Question

please help me to solve question #6 to #12. best wishes.

Universal Gadgets (UG) is a firm with the cost function is given by 20 50q 3q 0.1q C(q) + where q is the total quantity produced by the firm. Assume (for questions 1 to 5) this firm aims to minimize its average cost. 1. Write down the optimization problem of UG. 2. Write down the first-order derivative condition(s) 3. Find the value(s) of q that satisfy the first-order condition(s) 4. Write down the second-order derivative condition(s) and check whether they are satisfied 5. What is the value of the minimum average cost?. Assume from now on that UG is a profit-maximizing, perfectly competitive firm operating in a market with the market pricep (p>0). 6. Write down the optimization problem of UG. 7. Write down the first order condition(s). Do not solve it/them yet! 8. Use the first order condition(s) and the Implicit Function Theorem to find the derivative of the supply function with respect to the market price. (Remember, you have to calculate this derivative without deriving the formula for the supply function!) 9. Finally , find the supply function of UG, and use it to check your answer to question 8 10. Write down the second order condition(s) and check if it/they are satisfied. 11 How much would UG produce if p 30? How much would UG produce if p-25? 12 Investigate the possibility of a corner solution q 15. Do not forget to write down and check the first order condition Assume now that p-30 and UG is not allowed to produce more than 15 units.

Answer 1

I already solved this so I'll just copy it.

The optimization problem is that the profits are maximized that is Revenue-Cost is maximized=

pq-C(q) is maximized

Differentiating pq-(20+50q-3q² +0.1q³) and finding first order condition for maximizing we get

p-50+6q-0.3q²=0

finding second order condition,we get 6-0.6q which should be <0 ( for max), 6-0.6q<0 for q> 10

For finding supply function, we equate p=MC where MC is marginal cost=C'(Q)

p=50-6q+0.3q²

If p=30

-20+6q-0.3q²=0 which has zeroes at 15.7735 and 4.2265, we reject the latter because it doesn't satisfy the second order condition,and round up the former to get q=16 for maximum(alternatively check at both 15 and 16)

if p=25

-25+6q-0.3q²=0

we get zeroes at 14.0828 and 5.917 and similarly we reject the latter and choose q=14

If UG cannot produce more than 15 at p=30,we will have a corner solution of q=15 for profit maximization as the equation for profit (pq-(20+50q-3q² +0.1q³ ) is increasing between 4.2265 and 15.7735 (given its first order condition is positive) and also has the maximum at q>10 because second order condition is <0 at q>10

Do tell me if you have any questions,hope it helps