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\section{Introduction}\label{sec:introduction}

  These lecture notes provide a gentle introduction to a particular set of solution tools for the canonical consumption-saving/portfolio allocation problem.  Specifically, the notes describe and solve optimization problems for a consumer facing uninsurable idiosyncratic risk to nonfinancial income (e.g., labor or transfer income), first without and then with optimal portfolio choice,\footnote{See \cite{merton:restat} and \cite{samuelson:portfolio} for a solution to the problem of a consumer whose only risk is rate-of-return risk on a financial asset; the combined case (both financial and nonfinancial risk) is solved below, and much more closely resembles the case with only nonfinancial risk than it does the case with only financial risk.} with detailed intuitive discussion of various mathematical and computational techniques that, together, speed the solution by many orders of magnitude.  The problem is solved with and without liquidity constraints, and the infinite horizon solution is obtained as the limit of the finite horizon solution.  After the basic consumption/saving problem with a deterministic interest rate is described and solved, an extension with portfolio choice between a riskless and a risky asset is also solved.  Finally, a simple example shows how to use these methods (via the statistical `method of simulated moments' (MSM for short)) to estimate structural parameters like the coefficient of relative risk aversion (\textit{a la} Gourinchas and Parker~\citeyearpar{gpLifecycle} and Cagetti~\citeyearpar{cagettiWprofiles}).
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