7

It may be interesting to exploit the homothetic separability of the CES utility function in $x$. It implies that $$\frac{x_i}{x_j} = \left( \frac{\alpha_i}{\alpha_j}\frac{p_j}{p_i} \right)^\sigma $$ and after $log$-transformation: $$\ln(x_i) - \ln(x_j) = \beta_{ij} + \sigma (\ln(p_j) - \ln(p_i)). $$ After adding a random term, this specification could be ...


5

Note first that the function $r\mapsto A(r)^{\mu/p}$ is strictly increasing on $\mathbb{R}_+$. When you look at the minimum in the definition of quasi-concavity, you can therefore ignore this part and it suffices to show that the function $$(x_1,x_2,\ldots,x_n)\mapsto a_1x_1^p+a_2x_2^p+...+a_nx_n^p$$ is quasi-concave. Actually, the function is even concave. ...


5

I believe it can be found as "nested CES" See for example: Nested CES


5

This answer closely follows the logic of estimation of translog cost function presented in Section 4.7 of Fumio Hayashi's "Econometrics". Define for convenience the CES aggregate price index $P:=(\sum_i \alpha_i^\sigma p_i^{1-\sigma})^\frac{1}{1-\sigma}$. Then the Marshallian demand system in log form can be written as a system of linear equations $...


4

In this comment I simply show that Under certain assumptions the problem is not an estimation problem, there is an exact solution for $\sigma$ and a solution for the structural errors $\alpha_i$ up to a normalisation. The demand system considered is described by the Marshall demand function $$x_k^\star(p,M) = \left(\frac{\alpha_j}{p_j}\right)^{\sigma}\frac{...


4

What you have there are the preferences under an arbitrary policy -- what some call the prevalue function. The only thing missing is the max operator. Written with maximization (and making the state and choice explicit), the Bellman equation is \begin{align} U(K_t, \epsilon_t) =& \max_{C_t} \{ (1-\beta) C_t^{1 - \frac{1}{\eta}} + \beta [E_t(U(K_{t+1}, \...


3

This is going to be a long answer, and I'm not completely sure if it is going to answer your questions as I'm mostly going to focus on the derivations of the own and cross price elasticities. Most of the derivations can also be found in Houthakker (1960), "Additive Preferences" TLDR: If the utiltiy function is additively separable and if the share ...


3

These other answers seem to be citing some methods I haven't quite heard about yet however they all touch upon the idea developed by Czech economist named Jan Kmenta which has come to be known as the Kmenta approximation (an in depth explanation as well as detailed derivation can be found in the documentation of the R package,MicEconCES. A general form of a ...


3

There seems to be some confusion in the expression for $x^*_i$ in the question that whether $i$ is for consumer of for the good. Assuming $i$ is for consumer: Let $x^*_i = (x_1^i,x_2^i)'$ be the equilibrium bundle for consumer $i$. Since utility function is same for both, from MRS we have: \begin{align} \frac{x_1^i}{x_2^i}=\bigg(\frac{p_1}{p_2}\bigg)^{s-1} \...


2

Let $\delta_i=N_i\beta_i$. I think what you want is for generation $i$'s utility to be something like: \begin{equation} U_i=u(x_i)+\delta_{i+1}U_{i+1}+\cdots+\delta_{i+n}U_{i+n}, \end{equation} where $i$ derives utility from his own consumption $u(x_i)$ as well as from his descendants' utilities, $U_{i+t}$ for $t=1,\dots,n$, over their own consumption and ...


2

Here are two nice papers on recursive utility functions, a generalization of additive utility functions and compatible with "utility of the dynastic head to be partly a function of the utility of his children and grandchildren's utility, but where his children's utility is again partly a function of his grandchildren's and great-grandchildren's utility, ...


2

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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