7 votes
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Econ Intuition for Jacobian inverse in demand system

For the 2x2 case being considered, write $$\mathbf{B}=\left[\begin{array}{cc} b_{1,1} & b_{1,2}\\ b_{2,1} & b_{2,2} \end{array}\right].\quad$$ It follows that the element (1,1) in $B^{-1}$ is ...
dlnB's user avatar
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7 votes

How was CES utility function derived?

To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on constant elasticity of substitution. In particular, The CES aggregator is also ...
emeryville's user avatar
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6 votes

How was CES utility function derived?

The C.E.S functional has been introduced in Economics in the context of production theory, by Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and ...
Alecos Papadopoulos's user avatar
5 votes
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Contradictory FOC and maximizing solution

As alluded to by Bertrand in his +1 comments this is because FOCs do not tell you where maximum or minimum occurs. This is common misconception among some students but it simply does not hold. FOCs ...
1muflon1's user avatar
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5 votes
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If $X$ is a Giffen good then $Y$ must be a normal good

Reason: Both goods cannot be inferior. Let's say originally you consume $x$ and $y$. So your budget constraint looks like $$p_x x + p_y y = I.$$ If both X and Y are inferior, when income goes down ...
Art's user avatar
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4 votes

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

To derive the formula for the elasticity of substition I consider a function with two arguments $f(x_1,x_2)$. I then consider a level curve $f(x_1,x_2) = c$ assumed to define $x_2$ as a function of $...
Jesper Hybel's user avatar
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4 votes

Elasticity of Substitution of CRS Production Function

I am not sure if it is intuitive but this is because because CRS function is homogenous of degree 1. Full derivation: First, general formula for any arbitrary elasticity of substitution between $L$ ...
1muflon1's user avatar
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4 votes
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When do workers versus capital owners share of income increase?

From the shares equation, we obtain: $$ 1 + \frac{rK}{wL} = \frac{1}{s_L} \to tk = \frac{1 - s_L}{s_L} $$ where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$. then taking logs gives: $$ \ln(k) = -\...
tdm's user avatar
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4 votes

Different elasticities of substitution

This note only answers the last of your question. All these elasticities tend to disappear from the empirical literature since the publication of the influencial paper by Blackorby, C. and R. R. ...
Bertrand's user avatar
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3 votes

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

Let a function $y=f(x_1,x_2)$, and where $f_1$ and $f_2$ denote first partial derivatives and $f_{11}$ etc second partials. The elasticity of substitution was defined by Hicks in order to answer the ...
Alecos Papadopoulos's user avatar
3 votes
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How to compute elasticity of substitution in labor market of blacks and whites using experience-education groups?

I dont want to be rude but the only equation you copied correctly is the productivity augmented Cobb-Douglas production function. Equation 2 is equation 2 from Ottaviano, Peri (2008) (on page 8) it ...
Grada Gukovic's user avatar
3 votes

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

four things : The desired result is actually which is the answer you are getting if you multiply the negative into the denominator( you can cross-check on https://en.wikipedia.org/wiki/...
Poorvi's user avatar
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3 votes

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

Starting from $$Y = A\left( \alpha L^{\rho} + (1-\alpha)K^{\rho} \right)^{\frac{1}{\rho}}$$ $$...\implies MP_L = \alpha A\cdot\left( \alpha + (1-\alpha)\left(\frac {K}{L}\right)^{\rho} \right)^{\...
Alecos Papadopoulos's user avatar
3 votes
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Is optimal labour zero when (i) capital fixed and (ii) elasticity of substitution less than 1?

Ok, rather embarrassing, but the problem was in the formula for $Y_0$. In effect, for $\rho>0$, it is true that $$ Y_{_{L=0}} = AK(1-\alpha)^{\frac{1}{\rho}} $$ However, for the case of $\rho<...
luchonacho's user avatar
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3 votes

Hardcore elasticity of substitution (bad results)

I think that making the substitutions you overlooked some dependence on $z=x_3/x_1$. I mean, if you look at your calculations, you will find some $x_3/x_1$ which is not substituted by $z$.$^1$ By the ...
BakerStreet's user avatar
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2 votes

Empirical measurements of consumption and production elasticity of substitution?

The elasticity of substitution of production factors derived from empirical data depends on the data. In the case of simple production, you can estimate the production function (fit your data to the ...
Rokis1990's user avatar
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2 votes
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Derivation of the elasticity of substitution of a general production function with labor-augmenting technological progress

You have that in equilibrium each factor is paid its marginal product, so $$\tag1\frac{w_i}{p_i}=B_if'(l_i)$$ and $$\tag2\frac{r_i}{p_i}=f(l_i)-K_if'(l_i)\frac{B_iL_i}{K_i^2}=f(l_i)-l_if'(l_i)$$ so ...
Regio's user avatar
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2 votes
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Meaning of «intertemporal substitution effect dominates the income effect»

I suppose the right approach is to start with the statement: "Real wage is also a positive function of productivity". Let's start with the income effect: If both goods, free time and ...
Levittstyle's user avatar
2 votes

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

You could, for example take the function $f: \mathbb{R}^2_+ \times[0,1] \to \mathbb{R}$. $$ f(L,K, \rho) = L + K + (1-\rho) L K. $$ For $\rho = 1$, we have $f(L, K, 0) = L + K$ which is a production ...
tdm's user avatar
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2 votes

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

The OP has correctly defined the "isoquant elasticity", for which one could argue that it should be the basic definition of "elasticity of substitution". The reason why Hicks ...
Alecos Papadopoulos's user avatar
2 votes

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

Your Lagrangian has a $p_i$ in the first term that shouldn't be there.
VARulle's user avatar
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2 votes
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LSE EC417 2023: Markup as Elasticity Tends to Unity

A monopolist will always produce on the elastic part of the demand curve. The idea is the following: if the output level is on the inelastic pat of the demand curve, then increasing prices, by 1% will ...
tdm's user avatar
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1 vote
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Elasticity of substitution by regression: Biased results (simulation)

I am answering my question because I have found the solution. The thing is: I have started with the wrong premise and therefore reached poor conclusion. Due to this I will edit the question a little ...
Athaeneus's user avatar
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1 vote

Complements/substitutes estimation from data (Slutsky matrix)

If you neither observe the utility level nor the expenditure (or income) level, it seems not possible to identify separately Hicksian and Marshallian demands. So it is not advisable to impose symmetry....
Bertrand's user avatar
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1 vote

Derive and Decompose The Aggregate Elasticity of Substitution in CES Economy

I'm the one who asked the original question in another thread, but unfortunately, my narration is very unclear and hence the original one has been closed but I'm glad that there is a new question ...
Brandon Li's user avatar
1 vote

Elasticity of Substitution of CRS Production Function

1muflon1's answer is entirely correct. Let me give another alternative derivation, which might be a little bit easier to remember, although probably not more intuitive. Let $k = K/L$ be the capital to ...
tdm's user avatar
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1 vote

How to compute elasticity of substituion in general?

The formula you already have there is a general formula for elasticity of substitution, but I can see that it might be difficult to apply to your problem here given that $MU_{x_2}=1$. There is also ...
1muflon1's user avatar
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1 vote

Price elasticity of demand of CES

Assuming that $p_i \neq p_j$ you just apply the formula; $$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\...
1muflon1's user avatar
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1 vote

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

The elasticity of demand measures how changes in prices affect changes in quantities demanded So if you have the total quantity demanded at the original prices and reduced them by, say, $10\%$, then ...
Regio's user avatar
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1 vote
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Second order partial derivative and cross second-order partial derivative

Is there anyone who can help me with this? Here it is. Equations 1-3, and 5-6 are obtained in preparation for the 2nd derivatives of V with respect to L and K. Let me know if you have any questions.
Iñaki Viggers's user avatar

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