Below are screenshots of both the question I am answering and my work. I know my answer is wrong because I should be getting an equation for Demand. I suspect that my error has something to do with using ln, but I'm not sure. Any help would be appreciated.

The CARA (Constant Absolute Risk Aversion) utility function is a workhorse in the academic literature because it generates easy to use linear demands for normally distributed assets. For agent i, the function quantifies an individual's utility for a given level of wealth, w i , and risk tolerance, Ti (risk aversion coefficient is a reciprocal of risk tolerance, 3/,- = 1/ Ti ): T. 1 Ui (WI. )= _exp[_wf] Consider a one-period economy with a single risk asset and a risk-free rate equal to zero. The risky asset, present in positive fixed supply Q (with Q>0), is a claim on a single cash ow, Z, that is realized at the end of the period (t=1). Z is normally distributed with variance equal to 1. The risky asset is traded at price P at the beginning of the period (t=0). Agent 1' (who takes price P) as given) solves the problem of maximizing her expected utility at t=0, mDax E I. [U 1- (wt. )] SLIM/1.: w? +D,.(ZP] o where wi is agent i's initial level of wealth. Problem 1 Derive individual demand for the risky asset at t=0, Di , by writing out the first order condition (for an interior solution to) the maximization problem. Does the solution depend on the initial level of wealth? Why? Hint: (i) Maximizing EA _ is equivalent to maximizing - In(EX J. (ii) When X is lognormally distributed (i.e., In(4 ) is normally distributed, In(X ) ~ N(x,o? .), we have In (E[X ] =x+0, 12 E[U(w;)] = E[- e(;Wi)] = E[- e(7,)(W;+D;(Z-P))1 - In(- E[- e(7,)(W ;+D;(Z-P))1) = ElIn(- e(7,)(W;+D;(Z-P)> > ] E[(7)(w; + D;(Z -P)] E[U(Do)] = EL(!)(w; + D,(Z -P)] 0 =Z -P