Suppose the market value of a growing tree at time t is a function f(t, x), where a is expenditure on tree trimming at t = 0. (Think of a as an investment at t=0 that generates a return in the form of increased tree growth.) Assuming continuous compounding of the interest rater, the present discounted value of profit earned from harvest (or sale) of the tree at time t is V(t, x) = f(t, x)et - x. a. What are the first-order conditions for V(t, x) to have a maximum at t* > 0 and > 0? b. What are the first-order conditions if f(t, x) takes the separable form f(t, x) = g(t)h(x), with g(t) > 0 andh(x) > 0? c. In the separable case, show that g/(t*) < r²g(t*) and h(*) < 0 are sufficient conditions for a critical point (t*, *) to be a local maximum point for V. d. Find t* and ** when g(t) = evt and h(x) = ln(x + 1), and check the local second-order conditions.
Suppose the market value of a growing tree at time t is a function f(t, x), where a is expenditure on tree trimming at t = 0. (Think of a as an investment at t=0 that generates a return in the form of increased tree growth.) Assuming continuous compounding of the interest rater, the present discounted value of profit earned from harvest (or sale) of the tree at time t is V(t, x) = f(t, x)et - x. a. What are the first-order conditions for V(t, x) to have a maximum at t* > 0 and > 0? b. What are the first-order conditions if f(t, x) takes the separable form f(t, x) = g(t)h(x), with g(t) > 0 andh(x) > 0? c. In the separable case, show that g/(t*) < r²g(t*) and h(*) < 0 are sufficient conditions for a critical point (t*, *) to be a local maximum point for V. d. Find t* and ** when g(t) = evt and h(x) = ln(x + 1), and check the local second-order conditions.
Managerial Economics: A Problem Solving Approach
5th Edition
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Chapter5: Investment Decisions: Look Ahead And Reason Back
Section: Chapter Questions
Problem 5.2IP
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Suppose the market value of a growing tree at time t is a function f(t, x), where a is expenditure on tree trimming at t = 0. (Think of a as an investment at t=0 that generates a return in the form of increased tree growth.)
Assuming continuous compounding of the interest rater, the present discounted value of profit earned from harvest (or sale) of the tree at time t is V(t, x) = f(t, x)et - x.
a. What are the first-order conditions for V(t, x) to have a maximum at t* > 0 and > 0?
b. What are the first-order conditions if f(t, x) takes the separable form f(t, x) = g(t)h(x), with g(t) > 0 andh(x) > 0?
c. In the separable case, show that g/(t*) < r²g(t*) and h(*) < 0 are sufficient conditions for a critical point (t*, *) to be a local maximum point for V.
d. Find t* and ** when g(t) = evt and h(x) = ln(x + 1), and check the local second-order conditions.
Please do fast ASAP fast
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