Abstract
Let \(A = f_{A}(L_{A},K_{A})\) and \(B = f_{B}(L_{B},K_{B})\) be the aggregate production functions (twice continuously differentiable) of the two commodities, where \(L_{A} + L_{B} = L\) (the total amount of labour existing in the economy) and \(K_{A} + K_{B} = K\) (the total amount of capital existing in the economy).
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Notes
- 1.
Here, as everywhere in this book, we follow the commonly adopted practice of using the same symbol to denote both the commodity and its quantity.
- 2.
Consider for example the first equation and divide both sides by L, obtaining
$$\displaystyle{ \frac{A} {L}\text{d}a_{\mathit{LA}} + a_{\mathit{LA}} \frac{1} {L}\text{d}A + \frac{B} {L} \text{d}a_{\mathit{LB}} + a_{\mathit{LB}} \frac{1} {L}\text{d}B = \frac{\text{d}L} {L}.\ }$$Then multiply and divide the first term on the left by a LA and so on; the result is
$$\displaystyle{ \frac{a_{\mathit{LA}}A} {L} \cdot \frac{\text{d}a_{\mathit{LA}}} {a_{\mathit{LA}}} + \frac{a_{\mathit{LA}}A} {L} \cdot \frac{\text{d}A} {A} + \frac{a_{\mathit{LB}}B} {L} \cdot \frac{\text{d}a_{\mathit{LB}}} {a_{\mathit{LB}}\ } + \frac{a_{\mathit{LB}}B} {L} \cdot \frac{\text{d}B} {B} = L, }$$which is the first equation in (19.62).
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Gandolfo, G. (2014). Appendix to Chapter 3. In: International Trade Theory and Policy. Springer Texts in Business and Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37314-5_19
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DOI: https://doi.org/10.1007/978-3-642-37314-5_19
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