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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 ...


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}...


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 (...


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.


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 ...


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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