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Questions tagged [elasticity-of-substitution]

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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−...
4 votes
1 answer
244 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 $(...
3 votes
2 answers
108 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 ...
0 votes
1 answer
35 views

Simultaneously a substitute and a complement (validity of a claim)

I have read the following claim: Sometimes the relationship between products can be both substitute and complement; that is, two products may be complements for one purpose but substitutes for ...
5 votes
1 answer
184 views

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}{\...
0 votes
0 answers
94 views

substitution effect in slutsky equation

We know Slutsky equation decomposes the price effect into substitution effect and income effect, where the substitution effect is the partial derivative of Hicksian demand function against the change ...
0 votes
0 answers
29 views

Why does the price of firewood from Ardennes forests rise when the price of natural gas from the North Sea rises?

I have an exam question from last year that I don't understand. I have to explain this : why does the price of firewood from Ardennes forests rise when the price of natural gas from the North Sea ...
0 votes
0 answers
35 views

relation between elasticity of substituion and mrts

elasticity of substituion has been defined as the as the percentage change in capital labour ratio given change in mrts. but mrts itself tries to explain how a change in labour results in a change in ...
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 ...
4 votes
1 answer
157 views

Hardcore elasticity of substitution (bad results)

I have a following function and would like to find the elasticity of substitution between pairs: $$U = \left( x_1^\delta + x_2^\delta + x_3^\gamma + x_4^\gamma \right)^{\frac{1}{\delta + \gamma}}$$ ...
1 vote
1 answer
67 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 ...
3 votes
1 answer
97 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 ...
2 votes
0 answers
64 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 ...
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 ...
3 votes
0 answers
597 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 ...
0 votes
2 answers
167 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 ...
0 votes
1 answer
44 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 ...
0 votes
0 answers
80 views

Elasticity of substitution for 3 and more goods (interpretation)

Elasticity of substitution for 3 and more goods (interpretation) Hello everyone, I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function ...
2 votes
1 answer
129 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 $...
6 votes
2 answers
329 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 ...
7 votes
1 answer
97 views

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 $$Y = F(K,L)$$...
1 vote
1 answer
101 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 ...
4 votes
1 answer
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 ...
3 votes
1 answer
107 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 ...
4 votes
1 answer
115 views

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 ...
1 vote
1 answer
261 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 ...
5 votes
0 answers
55 views

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 ...
2 votes
1 answer
330 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, ...
3 votes
0 answers
109 views

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 ...
3 votes
0 answers
54 views

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. $A, a, b, > 0$,...
3 votes
2 answers
647 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.
1 vote
1 answer
481 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,...
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 (...
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 ...
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 ...
2 votes
0 answers
859 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 ...
0 votes
1 answer
492 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 ...
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}...
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 $...
1 vote
0 answers
105 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,...
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 ...
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 $Y$ ...
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 ...
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 ...
6 votes
1 answer
202 views

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 $\...
6 votes
1 answer
740 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 ...
0 votes
1 answer
127 views

Elasticity of substitution between leisure in two periods

This is a basic question, but I am new to macro models. The question is from Romer's text. Assume a household only has one member and has no initial wealth, and the household lives for two periods. ...
0 votes
1 answer
449 views

Second order partial derivative and cross second-order partial derivative

I'm following a paper (Full text available here!) where at some point (pag.17 and 20) the author get the following derivative: $$\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}...
1 vote
0 answers
79 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} & ...
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 ...