6

In general, it will not represent the same preferences. There seems to be confusion on what "monotonic transformation" means in this context. It does not have much to do with monotonic preferences. We say that the utility function $v:X\to\mathbb{R}$ is a monotonic transformation of the utility function $u:X\to\mathbb{R}$ if there exists a strictly ...


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

Note that $\exp(\ln(x)\ln(y))\ne \exp(\ln(x+y))$. Instead, \begin{equation} \exp(\ln(x)\ln(y))=x^{\ln y}=y^{\ln x}. \end{equation} The MRS of the monotonically transformed utility is still $\frac{y\ln y}{x\ln x}$.


2

For part (i), in complete rigor, you should also check the determinants of all the leading principal minors of the bordered Hessian and make sure that they have alternating signs. Your final conclusion looks correct though. For part (ii), recall that Walrasian demand is the solution to utility maximization subject to budget constraint. So you should setup a ...


1

The answer comes from looking at the Hicksian compensated demand: Since, WLOG, EV for only one good price change can be written as $$EV(p_1,p_0,u) = \int_{p_1}^{p_0} h(p,u_1)dp_1$$ and CV as $$CV(p_1,p_0,u) = \int_{p_1}^{p_0} h(p,u_0)dp_1,$$ and we know that $\frac{\partial h}{\partial u} \leq 0$ when the good is inferior (using the fact that $\frac{\partial ...


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