Question: In the lectures all differential equations have been defined in terms of time. That is, the derivatives in the differential equations have been with respect to time, t. This need not be the case. This is very natural and intuitive and the vast majority of applications will be of this type, but there is nothing in the mathematics that imposes this restriction. The following question, based on a famous concept from economic theory, uses the variable x, wealth, as that with respect to which differentiation takes place.

Let x be wealth and let u(x) be an individual’s utility function, depending on wealth.

Let \(u'\left( x \right)=\frac{du}{dx}\), \(u''\left( x \right)=\frac{{{d}^{2}}u}{d{{x}^{2}}}\). Define the function

\[\mu \left( x \right)=-x\frac{u''\left( x \right)}{u'\left( x \right)}\] (1)

µ(x) is called the Arrow-Pratt (AP) measure of relative risk aversion. This question addresses the issue of what type of utility functions have relative risk aversion that does not depend on the level of wealth. For the remainder of this question, consider only utility functions for which AP relative risk aversion is equal to a constant, k.

(a) Use equation (1) and the fact that relative risk aversion is constant to obtain a second order differential equation for utility, u(x). [10 marks]

(b) Rearrange your equation obtained in part (a) so it is of the form \(\frac{u''\left( x \right)}{u'\left( x \right)}=g\left( x \right)\),

where g(x) is a function of x and will include k. Then, by integrating both sides of this rearranged equation, obtain an equation for u’(x) (the derivative of the utility function).

You will find equations (2a) and (2b) below useful for this purpose, although you should certainly know (2b), and ought to know (2a):

\(\int{\frac{u''\left( x \right)}{u'\left( x \right)}dx}=\log \left( u'\left( x \right) \right)+C\)

\(\int{{{x}^{-1}}dx}=\log \left( x \right)+{{C}_{2}}\)

where c₁ and c₂ are constants of integration. [15 marks]

(c) Thus obtain a functional form for utility functions, u(x) that have relative AP risk aversion that does not depend on the level of wealth. [10 marks]

(d) By substituting back into equation (1), show that your solution does indeed yield functions of constant relative risk aversion. [5 marks]