This assignment comprises four Multiple Choice Questions (each worth 4%) and two long-answer questions (the first worth 36% and the second worth 48%). You should attempt all questions.

Please note that the multiple choice questions may have more than one answer. You do not need to show work for the multiple choice questions, but you must prepare full solutions for the long-answer questions.  We will mark your assignment partially on the correct solution and partially on   your process of finding the solution.  This means that you should show your work.  Just writing the correct answer will not get you full marks.  You need to show us, through your steps and brief explanations, that you have a good understanding of the material.

Multiple Choice Questions (4 Marks Each)

Q1    Which of the following utility function(s) represent a risk averse individual:

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(a)    u(x) = x

(b)    u(x) = ln(x)

(c)    u(x) = ex

(d)    u(x) = 

(e)    u(x) = x 2

Q2    The indexof absolute risk aversion is given by − . Which utility functions have constant absolute risk aversion (where Y is a positive constant)?

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(a)     U(x) = 

(b)    U(x) = Y ln(x)

(c)     U(x) = xln(x)

(d)    U(x) = −  e −Yx

(e)     U(x) = Y + Yx + Yx2

Q3    If (in one period of discounting) an individual has discount factor, δ  = 0.5, and their discount rate is T, then it must be true that:

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Q4                                                 f(x) = {200 30    if x(if x)  9(9)

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(a)    f(x) is continuous everywhere

(b)    f(x) can’t be differentiated anywhere

(c)    f(x) is undefined when x = 9

(d)    x(l)f(x) = 30

(e)    f(x) in an increasing function

Q5      (36 Marks)

Sam’s utility from drinking hot drinks is represented by:

1           1

u(c, t) = 2c2  + t 2

Where c is the number of cups of coffee they drink in a given week and t is the number of cups of tea they drink.

(a)      Carefully explain whether, according to Sam, coffee and tea are perfect substitutes?

(b)      Last week Sam’s utility from drinking hot drinks was equal to 6. Write down an

equation that expresses the different combination of drinks that might have been

drunk. Then rearrange this equation into a function, t(c), with cas the independent variable and t as the dependent variable.

(c)      How much tea would Sam have drunk last month if they had also drank 0,4, or 9 cups

of coffee? Explain why the domain of the t(c) function is 0 ≤ c  ≤ 9.

(d)     Show whether t(c) is aconcave function.

(e)      Sketch the function.

(f)       Now consider again the general utility function u(c, t) given at the start of the

question. Find the partial derivativesa(a)c(u)  anda(a)t(u) .  What other names do these have?

(g)      Compare Sam’s additional utility from drinking one more cup of coffee to drinking one more cup of tea.

(h)      Sam is offered the choice between either a cup of tea or a cup of coffee, but can’t decide which to take because they are indifferent. What can we infer about how   much tea and coffee they have already drank?

Q6      (48 Marks)

Assume that you have £20,000, which must be put into either (or split between) of the

following savings accounts, A and B. For both accounts, interest is calculated and paid daily (i.e. k=365).

Account A: Interest is 6% AER. If you withdraw before 180 days you pay a penalty sum

equivalent to the first 90 days of interest. If you withdraw on the 181st day,or later, there is no penalty.

Account B: Interest is 8% APR. Whenever you withdraw the money, you pay a penalty sum equivalent to the first 180 days of interest.

Assume that the first £1,000 of interest is tax free, and any interest earned over this is taxed at 20%.

(a)      Explain the difference between AER (annual equivalent rate) and APR (annual percentage rate), and express the APR as a function of the AER.

(b)      If you put £20,000 into account B, show that you would have less than £19,500 if you took it all out after exactly 60 days.

(c)      If you put £20,000 into account A, calculate how much money you would have if you took it all out after exactly 181 days.

Now assume you decide to keep an amount of your money,x, in bank account A (so that you also keep amount 20000 − x in bank account B).

(d)     Write down, G(x), your Gross (pre-tax) income you have after exactly 365 days, as a function of x.

(e)      Write down, N(x), your Net (post-tax) income after exactly 365 days, as a function of x.

(f)      Sketch N(x)

Now suppose there is a third savings account available, C, (interest is also calculated and paid daily):

Account C: Interest is 5% APR. You can take money out of this account whenever you like without any penalty.

(g)      Suppose you leave the money in one of the three accounts ford days. Carefully explain in detail why A is the best account iff  d  ∈  [181, 662].

Referencing

If you reference papers in your answers, you should reference them using a consistent referencing system, such as the Harvard  referencing  system; you should  normally  cite sources  in the text.   As  a  general  rule,  you  should  avoid  using footnotes to reference.

•     If you include a quote, it should be in quotation marks, and a page number included in the in-text reference.

•     Whilst you should normally avoid larger quotes, if you include them, you should also indent the text.

If you cite a paper in your essay, you should also include a full reference to the paper in the reference list at the end of the paper.

•     Do not list papers in your reference list that you have not referenced in the paper