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In the Ramsey-Cass-Koopmans model, the household sector is represented by infinitely-lived agents who maximize the present value of their utility from consumption over time, subject to a budget constraint that reflects income from labor and capital accumulation. The model provides a framework for understanding intertemporal consumption choices, capital accumulation, and the dynamics of economic growth. Household equilibrium is achieved when the representative household optimally chooses consumption and savings paths to maximize lifetime utility. The representative household seeks to maximize the intertemporal utility function: $$ U = \int_{0}^{\infty} e^{-\rho t} u(c(t)) \, dt $$ where $c(t)$ is per capita consumption at time $t$, $u(c)$ is the instantaneous utility function, typically assumed to be concave ($u'(c) > 0, u''(c) < 0$), and $\rho > 0$ is the subjective rate of time preference reflecting the household’s impatience. The household faces the intertemporal budget constraint derived from capital accumulation dynamics: $$ \dot{k}(t) = f(k(t)) - c(t) - (\delta + n) k(t) $$ Here, $k(t)$ is per capita capital, $f(k)$ is the production function per capita (assumed concave, $f'>0, f''<0$), $\delta$ is the depreciation rate of capital, and ________ _____ _____ _____ _____.
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