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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 ...


3

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 ...


1

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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