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Doing this more abstractly, let $Y_j\subseteq\mathbb{R}^n$ be a production set for $j=1,\ldots,m$ and let $$Y=Y_1+Y_2+\cdots+Y_N=\{y_1+y_2+\cdots+y_n|y_j\in Y_j, j=1,\ldots,m\}$$ be the aggregate production set. The standard result on when the aggregate production set is closed is the following: Theorem: Let $Y_j$ be closed and convex sets containing $0$ for ...


3

For a fully overview on the conditions for the sum of closed sets to be closed, see this note of Kim Border Recession cones I'll be working with subsets of $\mathbb{R}^n$. Let's start with some definitions. Def: A set $C$ is convex if for $x, y \in C$ and $\alpha \in [0,1]$, $\alpha x + (1-\alpha) y \in C$. Def: A set $K$ is a cone if for $x \in K$ and $\...


2

Touch wood that I did not make any mistakes. Consider the production function $X = (K + \alpha)(L + \beta)$. The elasticity of substitution is given by: $$ \frac{\partial \ln(K/L)}{\partial\ln(MP_L/MP_K)} = \frac{\partial(\ln(K) - \ln(L)}{\partial(\ln(K + \alpha)- \ln(L + \beta))} $$ Let's take the derivative of both numerator and denominator with respect to ...


2

Actually the Cambridge controversy sort of affected macroeconomic modeling. As explained in the article that is also linked in one of the comments, the aggregate production function, real business cycle, neoclassical general equilibrium models, and endogenous growth theories can be thought of as an retort to some of the criticism levied by the English side ...


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