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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 $$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 }... 5 Reason: Both goods cannot be inferior. Let's say originally you consume x and y. So your budget constraint looks like$$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 = ...

2

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 $$MP_L = \frac A {\left( \alpha + ... 2 Ok, rather embarrassing, but the problem was in the formula for Y_0. In effect, for \rho>0, it is true that$$ 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 ...

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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