7

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 given by $\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}}$. Notice that $$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 }...


7

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 sometimes called the Armington aggregator, which was discussed by Armington (1969). Then, the CES utility function was popularized by Dixit and Stiglitz (1977) ...


6

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 economic efficiency. The review of Economics and Statistics, 225-250. There you can find a discussion of how it was derived. A more pedagogic and detailed ...


5

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 from $I_0$ to $I'$, the quantity demanded for both has to go up (by definition) from $x$ to $x'$ and $y$ to $y'$. This implies $$p_x x' + p_y y' = I' < I = ...


5

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 give you location of stationary points. They give you points where $df(x,y)/dx = df(x,y)/dy= 0$. A maximum or minimum will occur at points where $df(x,y)/dx = df(...


4

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 $x_1$. From this it follows that $$\frac{\partial }{\partial x_1} f(x_1,x_2(x_1)) = c \Leftrightarrow \\[8pt] f_1(x_1,x_2) + f_2(x_1,x_2) \frac{\partial x_2}{\...


4

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 $L$ and $K$* is given by (see Sydsæter et al. EMEA pp 430): $$\sigma_{L,K} = \frac{-F_K'F_L'(xF_K'+ yF_L')}{xy \left( (F_L')^2F_{KK}^{''} - 2 F_K'F_L'F_{KL}'' + (F_K'...


4

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: $$ \ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L) $$ Then take the derivative of this expression with respect to $\ln(t)$: $$ \begin{align*} -\sigma &= -1 - \dfrac{\dfrac{\...


3

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 says: $$L=\left(\sum_{e=1}^{n} \theta_{e}L_{e}^{\frac{\delta - 1}{\delta}}\right)^{\frac{\delta}{\delta - 1}}$$. In the remaining equations $\theta$ is not under ...


3

Starting from $$Y = A\left( \alpha L^{\rho} + (1-\alpha)K^{\rho} \right)^{\frac{1}{\rho}}$$ $$...\implies MP_L = \alpha A\cdot\left( \alpha + (1-\alpha)\left(\frac {K}{L}\right)^{\rho} \right)^{\frac{1}{\rho}-1}$$ assume that $\rho <0$ (which gives an elasticity of substitution smaller than unity). Then we can write $$MP_L = \frac A {\left( \alpha + ...


3

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 $\rho<0$, the limit of $Y$ is zero. The proof is trivial. Rewrite the CES as: $$ Y = AL\left(\alpha + (1-\alpha)\left(\frac{L}{K}\right)^{-\rho} \right)^{\frac{1}{\...


2

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 functional form of the production function) and from it derive the elasticity of substitution and other functional parameters. The elasticity of substitution in ...


2

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 deviding (2) by (1) we have: $$\tag3\frac{r_i}{w_i}=\frac{f(l_i)-l_if'(l_i)}{B_if'(l_i)}.$$ Take the derivative of $r_i/w_i$ with respect to $l_i$: $$\tag4\...


2

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/Constant_elasticity_of_substitution) . However, you have made the following mistakes. NON CONSEQUENTIAL ERROR: Your first error is when you differentiate to get MP1 and MP2....


2

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 consumption are normal goods (This is the standard case; normal good simply means: if you are richer you want to consume more of them), then the individual wants to ...


2

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 function with perfect substitutes. The marginal product (of say $L$) is given by: $$ \frac{\partial f(L, K, \rho)}{\partial x} = 1 + (1-\rho)K, $$ which equals $...


1

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 labour ratio. Then we can define $f(k) = F(K/L, 1)$ to be the output per unit of labour. then by the CRS assumption we have: $$ F(K,L) = L f(K/L) = L f(k). $$ ...


1

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 another way how formula for elasticity of substitution can be expressed. You can use 'partial derivative formula' (e.g. see Sydsaeter et al. EMEA pp 430) which is ...


1

Assuming that $p_i \neq p_j$ you just apply the formula; $$\epsilon_i(p_i,I,P) =-\dfrac{\partial d(p_i,I,P)}{\partial p_i}\dfrac{p_i}{d(p_i,I,P)} \\ = -\left( \frac{-\sigma p_i^{-\sigma-1} I}{P^{1-\sigma}} \right)\frac{p_i}{\frac{p_i^{-\sigma} I}{P^{1-\sigma}}} =\sigma$$. The way how you gave the problem neither $P$ or $I$ contain $p_i$ so you treat them as ...


1

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 observe the total quantity demanded of both goods, you can calculate elasticity by simply using the arc formula, or the midpoint formula, etc. In fact, by just ...


1

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

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 that specific year. Typically we use GDP as a measure of output at country level. You can get it a.o. from the World Bank.


1

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 allocate for investment. And given the formulation and when $K>1$ we see that we economize on the amount of the intermediate good $Y$, $$Y_I = (I_{b}^K + I_{h}...


1

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: $$\sigma=\frac{e(x)f(x)f_1(x)f_2(x)}{x_1x_2|BH|}$$ where $x=x_1,x_2......x_n$ and $e(x)$ is elasticity of scale. You can use this information to solve your problem.


1

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 budget: $I=p_xu_1^{-1}(u_2p_x/p_y)+p_yy$ A sufficient condition is $u_1^{-1}$ be hod -1 to make $p_x$ drop out. But I don't think it's necessary. Now I'm not ...


1

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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