Problem 3
Suppose the cost function of producing Q > 0 units of a commodity is C(Q)=aQ2 + bQ + c Where a, b, c are all constants.
(a) Find the critical value of Q that minimizes the average cost function, AC(Q) = C(Q)/Q (this is called the minimum efficient scale in microeconomics).
(b) Find the marginal cost function MC(Q) = dC(Q)/dQ, and show that MC(Q) = AC(Q) at the critical value of Q you found in part (a).
Problem 4
Consider that a person has a utility of money, x, U(x) = ln(1+0.5x). For simplicity assume that we cannot have negative money, i.e. he cant borrow, so that x0. He is offered to enter a bet where there are two possible payouts, $5 with a probability of 0.25, and $25 with a probability of 0.75.
(a) Is this a risk averse, risk neutral, or risk loving individual? How do you know?
(b) If this were a fair game, what would the cost of the bet be?
(c) How much should this bet cost so that this particular individual would be indifferent between making the bet or not? (Hint: remember that an individual is indifferent when the utilities of the two options are the same.)
(d) Is the cost in part (c) higher or lower than if the bet was a fair game? Does this have to do anything with the persons attitude towards risk that you mentioned in part (a)?
Problem 5
Consider that we have a plantation of pines that currently have a value of $5,000. The value grows at a continuous rate of 4t1/4
(a) Write the expression for the present value, P V , of the plantation in terms of t and the interest rate, r.
(b) Write the expression for the optimal time, t*, to cut and sell the pine timber as a function of the interest rate, r.
(c) Check the second-order condition for a maximum at the optimal value of t* . Does it hold, knowing that r > 0?
(d) Assume That R=0.04. What is the value of t* and of PV* ?
Problem 6
Consider the function f(x) =2/(3x+1)
(a) Were going to consider a 2nd-order Taylor expansion around the point x = 2. What is the 2nd-order polynomial that approximates f(x)? That is, find the expression for P2 in this case.
(b) What is the general form of the Lagrange remainder for this case? That is, find the expression for R2 in this case. (Hint: this is a function of x and x*)
(c) Now consider that x =4. What is the value of f(4)?
(d) What is the value of P2 that you found in part (a) when evaluated at x= 4?
(e) What is, then, the value of the remainder R2, when x =4? (Hint: This is an actual number not a function of x* .)
(f) What is the value of x* that will make the function and the full expansion have the same value at x= 4?
Problem 7
Consider the function Y=[(x-7)x]2
(a) At what value(s) of x is it possible that we have a local maximum or minimum, or an inflection point?
(b) Use the general test we have seen in section 11.1 to determine whether we have a maximum, minimum, or inflection point, for each of the critical values you found in part (a).
