Micro doubts. Please help.

[MR]

Q1.A firm has the long- run cost function C(q)= 3q2+ 27. In the long run, it will supply a positive amount of output, so long as the price is greater than?
Q2. If output is produced according to Q= 4LK,the price of K is $10, and the price of L is $5, then the co st minimizing combination of K and L capable of producing 2 units of output is?
Q3. A firm uses a single input to produce its output, which is sold in a competitive market. It gets
quantity discounts on purchases of its input. If it buys x units of the input, the price it must pay per
unit of input is 289/x+ 3. If it buys no inputs, it doesn't have to pay anything. The firm's
production function is f ( x) = 45x-x2. If the price of the firm's output is 1, the profit-maximizing
amount of input to buy is?
Please share how to solve these.

[MI]

Assuming a perfect competition set-up, the quantity supplied in the long run will correspond to the minimised average total cost.

i.e. q= 3

by profit maximization you get p=MC => p>6q

so for price greater than 18 firm will supply non zero output

[MI]

Q3

profit= p*f(x) - cost per unit*x

differentiate and put equal to zero x=21

second differential in negative...so need its the optimal point

[MR]

Hey MI,
Thanks for replying. But how is the ATC 3? shouldn't it be 3q + 27/q?

[MI]

minimize ATC,

that happens at q=9

[MR]

Oh ok. Thanks.
Here is another question, let me know if you can solve it.
A profit-maximizing competitive firm uses just one input, x. Its production function is q = 8x1/2. The price of output is $24 and the factor price is $8. The amount of the factor that the firm demands is?

[MR]

It sqrt x.

[Devika]

Calculate MRPL i.e = MPL *P and equate it to W. Which is 8 here.