So here the optimal quantity would be 22.

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