7 votes
Accepted

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 ...
  • 595
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 ...
  • 6,672
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 ...
5 votes
Accepted

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 ...
  • 2,764
5 votes
Accepted

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 ...
  • 45.1k
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 $...
  • 3,186
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 $L$ ...
  • 45.1k
4 votes
Accepted

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) = -\...
  • 8,652
3 votes
Accepted

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 ...
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)^{\...
3 votes
Accepted

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<...
  • 8,487
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 ...
  • 151
2 votes
Accepted

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 ...
  • 4,168
2 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/...
  • 136
2 votes
Accepted

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 ...
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 ...
  • 8,652
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 ...
  • 8,652
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 ...
  • 45.1k
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-\...
  • 45.1k
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 ...
  • 4,168
1 vote
Accepted

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.
1 vote

CES production function application problem

Your y2 needs to be the output associated produced with those inputs. Given that you have a CES production function and you use World Bank Country data as inputs, you need output of the countries in ...
  • 2,330
1 vote
Accepted

Complementarity in CES Production Function

Total investment in terms of how much capital is augmented, is always $I = I_{b} + I_{h}$. $(I_{b}^K + I_{h}^K)^{\frac{1}{K}}$ is equal to the amount of the intermediate good $Y$ that we need to ...
1 vote

How do find elasticity of substitution for this function

Elasticity of subsitution ($\sigma$) can be found by using this formula $$\sigma=\frac{dln(\frac{x_1}{x_2})}{dln(MRS)}$$ alternatively you can use: $$\sigma=\frac{e(x)f(x)f_1(x)f_2(x)}{x_1x_2|BH|}$$ ...
  • 7,880
1 vote

Obj function yielding independent goods demand functions

So if we take a simple example with 2 goods and consider the consumer's problem: $L=u(x,y)+\lambda(I-p_xx-p_yy)$ This yields the ratio of FOC: $u_1/u_2=p_x/p_y$ Now plugging these guys into the ...
  • 756
1 vote

Estimating elasticity of substitution in nested CES functions

Neglecting technology parameters and assuming constant returns to scale, the parameters $\sigma$ and $\gamma$ are jointly estimable via dynamic (least squares) programming. See this paper.

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