Questions tagged [ces-function]

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Estimating CES utility (not production) function parameters

The CES utility function has the form \begin{equation} u(x_1,\dots,x_n)=\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho}, \end{equation} where $\alpha_i$ is the consumption share parameter and $\...
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4 votes
1 answer
67 views

Convex CES Aggregator

I just find it seems that a CES aggregator e.g. $\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}$ with $\sigma<0$ is called a general convex aggregator and its limit as ...
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4 votes
1 answer
78 views

Is there a name for this type of CES look-alike utility function?

I do not know whether the following utility functions are used a lot but I was wondering whether there was a common name for this type CES like utility function $$u(z) = \left[ \left( \sum_{j \in J_1}...
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4 votes
1 answer
60 views

Homogeneous Utilities: Anything other than CES?

Does there exist any homogeneous utility function, i.e., $u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$, that is not a special case of the CES (or nested CES) family of utility functions or its ...
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0 answers
45 views

CES utility with negative values

We know that the CES utility function approximates a Leontief utility function as in the following: $$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\...
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4 votes
0 answers
95 views

CES aggregator for intermediates with heterogeneous productivity

I'm following Atkeson, Burstein (2019 JPE), and cannot understand the aggregation result. There is measure $M(z)$ of firms with productivity $z$, with production function $$ y(z) = z k(z)^\alpha l(z)^...
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3 votes
1 answer
206 views

How to prove a generalised function is quasiconcave?

I have a question I have been asked to solve: Given that $(a_1, a_2,...,a_n)\in R_{++}^n$ and $(x_1, x_2,...,x_n)\in R_{++}^n$, and $A>0, \mu >0, p \neq 0$, if there is a function $f(x)=A(a_1x_1^...
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3 votes
1 answer
128 views

The Intuition of CES Utility

Suppose a (symmetric) CES utility function $$U(\mathbf{x})=\left[\int_{\Omega}\left(x_{\omega}\right)^{\frac{\sigma-1}{\sigma}} d \omega\right]^{\frac{\sigma}{\sigma-1}}, \sigma>1$$ 1 The indirect ...
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  • 2,011
3 votes
1 answer
837 views

Cobb-Douglas nested in CES model

Assume, using the following equation $Y=[((A_1L)^\alpha K^\beta)^\sigma+ (A_2X^\gamma)^\sigma]^{1/\sigma}$, we back out (obtain) the evolution of $A_1$ & $A_2$. Can we interpret $A_1$ as labour-...
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  • 1,950
3 votes
1 answer
73 views

A utility function (neither perfect substitues nor perfect complements) which stems from a CES f. and leads to gross complements or gross substitutes

So the most prominent preferences are perfect substitutes, perfect complements and cobb-douglas preferences. Perfect complements and perfect substitutes are extreme cases and I was asked whether there ...
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2 votes
1 answer
81 views

CES utility function in an Edgeworth box

Two consumers have the CES utility function $x_1^\beta +x_2^\beta$, for $0<\beta<1$, their initial endowments are $w^1=(1,0)$, $w^2=(0,1)$ Draw the Core of this economy in an Edgeworth box. Note ...
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2 votes
1 answer
762 views

CES production function profit and supply function

I need to derive the profit function for the following CES function: $$ f(z) = (\sqrt{z_{1}^{\rho} + z_{2}^{\rho}})^{1/ \rho}$$ where $\rho \leq 1$. This is the answer that I am supposed to be getting:...
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2 votes
1 answer
56 views

Non CES Production Functions

I know CES production functions dominate economics, but I was curious, why? I've never seen a research paper or presentation utilize any form of a production function that is not CES. My question is ...
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2 votes
1 answer
124 views

CES function estimation

For a paper I was using the micEconCES package to estimate the CES production function for a country at the aggregate. For a two-input function with capital and labour I used for the variables the ...
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2 votes
1 answer
95 views

Choosing Data for CES Production Function

So I am currently trying to write a paper estimating the Constant Elasticity of Substitution Production Function of the USA. I am using the simple version with two inputs capital and labour. Since the ...
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2 votes
1 answer
42 views

Can you set $C / P^{\eta}$ to be the numeraire in a NK model?

Consider a static model with CES demand. Real aggregate demand is $$ C = \left(\sum_j c_j^{(\eta-1)/\eta}\right)^{\eta/(\eta-1)} $$ and labor supply is inelastic, so the budget constraint is $ \sum_j ...
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2 votes
1 answer
285 views

CES production function application problem

I'm currently trying to do some estimations using the micEconCES package in R by Henningsen/Henningsen (2011). My issue is that I am not very familiar with R and I'm trying to implement my own dataset ...
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2 votes
0 answers
56 views

Solving Cost Minimization with CES Production Function and Two Types of Input

A CES aggregator: $y=\left(\int_{0}^{1} y(i)^{\frac{\eta-1}{\eta}} d i\right)^{\frac{\eta}{\eta-1}}$, where $i \in[0,1]$. For each intermediate good: $y(i)=\ell_{r}(i)+\ell_{c}(i) / a(i)$, where $a(i)$...
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2 votes
0 answers
43 views

What's special about Cobb–Douglas utility relative to the rest of the CES family?

My question concerns CES utility functions, which have the form $$u(x) = \left(\sum_{j=1}^n a_j x_j^{\rho} \right)^{1/\rho}$$ for utility parameters $a_j$, an elasticity parameter $\rho$, and some ...
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  • 213
2 votes
0 answers
236 views

Log-linearization of CES production function

I am trying to recover the Log-linearisation of a CES production function in a paper. Although I am fairly confident with Log-linearisations, I simply do not find the supposed result. The production ...
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2 votes
0 answers
350 views

Negative elasticity of substitution in a CES production function

I have empirically estimated the elasticity of substitution parameter in the following model: $$Y_t=[(A_1L_tK_{t})^{\rho} +(A_2M_{t})^{\rho}]^\frac{1}{\rho} $$ here, $Y_t$ is output, $A_i$ is a ...
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  • 1,950
2 votes
0 answers
119 views

Derivation of demand function

Hello. I'm graduate student in Japan. This time, what I want to ask is how to solve the profit maximization problem using the image production function and derive the demand function. This ...
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2 votes
0 answers
68 views

Relationship between Elasticity of substitution of sectoral outputs and elasticity of substitution of inputs

There are two sectors Y1 and Y2. Composite output is given by CES form - Each sector employs Capital and Labor in combination through Cobb-Douglas Production Technology. The paper mentions that ...
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1 vote
2 answers
83 views

Nested/Recursive Dynastic Utility Functions

I want to find a way of representing a dynastic utility function in which not only the head of the dynasty's utility is dependent on its descendants' utility, but all members of the family tree gain ...
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  • 13
1 vote
1 answer
93 views

Bellman equation corresponding to stochastic EZW recursive utility

A bit of a basic question maybe, but I am confused how to write down the Bellman of this recursive utility function based on Epstein-Zin-Weil preferences and some stochastic constraints. $$ U_t = \...
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  • 1,063
1 vote
1 answer
591 views

Nested CES Production Function

If I have four input factors (a, b, c, b) and I want to construct a nested CES production function such that (a, b) are substitutes, (c, d) are substitutes and [(a, b), (c, d)] are complements, I.e. a,...
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1 vote
0 answers
15 views

on nested CES--derivation issue

I am reading Sachs and Kotlikoff (2012 NBER) and am wondering how they derive one equation in their paper. The setting is the following: Their model is a variant of the standard two-period ...
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1 vote
0 answers
39 views

Parameter value for a CES production function

Consider a firm with the following CES production function, which utilizes only two production factors (capital and labor) whose prices are, respectively, $r > 0$ and $w > 0$: $$ y = \gamma \...
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1 vote
0 answers
305 views

On complements and substitutes with a CES function

Define the CES function $q : \mathbb R_+^n \to [0,1]$ by \begin{align} q(x) = \left[\frac{1}{n}\sum_{j=1}^n{x_j^\frac{\sigma-1}{\sigma}}\right]^\frac{\sigma}{\sigma-1} \end{align} where $x \in \mathbb ...
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  • 1,529
1 vote
0 answers
99 views

Elasticity of substitution meaning

If I computed an elasticity of substitution of f.e. 0.9 between capital and labour, does this implicate that the factors are rather well substitutable or not? Since for 0 they are perfect complements ...
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1 vote
0 answers
23 views

Raising the elasticity of substitution in an economy

Papers such as de la Grandville (1989) and Klump/de la Grandville (2000) have shown that a higher elasticity of substitution leads to higher economic growth. My question is, if there is a way to ...
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0 votes
1 answer
49 views

Confusing on the CRS Property of CES Function

Say a CES function is that $$Y = A\left[\alpha K^{\rho}+ \beta L^{\rho}\right]^{\frac{1}{\rho}}$$. Clearly this function is constant return to scale whatever the values of $\alpha$ and $\beta$ take. ...
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0 votes
0 answers
11 views

When does a CES production function have a weakly convex isoquant

Some CES production functions have weakly (but not strongly) convex isoquants; e.g., perfect substitutes. Under what general conditions do CES functions have weakly convex isoquants?
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0 answers
31 views

Complementary input production function

I was looking for some complementary functions to work with. Let me state my problem first. I have two different households and I want to distribute them some money based on their characteristics. I ...
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