How to find expression for MR if Qd=490-7p?

Question

IF Qd=490-7p what is the expression for MR in this case

and secondly what would be the quantity is firm produced MC = 10

Answer 1

1. IF Qd=490-7p what is the expression for MR in this case

Ans: Total Revenue = TR= p* Qd,

but Qd = 490- 7p or p= 490/7- Qd/7=70 - Qd/7.

So TR= [70 - Qd/7]* Qd = 70*Qd- Qd^2/7

Marginal Revenue = First derivative of TR with respect to Qd

= 70 - 2Qd/7

--and secondly what would be the quantity is firm produced MC = 10

For profit maximization, the firm will produce that quantity where marginal cost MC= MR But you have given MC= 10,

So, 70 - 2Qd/7= MR = MC = 10, or,

2 Qd/7 = 60 , or, Qd= 60*7/2= 420/2 =210. So the firm will produce 210 units at MC=10

at a price p = 70 - 210/7= 70 - 30= 40

---If MC is constant what does it tells about firm's average total costs (assuming positive fixed costs)?

A constant MC means that the average variable cost (AVC) is equal to the MC or  variable cost per unit of output remains the same at all and any level of output and the fixed costs (FC) being poitve and fixed irrespective of the output level, the total cost is exressed as follows: TC=  AVC*Q+F

or, TC= MC*Q +F which means that Average Total Cost

= ATC = TC/ Q = MC+ F/Q . So, the firm's average total cost  continually declines as output increases.

- -  i indicated MC=MR

MR=P-ATC where MR is constant

so if we dont change P then ATC will be constant.

So the marginal revenue is above the average total cost price the firm is deriving an economic profit.

Is this correct answer?

---- This part of the question is not clear? If MC= MR at equlibrium level of output and MC is constant, then AVC is constant but ATC cannot be constant as shown earlier, it will continuously decline as output increases..

But irrespective of that, we can examine if' the marginal revenue is above the average total cost price the firm is deriving an economic profit." Please note that there is nothing like average total cost price !!. If Marginal revenue is higher than the average total cost, then the firm does make an economic profit because the average revenue or price per unit is higher than or equal to the marginal revenue. But that would not give us the profit maximization output unless the marginal cost equals marginal revenue. Please think thru again and post the exact question you have in mind.

Answer 2

MR = δTR / δQ = (TR)'

TR = Q*P

Q=490-7P

7P=490-Q

P=70 - Q/7

TR= (70 - Q/7) * Q = 70Q - Q²/7

MR=(TR)'= (70Q - Q²/7)' = 70 - 2Q/7

MC=MR

MC=10

70-2Q/7=10

70-10=2Q/7

60=2Q/7

7*60/2=Q

Q=210