A professor chooses how much time to spend on three activities: re- search, teaching, and administrative...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
A professor chooses how much time to spend on three activities: re- search, teaching, and administrative duties. The amount of time spent conducting research is written x, the amount of time spent teaching y, and the amount of time spent attending to administrative duties z. The professor has a total amount of time t > 1, and has preferences over these three activities represented by a utility function u: R³ →R, with u(x, y, z) = lnx + Iny+ainz, where a > 0 is a fixed parameter. The time spent conducting research must be at least 1 or else the professor will be fired; y and z cannot be negative. (a) Write down the professor's constrained optimization problem. [4] (b) Prove that the professor's objective function is concave and that [8] the professor's constraint set is convex. (c) Why will a non-degenerate constraint qualification (NDCQ) hold? [2] (d) Write down the Lagrangian for this problem. Compute the first- [4] order conditions. Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions. (e) Briefly explain why the solution to the equations in your answer to [2] part (d) will generate the solution to the problem in part (a). (f) Find the optimal amount of time spent on each activity (x,y,z) [8] when t>a + 2. What happens when t≤a + 2? (g) Derive the marginal utility of time as a function of t and plot it in a [8] graph. Is this function continuous at t= a + 2? Is it differentiable? (h) Show that the multiplier associated with the constraint that at least [4] time 1 must be spent conducting research is given by ax {0, a +1 -1}. Interpret this multiplier both when t>a +2 and when t ≤a + 2. A professor chooses how much time to spend on three activities: re- search, teaching, and administrative duties. The amount of time spent conducting research is written x, the amount of time spent teaching y, and the amount of time spent attending to administrative duties z. The professor has a total amount of time t > 1, and has preferences over these three activities represented by a utility function u: R³ →R, with u(x, y, z) = lnx + Iny+ainz, where a > 0 is a fixed parameter. The time spent conducting research must be at least 1 or else the professor will be fired; y and z cannot be negative. (a) Write down the professor's constrained optimization problem. [4] (b) Prove that the professor's objective function is concave and that [8] the professor's constraint set is convex. (c) Why will a non-degenerate constraint qualification (NDCQ) hold? [2] (d) Write down the Lagrangian for this problem. Compute the first- [4] order conditions. Write down the associated complementary slack- ness, feasibility, and non-negative multiplier conditions. (e) Briefly explain why the solution to the equations in your answer to [2] part (d) will generate the solution to the problem in part (a). (f) Find the optimal amount of time spent on each activity (x,y,z) [8] when t>a + 2. What happens when t≤a + 2? (g) Derive the marginal utility of time as a function of t and plot it in a [8] graph. Is this function continuous at t= a + 2? Is it differentiable? (h) Show that the multiplier associated with the constraint that at least [4] time 1 must be spent conducting research is given by ax {0, a +1 -1}. Interpret this multiplier both when t>a +2 and when t ≤a + 2.
Expert Answer:
Answer rating: 100% (QA)
a The professors constrained optimization problem is Maximize ux y z Inx Iny aInz Subject to x 1 y 0 ... View the full answer
Related Book For
A Survey of Mathematics with Applications
ISBN: 978-0134112107
10th edition
Authors: Allen R. Angel, Christine D. Abbott, Dennis Runde
Posted Date:
Students also viewed these accounting questions
-
The file P10_49.xlsx contains the amount of money spent advertising a product and the number of units sold for eight months. a. Assume that the only factor influencing monthly sales is advertising....
-
The graph below shows the amount of money spent in the United States on video games played on mobile devices in 2013 and the projected amount for the years 20142018. (a) Assuming this trend...
-
The amount of time spent by North American adults watching television per day is normally distributed with a mean of 6 hours and a standard deviation of 1.5 hours. a. What is the probability that a...
-
Question 7: The owner of a small social services management consulting firm wants to minimize the total number of hours it will take to complete four projects for a new client. Accordingly, she has...
-
a. Which types of capital expenditures incurred specifically for medical purposes are deductible? b. What limitations, if any, are imposed on the deductibility of these expenditures?
-
Will you be hiring employees during your first year of operations? If so, name their positions and describe the required qualifications, anticipated compensation, and how they will help your business.
-
In 2013, Thomas DePrince, a passenger aboard a cruise ship, visited the ships jewelry boutique, operated by Starboard Cruise Services, Inc. DePrince told the employees of the boutique that he was...
-
1. Although the case with OFCCP is closed, we wonder if there are any less discriminatory explanations possible for why our women sales reps on average earned less than men. If so, what are they? 2....
-
Tina is aware that thousands of children starve to death every day, but only after reading a story about a single starving child did she decide to donate money to a charitable organization. Tina's...
-
There is an array A made of N integers. Your task is to choose as many integers from A as possible so that, when they are put in ascending order, all of the differences between all pairs of...
-
JDS Shipyard's projected benefit obligation, accumulated benefit obligation, and plan assets were $55 million, $45 million, and $28 million, respectively, at the end of the year. a. What, if any,...
-
I owe $46k on a $50k business loan that I got and purchased a home with. Tree fell on home got substantial insurance claim. I just sold home home for $85k. I made a $35k profit after closing costs....
-
Reading the Admonitions for women, chose one of the following questions and answer it as an initial post (and be sure to comment on classmates who have answered questions of their own): What does Ban...
-
On 4 January 2 0 2 0 Thabo deposited P 7 8 0 0 into savings account. The simple interest rate agreed upon was 6 . 2 % per annum. What will be accumulated amount in the savings account on 2 1 November...
-
Assume before the June 2022 floods, Pakistan could produce only two commodities: electric cars and wheat. Also assume that Pakistan can produce at any of the positions represented by the production...
-
Share your experience in regards to your transition from High School to University. What were your biggest challenges and accomplishments?
-
QUESTION 8 8 How is data at rest protected? a . . Encryption b . . Decryption c . . Read / / Write permissions d . . Network router
-
How is use of the word consistent helpful in fraud reports?
-
One square yard equals 9 ft 2 . Use this information to convert the following. 13.5 yd 2 to square feet.
-
Add in the base indicated. 30245 + 40235
-
Use a graph to represent (a) The floor plan of your home. (b) The streets in your neighborhood within 1/2 mile from your home in all directions.
-
Consider the dynamic system, a mass, spring and damper structure, shown in Figure 2.2. (a) Draw a free-body diagram for the dynamic system (b) Derive the Input-Output model of the dynamic system k...
-
Two connected cars with an applied input force \(u(t)\) and negligible rolling friction can be represented by a translational mechanical system as shown below. (a) Draw the free-body diagrams of this...
-
Consider an RLC circuit consisting of a resistor \((R)\), an inductor \((L)\), and a capacitor \((C)\), connected in series, as depicted in Figure 2.4. Derive the Input-Output model of the network....
Study smarter with the SolutionInn App