There is a lot of notation in the paper that might get you bogged down. These suggestions are aimed at helping with that.
The model has a random variable, n, with a probability density function, f(n). This random variable is the random component of wealth. As stated in the paper, n is a continuous random variable. This assumption is for slickness in the notation only. It doesn't materially impact the results at all. It might help you to understand the economics if instead n is a discrete random variable with a two point support, nL and nH. In other words, nL denotes a low income shock and nH denotes a high income shock. Then let pL and pH denote the corresponding probabilities. The expectation of n is given by E(n) = pLnL + pHnH. Also for simplicity, treat the agent's action, a, as a scaler.
Once you've done these substitutions, you can try to derive the solution via the method of Lagrange multipliers. Use the separable form of the utility function that is in equation (10) in the paper. It is the easiest to interpret. I've written up some notes for you so that it is not too hard to reproduce the results. The essence of the article is in contrasting the solution to Case I (No Individual Choice) to Case III (Individual Chooses Before Nature - Insurer Monitors Only R). Time willing, we'll then relate the other cases to these two.
Also note that there is a typo in equation (3). At present it say the utility, u, equals the Lagrange multiplier, lambda. It should say the the marginal utility of income equals lambda.
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