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    I have a question about uncertainty and I'm not really sure if I'm doing it right so would love some help.

    Q: Hilary is an expected utility maximiser with a utility over outcomes (in terms of wealth) of u(z) = 2√ z. Of her wealth of 100,000 she can invest at most 80,000 in a risky asset. She knows that with a chance of 50 percent, the asset performs very well and yields a return of 50%, while with the same chance it performs badly and yields a (negative) return of −40%. So if she invests x, her final wealth would be either 100, 000 − x + 1.5x (if the asset does well), or 100, 000 − x + (1 − .4)x (else).
    (a) What is her expected wealth as a function of x?

    For this question I wrote:
    E(X) = 0.5(100,000-x+1.5) +0.5(100,000-x+(1-0.4)x
    = 10,000 + 0.05x

    The notes in my textbook aren't very clear on this. Is this correct?

    (b) What is her expected utility as a function of x? Denote this by U(x)

    as u(z) = 2√ z
    I assumed U(x) = 2√(10,000 + 0.05x)

    (c) What is the sign of U''(x)?

    I differentiated the above function with respect to x twice and got:
    U'(x) = 1/20(10,000+0.05x)^(-1/2)
    U''(x) = -1/800(10,000+0.05x)^(-3/2)
    Which gives a negative sign. I'm not sure what the significance of the negative sign is?

    (d) How much will Hilary invest?
    (e) Donald is in the exact same situation, with the only difference that his utility over outcomes is u(z) = 42 + z 2 . How much will he invest?

    With these last two questions I really had no idea. Would anyone be able to tell me whether I have this completely wrong and if so give me some hints? I don't major in Economics so it's a very weak subject for me.

    a) You have the right idea, but you've missed an x (should be 1.5x as stated by the question) and have made some silly mistakes simplifying.

    b) Right, but plug in new value from a).

    c) Just differentiate twice. This is significant for the next questions.

    d) You have been given a utility curve. Now you need to move away from a purely mathematical mindset. Hilary wants to maximise utility (basic assumption, and given in the question). Remember the differential shows the gradient. How can you use the differential U'(x) to find a maximum (or minimum, or stationary point)? Next, how can you use U''(x) to prove this is a maximum, not a minimum?

    e) Same as Hilary, but with a different utility curve.
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