ECO1005 exam 2018-9 - Past Paper PDF

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ECO1005/3/SEMR2/18/19 (1 hand-out)

UNIVERSITY OF SURREY© Faculty of Arts & Social Sciences School of Economics Undergraduate Programmes in Economics

ECO1005: Mathematics for Economics FHEQ Level 4 Examination

Time allowed:

2 hours

Semester 2 2018/9 Answer all FOUR questions Each question carries equal marks

Where appropriate the mark carried by an individual part of a question is indicated in square brackets [ ].

Calculators are permitted providing they are non-programmable and not wireless enabled

Additional materials: 1 hand-out (Formula sheet)

©Please note that this exam paper is copyright of the University of Surrey and may not be reproduced, republished or redistributed without written permission 1

ECO1005/3/SEMR2/18/19 (1 hand-out)

Question 1 A consumer’s utility function is 𝑈 = 𝑥10.25 𝑥20.75, where 𝑥1 and 𝑥2 are the quantities consumed of goods one and two, respectively. The prices of goods one and two are £2 and £4, respectively, and the income available to spend is £100. (a) Using the Lagrangean method, determine the maximum utility.

[70%]

(b) Interpret the optimal value of the Lagrange multiplier.

[30%]

Question 2 Consider an investment that involves an initial cost of £150, and produces cash inflows of £𝑥 at the end of year 1 and of £𝑥 at the end of year 2. (a) Derive an expression for the Internal Rate of Return (IRR) as a function of 𝑥.

[50%]

(b) Suppose now that the yearly available interest rate is 2% compounded daily, and 𝑥 equals 200. What is the Net Present Value (NPV) of the investment? [50%]

Question 3 Consider the series 𝑆𝑛 associated to the sequence 𝑎𝑛 = 𝐴(1 + 𝑟)𝑛−1 , for 𝑛 = 1,2, …, where 𝐴 and 𝑟 are constants. (a) Derive 𝑆𝑛 .

[70%]

(b) Suppose now that 𝑛 is large. What is a good approximation to 𝑆𝑛 if −1 < 𝑟 < 0? What if 0 < 𝑟 < 1? [30%]

[SEE NEXT PAGE] 2

ECO1005/3/SEMR2/18/19 (1 hand-out)

Question 4

(a)

Evaluate the integral

3

(ln(𝑥 + 1)) 𝑑𝑥 ∫ 𝑥+1 1 2

[35%] (b)

Evaluate the integral ∫ 𝑥 √1 − 𝑥 𝑑𝑥 [35%]

(c)

Suppose the function 𝑓(𝑥, 𝑦) is homogeneous of degree 𝑘, and has continuous secondorder partial derivatives 𝑓𝑖𝑗 (𝑥, 𝑦), for 𝑖, 𝑗 = 1,2. Show that 𝑥 2 𝑓11(𝑥, 𝑦) + 2𝑥𝑦𝑓12(𝑥, 𝑦) + 𝑦 2 𝑓22(𝑥, 𝑦) = 𝑘(𝑘 − 1)𝑓(𝑥, 𝑦). (Hint: apply Euler’s theorem to the first-order partial derivatives of 𝑓(𝑥, 𝑦))

[30%]

Internal Examiner: Federico Martellosio [END OF PAPER]

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