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Questions tagged [elasticity-of-substitution]

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When do workers versus capital owners share of income increase?

The share of total income obtained by workers rather than capital owners is for obvious reasons of interest. Assuming that the economy can be desribed by an aggregate production function $$Y = F(K,L)$$...
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6 votes
1 answer
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Econ Intuition for Jacobian inverse in demand system

Consider the following simple linear demand system (in vector notation) with n different products Demand: $\quad\mathbf{q=B\left(a-p\right)}$ Inverse demand: $\quad\mathbf{p=a-B^{-1}q}$ where $\...
bbecon's user avatar
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317 views

Elasticity of Substitution of CRS Production Function

Suppose that $F(\cdot)$ has CRS in $K$ and $L$, the elasticity of Substitution is $\sigma_{K L} \equiv F_{L} F_{K} / F F_{L K}$. I once derived this equation but I remember that it takes me quite ...
Alalalalaki's user avatar
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6 votes
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730 views

Estimating elasticity of substitution in nested CES functions

I have aggregate data on $L_t, K_t$ and $X_t$, and want to estimate elasticity of substitution parameters, $\gamma$ and $\sigma$ for these factors. Assuming the production function takes the following ...
london's user avatar
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5 votes
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Derive and Decompose The Aggregate Elasticity of Substitution in CES Economy

The paper is Oberfield & Raval 2021. Consumers have standard Dixit-Stiglitz preferences consuming the bundle $$Y= \left(\sum_{i \in I} D_{i}^{\frac{1}{\varepsilon}} Y_{i}^{\frac{\varepsilon-1}{\...
Alalalalaki's user avatar
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In the BLP paper, why interacting consumer characteristic with product characteristic can generate more desirable substitution pattern

Got a question about the famous BLP paper (http://people.stern.nyu.edu/wgreene/Econometrics/BLP.pdf). When there is no interaction between product characteristic and consumer characteristic, the ...
ExcitedSnail's user avatar
4 votes
2 answers
195 views

Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1?

I am getting some "weird results". I find that in a CES, with short term fixed capital and elasticity of substitution smaller than one, it is optimal for firms to hire zero labour, which seems at odd ...
luchonacho's user avatar
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4 votes
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378 views

How does one derive the elasticity of substitution with implicit functions?

I would like to derive the elasticity of substitution. I'm aware that such a thread with a very straightforward explanation already exists, but my case is slightly different and I'm not sure how to ...
Louki's user avatar
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1 answer
155 views

Hardcore elasticity of substitution (bad results)

I have a following function and would like to find the elasticity of substitution between pairs: $$U = \left( x_1^\delta + x_2^\delta + x_3^\gamma + x_4^\gamma \right)^{\frac{1}{\delta + \gamma}}$$ ...
Athaeneus's user avatar
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How to compute elasticity of substitution in labor market of blacks and whites using experience-education groups?

I'm reading "Competition in the Promised Land – Black Migrants in Northern Cities and Labor Markets", by Leah Boustan, and I'm trying to understand her computation of black and white ...
Lucas's user avatar
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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 $(...
Joseph Basford's user avatar
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1 answer
791 views

Complementarity in CES Production Function

I'm reading Fisher (1997, Journal of Monetary Economics). From the intermediate goods produced ($Y_t$), the final goods firm allocates into consumption ($C_t$), business capital investment ($I_{b,t}$),...
hipHopMetropolisHastings's user avatar
3 votes
2 answers
641 views

How was CES utility function derived?

Is there any book/papers that I can refer to the proof (derivation) of the CES utility function? Or if anyone could help me with the derivation, I will be so much grateful to you.
stochastic learner's user avatar
3 votes
2 answers
103 views

Different elasticities of substitution

I have been reading into generalizations of the concept of elasticity of substitution for more goods/inputs and three main possibilities emerged: Hicksian EOS Allen-Uzawa EOS Morishima EOS HICKS As ...
Athaeneus's user avatar
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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 ...
Athaeneus's user avatar
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1 answer
106 views

How to compute elasticity of substituion in general?

We know that the elasticity of substitution is defined as $$ e=\frac{d \ln(x_2/x_1)}{d \ln(MRS_{12})}=\frac{\frac{d(x_2/x_1)}{x_2/x_1}}{\frac{d(MU_1/MU_2)}{MU_1/MU_2}} $$ When we compute ES for CES ...
Andy Xu's user avatar
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1 answer
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Obj function yielding independent goods demand functions

I know that if the objective function (aka utility) is homothetic, demand functions will be linear in income. So for an homothetic demand function to give goods independent of prices other than their ...
user_newbie10's user avatar
3 votes
0 answers
572 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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Why is my elasticity of substitution wrong?

I am calculating elasticity of substitution for the following production function: $$F(K,L) = A(aK^{-\gamma}+bL^{-\gamma})^{-\mu/\gamma}$$ where $A, a, b, \mu, \gamma$ are constants. $A, a, b, > 0$,...
WilliamT's user avatar
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How can you interpret one of the parameters of optimal consumption at the Merton portfolio problem?

Statement: Let the dynamics of wealth of the agent satisfy $$dX_{t} = \pi_tX_t\Big(\mu dt+\sigma dB_{t}\Big)- c_t X_t dt, \qquad \textrm{with}\quad X_0=x_0 \in \mathbb{R},$$ where $(\pi,c)$ is an ...
epine_se's user avatar
3 votes
0 answers
157 views

Wages, capital: Substitution and Output Effects

Consider a CES production function $Y=f(K,L)$ with elasticity of factor substitution $\sigma>0$. The substitution effect of higher real wages naturally implies a shift along the isoquant to more $...
user19132's user avatar
2 votes
1 answer
301 views

Meaning of «intertemporal substitution effect dominates the income effect»

Assume a household intertemporal optimisation problem, where they only either consume or labour, and one of the equilibrium conditions states that labour will be a positive function of productivity, ...
An old man in the sea.'s user avatar
2 votes
1 answer
122 views

Derivation of the elasticity of substitution of a general production function with labor-augmenting technological progress

I am following and trying to fully understand a famous and interesting work of Bentolila and Saint-paul (2003). They try to explain movements of the factor's share in terms of a relationship between ...
Alessandro's user avatar
2 votes
1 answer
58 views

Empirical measurements of consumption and production elasticity of substitution?

The consumption elasticity of substitution (ES) is the ES between different consumption goods. The production ES is the ES between factors of production. My question is: are there any serious ...
user56834's user avatar
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2 votes
1 answer
126 views

Complements/substitutes estimation from data (Slutsky matrix)

Estimation of complements/substitutes by Slutsky matrix from observable data Hello everyone, I was curious about the following problem: I can observe price $P_i$ of $n$ goods and the amount of goods $...
Athaeneus's user avatar
  • 822
2 votes
1 answer
327 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
2 votes
0 answers
62 views

Relation between complements and substitutes (for multiple goods)

I am a little bit curious about the following problem: If we have multiple goods (at least 3 or more)... And we know that $x_1$ and $x_2$ are substitutes and $x_1$ and $x_3$ are also substitutes, does ...
Athaeneus's user avatar
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2 votes
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852 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
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2 votes
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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 ...
Elina Gilbert's user avatar
2 votes
0 answers
139 views

Linearization of VES production funtions

I know that the linearization of a CES (constant elasticity of substitution) funtion is a bit complicated. There is even an R package dedicated just for that - the econometric estimation and ...
Rokis1990's user avatar
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1 vote
1 answer
260 views

Contradictory FOC and maximizing solution

I have to maximize the following function - $\max_{x \in (0,1)} (((p_1x)^{2r} + (p_2(1-x))^{2r})/2)^{1/r}$ where, $p_1$ and $p_2$ are drawn from uniform distribution [0,1] and are considered to be ...
Elina Gilbert's user avatar
1 vote
1 answer
66 views

Krugman Model: Profit Maximization (relation price elasticity of demand and elasticity of substitution)

the pictures outline the situation and my problem. The only thing I dont understand, is why the inverse of the price elasticity is equal to the negative elasticity of substitution? (price elasticity ...
Paul Herrmann's user avatar
1 vote
1 answer
100 views

Example of a (not quasi-linear) production function whose inputs are not perfect substitutes but are not asymptotic at the axes

I'm looking for an example of a family of production functions indexed by, say, rho, where the inputs become closer and closer to perfect substitutes as rho approaches 1, and yet, the marginal product ...
Leo Simon's user avatar
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1 vote
2 answers
1k views

If $X$ is a Giffen good then $Y$ must be a normal good

While going through some problems as part of self-study I encountered the following True/False question: Q. Steven only consumes two goods: $X$ and $Y$. If $X$ is a Giffen good for Steven, then $Y$ ...
Vizag's user avatar
  • 224
1 vote
1 answer
461 views

Price elasticity of demand of CES

Anyone would like to help me show the following ( or a book/paper reference would be a great help) " The price elasticity of demand is equal to $\sigma$ for the demand function of CES preference,...
stochastic learner's user avatar
1 vote
2 answers
66 views

Calculating price elasticity by asking a "this" or "that" question

Let's say 2 goods exist: A for 10 and B for 20 (B is more premium in this example) If I decrease the price of both goods by 10% and ask: "Do you prefer A for 9 or B for 18 " What sort of insights (...
Frederick's user avatar
1 vote
0 answers
41 views

Intertemporal elasticity of substitution for leisure with log utility

I am interesting in proving that the elasticity of substitution between leisure in two periods is equal to one for the following utility function: $$U = \ln(c_1)+b\ln(1−l_1)+e^{−\rho}[\ln(c_2)+b\ln(1−...
Rana Barakat's user avatar
1 vote
1 answer
59 views

Derivative to ln(K(t)) in the RBC model

In the calculation of the equation of motion for capital in the RBC model, I came across this equation: Can someone explain what are the mathematical steps in between? I don't see how exactly the ...
Delia's user avatar
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1 vote
0 answers
12 views

How does the market respond when some suppliers of one product are forced to switch to supplying a similar product?

I'm interested in coming up with a basic model to describe the following simple situation. Initially there is an equilibrium in the market for the consumption of product A and product B. Product A and ...
user45757's user avatar
1 vote
0 answers
53 views

Capital in terms of labor

I have a question that asks to find $\frac{\partial K}{\partial L} $ from $Q=cL^aK^b$, when $Q$ and $c$ are constants. It lists 4 answer choices but I’m just not sure how to approach it. Implicit ...
lampshade's user avatar
1 vote
0 answers
140 views

Constant Elasticity of Substitution- profit maximization vs cost minimization

Consider the following production function: $$ Y=A\left[\alpha K^{\rho}+\left(1-\alpha\right)L^{\rho}\right]^{\frac{1}{\rho}} $$ In the above, $\rho$ is the substitution parameter, and $\sigma=\frac{1}...
ChinG's user avatar
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1 vote
0 answers
104 views

Elasticity of substitution

So, this is an economics question but the problem I have is a pure math problem I guess. So I have the following equation:f(x,y) this function have the elasticity of substitution(EOS): 1/(1-beta). a,...
Fridein's user avatar
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1 vote
0 answers
31 views

How to simulate changes in quantity demanded when several prices change?

I am not sure if this is a straightforward question or not. Basically, I have a set of products along with their prices and quantities demanded. I also have own and cross price elasticities for each ...
ninapatari's user avatar
1 vote
0 answers
77 views

Demand Elasticity, Factor Substitution: Independent?

Given $ Y=f(K,L;\sigma) $, the effect on labor from a change in the price of capital can be gauged through a substitution effect and a scale effect: \begin{align*} \frac{\partial L}{\partial r} & ...
user19132's user avatar
1 vote
0 answers
128 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
1 vote
0 answers
111 views

CES utility in dynamic setting

Suppose I have a multiperiod consumption-saving problem with two or multiple goods able to be consumed. If the utility within a period is Constant Elasticity of Substitution, ie. $C = (c_1^\frac{\...
econovan's user avatar
0 votes
2 answers
160 views

Is elasticity of substitution defined for non-homogeneous production functions?

The elasticity of substitution between two inputs $x_1$ and $x_2$ is typically given as $$\frac{d \ln \left( \frac{x_2}{x_1} \right)}{d \ln(\mathrm{MRTS}_{21})}.$$ As these notes show, if the ...
tparker's user avatar
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0 votes
1 answer
474 views

How to find the elasticity of substitution for the general CES-function?

I have to find the EOS for the general CES-function defined as $$ y = (a_1x_1^p+a_2x_2^p)^{1/p} $$ I first find $MP_1$ and $MP_2$ to find $MRTS$. We have that $MP_1 = \frac{\partial y} {\partial ...
Mathias's user avatar
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0 votes
1 answer
42 views

Is there a standard term for the elasticity of an isoquant?

Isoquants - the level sets of a production function $f$ - are very useful in microeconomics. For example, if we hold all but two inputs fixed, then the isoquant is a plane curve that quantifies the ...
tparker's user avatar
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