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4

Perhaps I misunderstand the question, because it seems trivial coming from such an established researcher. As @HerrK. points out, utility functions that represent intertemporal discounting are generally of the form $$U\left((x_i)_{i=1}^T\right) = u(x_1) + \delta_1 u(x_2) + \delta_2^2 u(x_3) + \dots$$ where $\delta_i$ is the discount factor and $x_i$ is the ...

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The problem can be solved using two stage budgetting. In stage 1 total income $m = \sum_t p_{at} c_{at} + \sum_t p_{bt} c_{bt}$ is allocated across periods. In stage two the optimal expenditure $E_t$ in period $t$ is divided between $c_{at}$ and $c_{bt}$. This second stage problem can be written as: $$\max \left((c_{at})^\rho + (c_{bt})^\rho\right)^{\frac{1}... 2 Yes, it is. First, what you describe is not as much binary choice, but situation where you have discrete quantities where any quantity higher than 1 does not bring any benefits (a person can have 2 beds just the second bed is useless). You can calculate marginal utility for such case normally how you would do for other goods that come in discrete quantities (... 2 Marginal utility is the increase in utility per unit increase in a good. Marginal rate of substitution is like the exchange rate between two goods given a level of utility. 2 To Answer your question, suppose that you have the following utility function$$ U ={U(x,y)}$$Then the marginal Utility of good x will be MU_x = \frac{\partial U}{\partial x}  and marginal Utility of good y will be MU_y = \frac{\partial U}{\partial y}  The Marginal rate of substitution will be$$ MRS = \frac{MU_x}{MU_y}  In summary Marginal utility ...

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Try cases: $x_2 = 0$ or $x_3 = 0$, then consume $x_1$ only. If only $x_1 = 0$, solve for the Walrasian for Cobb Douglas All $x$ are non-zeros. I have a feeling you might reach a contradiction in the Kuhn-Tucker conditions, so you will be able to rule out this case. Anyhow, you need to work this out on your own.

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