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Why can the Lagrangian Multiplier be dropped in the inverse demand function?

I'm deriving the Antras and Helpman (2004) paper. The model assumes a nested CES utility function $$ U = x_0 + \frac{1}{\mu} \sum_{j=1}^{J} X_j^\mu $$ where $X_j = \left[ \int x_j(i)^\alpha \,di \...
Hopeless Economist's user avatar
2 votes
1 answer
45 views

CES Price Index in Melitz (2003)

I am reading Melitz (2003), and I cannot understand why the CES price index can be rewritten as an integral including $M$ and $\mu(\phi)$. I find the lecture notes by Dave Donaldson (https://dave-...
LEcon's user avatar
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4 votes
1 answer
242 views

LSE EC417 2023: Markup as Elasticity Tends to Unity

I'm going over a macro past paper and am stuck deriving and interpreting a result. The question begins with the CES aggregator for aggregate output $Y$, based on a continuum of intermediate goods $(...
Joseph Basford's user avatar
0 votes
3 answers
76 views

Tastes in Cobb-Douglas vs CES: Different degree of homogeneity

The question has been reworded based on Giskard's comment If we have a Cobb-Douglas function: $$U_1 = x_1^{\alpha_1} x_2^{\alpha_2}$$ The degree of homogenity depends on tastes $\alpha_1$ and $\...
Athaeneus's user avatar
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1 vote
1 answer
68 views

Cost function from a weighted CES production function

I want to find the cost function given the CES production function: $$ Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho} $$ with $0<\rho<1$. So far I have set up the ...
fabs's user avatar
  • 11
2 votes
1 answer
68 views

Balanced growth path in the Hicks neutral technology and CES function

In an another post (Balanced growth path definition in the Hicks neutral steady state with technology), the problem of having a not stationary steady state in a Solow model with Hicks neutral ...
Veronica's user avatar
1 vote
1 answer
102 views

Are homothetic additively separable preferences always equivalent to CES?

Are homothetic additively separable preferences always a monotonic transformation of CES preferences? In technical language, the question is the following: Let $n>1$, and let $f:\mathbb{R}^n_{\ge 0}...
cfp's user avatar
  • 252
1 vote
2 answers
266 views

What is an example of utility function where the proportion of one goes to zero?

I would like an example of an utility function with 2 goods where the proportion of one goes to zero (and the other goes to one). I am thinking of a problem where the household receive an endowment y ...
Rodrigo Miyamoto's user avatar
3 votes
1 answer
323 views

CES production function: How to show that $\sigma < 1$ implies essentialness?

Consider the CES production function: $$Y = f(K, L) = (a \cdot K^\rho + (1 - a) \cdot L^\rho )^{1/\rho}$$ The elasticity of substitution is $\sigma = 1/(1 - \rho)$. I remember that, if the elasticity ...
MoreQuestionsThanAnswers's user avatar
3 votes
1 answer
96 views

Elasticity of substitution by regression: Biased results (simulation)

I have the following simulation problem: Consumers, whose utility I know, go shopping for two goods. However, prices differ each time they visit a shop. Therefore, these consumers always purchase ...
Athaeneus's user avatar
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3 votes
1 answer
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Explain the definition of a primal shifter versus an input shifter parameters in the standard CES function

I have run into a CES function that seems to be very closer to standard but with a small disaggregation of the share parameter into two parameters (primal share) and (input shift). I am hoping someone ...
user42955's user avatar
3 votes
0 answers
594 views

Utility function distinguishing between complements/substitutes

Distinguish between complements/substitutes in utility function or production function Hello everyone, I would like to know if there exists some utility function $U(x)$ for $n$ goods that is able to ...
Athaeneus's user avatar
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2 votes
1 answer
342 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. ...
Alalalalaki's user avatar
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1 vote
0 answers
41 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 ...
embeconp's user avatar
3 votes
1 answer
578 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 ...
Sceptical_Economist's user avatar
4 votes
1 answer
157 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 ...
Alalalalaki's user avatar
  • 2,474
2 votes
0 answers
363 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 $a(i)$...
Alalalalaki's user avatar
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4 votes
0 answers
133 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\...
jabberwocky's user avatar
4 votes
1 answer
183 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 ...
Michael Gmeiner's user avatar
3 votes
1 answer
619 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 ...
Alalalalaki's user avatar
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3 votes
1 answer
324 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^...
DoubleRainbowZ's user avatar
3 votes
1 answer
224 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 ...
Max's user avatar
  • 223
2 votes
0 answers
628 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 ...
EconRider's user avatar
1 vote
2 answers
166 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 ...
1.618's user avatar
  • 13
1 vote
1 answer
252 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 = \...
Papayapap's user avatar
  • 1,843
4 votes
1 answer
112 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}...
Jesper Hybel's user avatar
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2 votes
1 answer
133 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 ...
Ana Ellis's user avatar
4 votes
1 answer
98 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 ...
Denizalp's user avatar
  • 105
1 vote
0 answers
75 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 \...
Pedro Cunha's user avatar
2 votes
0 answers
857 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 ...
london's user avatar
  • 2,010
4 votes
0 answers
166 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)^...
JaySingsBlue's user avatar
2 votes
1 answer
61 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 ...
Levi Crews's user avatar
10 votes
4 answers
1k views

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 $\...
Herr K.'s user avatar
  • 15.4k
2 votes
0 answers
128 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 ...
Kenta's user avatar
  • 21
2 votes
1 answer
1k 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:...
Ali's user avatar
  • 862
2 votes
0 answers
686 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 ...
clueless's user avatar
  • 1,579
2 votes
0 answers
96 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 ...
Elina Gilbert's user avatar
1 vote
1 answer
1k 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,...
user10158324's user avatar
1 vote
0 answers
130 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 ...
macro123's user avatar
1 vote
0 answers
27 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 ...
macro123's user avatar
2 votes
1 answer
136 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 ...
macro123's user avatar
2 votes
1 answer
99 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 ...
macro123's user avatar
2 votes
1 answer
331 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 ...
macro123's user avatar
3 votes
1 answer
1k 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-...
london's user avatar
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