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Richard acts to maximise utility at the end of one year, where the utility is given by the function $$u(w) = -\frac{1}{2w^2}$$ where $w > 0$ is wealth.

Richard is given the choice between a risk-free asset, with rate of return of zero, or a risky asset, with a random annual rate of return $X$ such that $1 + X = S$. The distribution of $S$ is given by

$$f_{S}(s) = 8s^3e^{-2s^2}$$ for $s > 0$, and $0$ otherwise.

The government wants to encourage investment in risky assets, in order to benefit the economy. Assume that Richard is permitted to invest a proportion $\alpha$ in the risky asset above and the remainder of his wealth in the risk-free asset. Furthermore, assume the government will create an annual wealth tax at rate $t$. The risky asset is exempt from the wealth tax - it applies only to the risk-free asset.

  1. How high must the government set the tax rate in order to encourage Richard to invest at least some proportion of his wealth in the risky asset?

  2. How high does the tax rate have to be for Richard to want to invest all his wealth in the risky asset?

So we aim to maximize

$$\mathbb{E}[u((1-t)(w-\alpha) + \alpha S)] = \int_{0}^{\infty} \frac{-f_S(s)}{2((1-t)(w-\alpha)+\alpha s)^2}ds$$

or more precisely, to see for which $t$ the maximum, as $\alpha$ varies, occurs internally (strictly in $(0,w)$) and when at $w$. Is my modelling correct? If yes, how to deal with the maximization of this integral?

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