## Problem Set 4

### Instructions: 经济学家的优化技术代写

Full work must be shown and the reasoning underlying a conclusion must be made clear. Reporting a result without showing your full work (i.e the calculations leading to that result) will not be accepted.

Some of the questions make use of the digits in your student ID number. Your answer will only be accepted if you use the correct digits.

1. (45 points) UBS (formerly Peckham Polytechnic) is one of the leading universities in online education. The executive board is making plans for its new Fasttrack E-Degree programme for the next academic year. Market research shows that the demand function for the programme is given by

Q = 8, 000 (0.2 + 0.05 S8) P

where S8 is the eighth digit of your student ID number, Q is the number of students enrolled in the programme and P is the tuition fee. The total cost function of the university is given by

T C = 6, 000, 000 + 2.5Q 2

#### (a) (20 points) Determine the number of students and the tuition fee that maximize the university’s profits. (Note: Assume that the number of students the university admits does not have to be an integer.) 经济学家的优化技术代写

(b) (25 points) The government decides that every student enrolled in an online programme in the UK must take an English test before enrolment. The cost of this test is £1, 000 payable directly to the government by the students. Although the fee is payable to the government, students will perceive this Optimisation Techniques for Economists University of Exeter, Fall 2022 fee as if it was part of their tuition fee and their demand will change accordingly. How will the introduction of this fee affect the optimal value of the university’s profits approximately?

1. (55 points) An investor is considering to invest £3, 000, 000 in the following assets:
1. The first is a US government bond. Each bond purchased pays a fixed amount of £1, 050 after holding it for a year and the unit price of the bond is £1, 000 (i.e. we assume that the investor is small compared to the size of the US bond market and any purchase will not affect the market price). We will denote the number of US government bonds purchased by the investor by dUS. Note that negative amount of bonds cannot be purchased.
1. The second asset is a UK government bond. Each bond purchased pays a fixed amount of £840 after holding it for a year. We will assume that the investor is influential in the UK bond market and any purchase will influence others to invest in UK government bonds as well. Consequently, the price of the UK government bond depends on the number of bonds purchased by the investor according to the function

pUK(dUK) = 735 + 10 S6 + 0.005dUK,

where S6 is the sixth digit of your student ID number, and pUK and dUK denote the purchase price and the number of bonds purchased by the investor, respectively. Note that negative amount of bonds cannot be purchased.

#### (a) (25 points) Determine the portfolio that maximizes the investor’s profits. 经济学家的优化技术代写

(Hint: Assume that the budget constraint is satisfied with equality; i.e. it is always binding. Also, you can eliminate prices from the problem altogether and set the Lagrangian up in terms of quantities dUK and dUS.)

(b) (30 points) Interpret the Lagrange multiplier associated with the investor’s budget constraint.