If I am given TC(q)=11+q,how do I solve for marginal revenue?

Question

If a firm earns a constant price of 2 for each unit they sell and can choose to produce a quantity in the range [0,10] to produce. the firms cost function is represented by TC(q)= 11+q..

what is the firms marginal revenue,fixed cost,marginal fixed cost, profit maximizing choice of output and optimal profit??

Answer 1

Marginal Revenue is: $2 at all output levels

- Selling one more unit yeilds two dollars at all output levels

Fixed cost is: $11

-look at cost equation and see that when output is zero cost is $11

No such thing as Marginal fixed cost

Optimal output is infinity, this is very unusual but MR is constant at $2 and MC is costant at 1$ so it is profitable to increase production forever. However, your question limits output between zero and ten, meaning the company would produce ten units.

Answer 2

I respond as I received a msg. My response is:

Since the price is constant irrespective of the quantity sold in the range (o,10), marginal revenue is always 2 in that range and Total revenue is given by TR(q)= 2q. Note,marginal cost (MC) is given by dR/dq= 2 here.

The cost equation being TC (q) = 11+q, Fixed cost is the constant term 11: the fixed cost of 11 does not change whatever the level of output. Even if output q=0, the fixed cost of 11 has to be incurred.

Marginal Cost is 1. Check : at q=1 TC= 12, at q=2, TC=13, at q=3, TC= 14, at q=4, TC= 15, at q=10, TC= 21. Increase output by i unit, Total cost increases by 1. So, marginal (MC) cost is constant at 1. Mathematically, dC/dq=1

Since MR=2 and MC=1, on each extra unit the firm produces it makes a profit of 1 (@-1). Therefore the more it produces, more will be its profit. But it can produce maximum of 10 units as specified in your question. So, the profit max. output will be 10.

Optimal profit=TR(q=10) - TC(q=10) = 2*10 - [ 11+ 1*10] = 20- (11+10) = 20-21= -1. So, the optimal profit is actually a loss of 1. But if the firm produced any thing less it would make a higher loss. For example at q= 0, TR is zero but TC is 11, so loss is 11, at q=9, TR is 18, TC= 20 so loss is 2.

Note here the principle of MR=MC is not applicable as both the MR and MC are constant numbers. The profit is maximum when the output is maximum in this case. The firms maximum output is 10. So, it's optimal output is 10

Note also, since Fixed costs are Fixed and does not vary with output marginal fixed cost is zero.

If howevet the firm has not incurred any fixed costs meaning that it has not yet set up the facility  to produce q upto 10 at a fixed cost of 11, it should not get into that business because the business is not profitable at a production capacity of maximum 10. It should redesign its business project.to scale down its fixed costs and increase the capacity.