Questions tagged [ces-function]
The ces-function tag has no usage guidance.
45
questions
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Normalized CES supply systems: Klump, McAdam, and Willman (2007)
I am trying to use EViews to replicate the article that used the normalized CES supply system from Klump, McAdam, and Willman (2007).
When I set technology progress as an exponential function, I can ...
- 11
3
votes
1
answer
60
views
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 \...
2
votes
1
answer
60
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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-...
- 21
4
votes
1
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286
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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 $(...
- 837
0
votes
3
answers
92
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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 $\...
- 834
1
vote
1
answer
82
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 ...
- 11
2
votes
1
answer
72
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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 ...
- 21
1
vote
1
answer
120
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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}...
- 252
1
vote
2
answers
285
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 ...
3
votes
1
answer
409
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 ...
3
votes
1
answer
103
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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 ...
- 834
3
votes
1
answer
119
views
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 ...
- 31
3
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0
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656
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 ...
- 834
2
votes
1
answer
399
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.
...
- 2,506
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0
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60
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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 ...
- 11
3
votes
1
answer
639
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 ...
4
votes
1
answer
160
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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 ...
- 2,506
2
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0
answers
388
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 $a(i)$...
- 2,506
4
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0
answers
140
views
CES utility with negative values
We know that the CES utility function approximates a Leontief utility function as in the following:
$$u(x_1,x_2) = (x_1^{-r}+x_2^{-r})^{-1/r}\overset{r\rightarrow\infty}{\longrightarrow}\min\{x_1, x_2\...
- 41
4
votes
1
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195
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Non CES Production Functions
I know CES production functions dominate economics, but I was curious, why? I've never seen a research paper or presentation utilize any form of a production function that is not CES.
My question is ...
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712
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The Intuition of CES Utility
Suppose a (symmetric) CES utility function $$U(\mathbf{x})=\left[\int_{\Omega}\left(x_{\omega}\right)^{\frac{\sigma-1}{\sigma}} d \omega\right]^{\frac{\sigma}{\sigma-1}}, \sigma>1$$
1 The indirect ...
- 2,506
3
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1
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333
views
How to prove a generalised function is quasiconcave?
I have a question I have been asked to solve:
Given that $(a_1, a_2,...,a_n)\in R_{++}^n$ and $(x_1, x_2,...,x_n)\in R_{++}^n$, and $A>0, \mu >0, p \neq 0$, if there is a function $f(x)=A(a_1x_1^...
4
votes
1
answer
268
views
What's special about Cobb–Douglas utility relative to the rest of the CES family?
My question concerns CES utility functions, which have the form
$$u(x) = \left(\sum_{j=1}^n a_j x_j^{\rho} \right)^{1/\rho}$$
for utility parameters $a_j$, an elasticity parameter $\rho$, and some ...
- 233
2
votes
0
answers
655
views
Log-linearization of CES production function
I am trying to recover the Log-linearisation of a CES production function in a paper. Although I am fairly confident with Log-linearisations, I simply do not find the supposed result.
The production ...
- 63
1
vote
2
answers
172
views
Nested/Recursive Dynastic Utility Functions
I want to find a way of representing a dynastic utility function in which not only the head of the dynasty's utility is dependent on its descendants' utility, but all members of the family tree gain ...
- 13
1
vote
1
answer
262
views
Bellman equation corresponding to stochastic EZW recursive utility
A bit of a basic question maybe, but I am confused how to write down the Bellman of this recursive utility function based on Epstein-Zin-Weil preferences and some stochastic constraints.
$$
U_t = \...
- 1,898
4
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1
answer
116
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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
$$u(z) = \left[ \left( \sum_{j \in J_1}...
- 3,798
2
votes
1
answer
138
views
CES utility function in an Edgeworth box
Two consumers have the CES utility function $x_1^\beta +x_2^\beta$, for $0<\beta<1$, their initial endowments are $w^1=(1,0)$, $w^2=(0,1)$ Draw the Core of this economy in an Edgeworth box. Note ...
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100
views
Homogeneous Utilities: Anything other than CES?
Does there exist any homogeneous utility function, i.e., $u(\lambda \mathbf{x}) = \lambda u(\mathbf{x})$, that is not a special case of the CES (or nested CES) family of utility functions or its ...
- 105
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80
views
Parameter value for a CES production function
Consider a firm with the following CES production function, which utilizes only two production factors (capital and labor) whose prices are, respectively, $r > 0$ and $w > 0$:
$$ y = \gamma \...
- 400
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0
answers
916
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 ...
- 2,020
4
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0
answers
169
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
$$
y(z) = z k(z)^\alpha l(z)^...
- 41
2
votes
1
answer
64
views
Can you set $C / P^{\eta}$ to be the numeraire in a NK model?
Consider a static model with CES demand. Real aggregate demand is
$$
C = \left(\sum_j c_j^{(\eta-1)/\eta}\right)^{\eta/(\eta-1)}
$$
and labor supply is inelastic, so the budget constraint is
$
\sum_j ...
- 31
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votes
4
answers
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views
Estimating CES utility (not production) function parameters
The CES utility function has the form
\begin{equation}
u(x_1,\dots,x_n)=\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho},
\end{equation}
where $\alpha_i$ is the consumption share parameter and $\...
- 15.8k
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0
answers
128
views
Derivation of demand function
Hello.
I'm graduate student in Japan.
This time, what I want to ask is how to solve the profit maximization problem using the image production function and derive the demand function.
This ...
- 21
2
votes
1
answer
1k
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CES production function profit and supply function
I need to derive the profit function for the following CES function: $$ 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:...
- 860
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716
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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 ...
- 1,579
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98
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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 ...
- 499
1
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1
answer
1k
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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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0
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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 ...
- 93
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0
answers
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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 ...
- 93
2
votes
1
answer
142
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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 ...
- 93
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1
answer
100
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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 ...
- 93
2
votes
1
answer
337
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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 ...
- 93
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votes
1
answer
1k
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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-...
- 2,020