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7

To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on constant elasticity of substitution. In particular, The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969). Then, the CES utility function was popularized by Dixit and Stiglitz (1977) ...


6

Quadratic utility is given by $$u(w) = w - b w^2$$ which has derivative $$u'(w) = 1- 2b w$$ such that for high levels of $w, u'(w)<0$. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less. The second derivative is $$u'(w) = -2b$$ such that absolute risk aversion is $$\frac{- u'...


6

The C.E.S functional has been introduced in Economics in the context of production theory, by Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The review of Economics and Statistics, 225-250. There you can find a discussion of how it was derived. A more pedagogic and detailed ...


4

Just for sake of acknowledgement, please note that the game described in the question is a variation of the famous Ultimatum game. Knowing this can help you get a ton of literature on such games. Further note that your professor has made an extremely important point that coming up with answer is sufficient, solving is not necessary. My answer is also limited ...


3

$u = \max(x, y)$ represents the preferences over two substitute good that cannot be consumed together. For example - tea and coffee. In the event that the consumer gets x quantity of tea and y quantity of coffee, consumer choose to consume only one of the them depending on the quantity. He always choose the one that is offered in larger quantity and throws ...


3

As far as I can see this comes just from definitions: As given in MWG definition 1.C.1: The choice structure $(\mathscr{B},C(\cdot))$ satisfies the weak axiom of revealed preference if the following property holds: If for some $B \in \mathscr{B}$ with $x,y \in B$ we have $x\in C(B)$, then for any $B'\in \mathscr{B}$ with $x,y\in B'$ and $y\in C(B')$, we ...


2

You have added inequality constraints $x_1 \geq 0$ and $x_2 \geq 0$. Hence, you now have Kuhn-Tucker conditions for complimentary slackness: $$ \mu_1 x_1 = 0 \text{ and } \mu_2x_2 = 0 $$ So whenever $x_i > 0$, by the complimentary slackness condition, $\mu_i = 0$. See Karush-Kuhn-Tucker conditions. It also contains regularity conditions you ask for.


2

There are no simple methods for estimating cardinal utility (ordinal utility would be a different matter - you could just observe few of your choices). This is not because cardinal utility would necessary be unmeasurable. Although this is not completely settled question (see Moscati (2018) Measuring Utility: From the Marginal Revolution to Behavioral ...


2

Your thinking is correct that, in some ways, $x_1, x_2$ are substitute goods. We define substitute goods which have the following property: $$\left.\frac{\partial x_i}{\partial p_j}\right|_{u=\bar u}>0$$ The case of $U(x_1,x_2)=\max\{x_1,x_2\}$ is that of a boundary solution as the indifference curves are now concave to the origin. So equilibrium solution ...


2

Note that $$ln(x_1)+2*ln(x_2) = ln(x_1)+ln(x_2^2) = ln (x_1 * x_2^2),$$ and note that $$MRS_v = \frac{g'(x_1 * x_2^2) x_1 * 2 x_2}{g'(x_1 * x_2^2) x_2^2}$$ such that the derivative of $g$ cancels out. For more intuition, see here.


1

The intuition you have is correct. Mathematically you can show it by first deriving the optimal choices with the lump sum income tax. So you will set up the following lagrangian: $$\mathcal{L} = x^{1/2}_1 x^{1/2}_2 - \lambda [x_1p_1+x_2p_2 - m + T] $$ This gives you 3 FOC's the budget constraint and: $$ 0.5x_1^{-0.5} x_2^{0.5} = \lambda p_1 \\ 0.5x_2^{-0.5} ...


1

Suppose $c_t$ is some composite function of interest rate $r$, e.g. $c_t(G(r_t))$. First the intertemporal elasticity of substitution is actually (IES) given by $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial r}$ (or also $\frac{\partial \ln(c_{t+1}/c_{t})}{\partial \ln( u'(c_{t+1})/u'(c_t))}$). You can take the derivative above using chain rule for composite ...


1

There is an infinity of such functions. You can for instance construct a linear homogeneous function $u$ from any utility function $U$ by using a linear homogeneous function $h: \mathbb{R}^J \rightarrow \mathbb{R} $ as follow: $$ u(x) = h(x)U(x/h(x)). $$ Example: $U(x)= \alpha x_1^2 + \beta x_1x_2 + \gamma x_2^5 $, $h(x)=x_1+x_2$ yields $$u(x) = (x_1+x_2) \...


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