I am trying to get through practice problems but want to know step-by-step how to get to the solutions shown in the book and why we need to do each of those steps. Please include all the steps required to solve the problem, including any mathmatical rules or "shortcuts" used to reach the transformation of the equation.

I've included some additional, specific questions I'd liked addressed below some of the problems.

The questions & answers provided are as follows:

1-Given a price of $10 and a marginal cost curve of f(x)=2x2 what does profit equal at an output of 3 with a fixed cost of $3?

Answer:

• Profit = Total Revenue - Total Cost
• Revenue = Price x Quantity = 10 x 3 = 30
• Cost =2x2dx=32x3+3=18+3=21
• Final Solution: 30-21=9

2-What is the marginal product of labor given the production function f(K,L)=yK0.7L0.6

Answer:

(L)(f) =0.6(K0.7)(L0.4)

3a- Suppose you are given a profit function of f(x)=x3+12x what is the optional input of x?

Answer:

• 0=3x2+12=x2=4=x=2
• 6x=0= x=0 and the second derivative is nonpositive

3b- Given the same profit function as the previous problem, what is the optimal profit?

(Why do we use x=2 and not x=0 or some other number?)

Answer:

When x=2 then f(x)=x3+12x=8+24=16

4-Suppose the profit function has changed and is now f(x)=(x+3)(x+2)(x5) , what is your optimal profit?

(How do we get to x=4 to know to substitute?)

Answer:

At x=4, Profit- 6 = (-4+3)(4+2)(4-5) = (-1)(6)(-1)

5- Given a utility function of f(x)=7x7 what is the absolute risk aversion?

(Why do we need to take ratio of the 1st and second derivative?)

Answer: f(x)=49x6f"(x)=249x5f"(x)f(x)=49x6249x5=6x1

6-What is the price-cost margin if price elasticity of demand is 0.45 for a monopoly?

Answer: 2.22

7-When elasticity is 1.5 by what factor should we expect a monopolist to markup marginal cost?

Answer: 3