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[1] A linear marginal utility is essential for deriving [(6)](#equation-sprob5) from [(5)](#equation-sprob4). Suppose instead that we had imposed the following more standard assumptions on the utility function: $ u'(c) >0, u''(c)<0, u'''(c) > 0 $ and required that $ c \\geq 0 $. The Euler equation remains [(5)](#equation-sprob4). But the fact that $ u''' <0 $ implies via Jensen’s inequality that $ \\mathbb{E}_t [u'(c_{t+1})] > u'(\\mathbb{E}_t [c_{t+1}]) $. This inequality together with [(5)](#equation-sprob4) implies that $ \\mathbb{E}_t [c_{t+1}] > c_t $ (consumption is said to be a ‘submartingale’), so that consumption stochastically diverges to $ +\\infty $. The consumer’s savings also diverge to $ +\\infty $.\n", "\n", "
[2] An optimal decision rule is a map from current state into current actions—in this case, consumption.\n", "\n", "
[3] Representation [(3)](#equation-sprob15ab) implies that $ d(L) = U (I - A L)^{-1} C $.\n", "\n", "
[4] This would be the case if, for example, the [spectral radius](../tools_and_techniques/linear_algebra.html#la-neumann-remarks) of $ A $ is strictly less than one.\n", "\n", "
[5] A moving average representation for a process $ y_t $ is said to be **fundamental** if the linear space spanned by $ y^t $ is equal to the linear space spanned by $ w^t $. A time-invariant innovations representation, attained via the Kalman filter, is by construction fundamental.\n", "\n", "
[6] See [[JYC88]](../zreferences.html#campbellshiller88), [[LL01]](../zreferences.html#lettlud2001), [[LL04]](../zreferences.html#lettlud2004) for interesting applications of related ideas." ] } ], "metadata": { "date": 1591310616.7723167, "download_nb": 1, "download_nb_path": "https://julia.quantecon.org/", "filename": "perm_income.rst", "filename_with_path": "dynamic_programming/perm_income", "kernelspec": { "display_name": "Julia 1.4.2", "language": "julia", "name": "julia-1.4" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.4.2" }, "title": "Optimal Savings I: The Permanent Income Model" }, "nbformat": 4, "nbformat_minor": 2 }