Developments in International Trade Theory and Gravity Modelling

Abstract

This chapter discusses advances in international trade theory and gravity modeling with an explanation of the reasons behind gains from trade. The changing pattern of trade over time has also changed the explanation of the emergence of gains from trade, which provides room for new trade theories. Initial theories of trade, known as traditional trade theories, explain the pattern of trade in terms of comparative advantage. But with the passage of time, the emergence of trade in intermediates and services has provided new reasons for trade and hence has led to the advent of new trade theories. This chapter will explain the different reasons behind international trade.

Appendix

We have the utility function:

$$ {\left[\sum \limits_{i=1}^N{c}_i^{\rho}\right]}^{\left(1/\rho \right)} $$

This utility should be maximized subject to the income constraint:

$$ I=\sum \limits_{i=1}^N{p}_i{c}_i $$

Now, we have Langragian (ℒ) with multiplier (λ):

$$ \mathrm{\mathcal{L}}={\left[\sum \limits_{i=1}^N{c}_i^{\rho}\right]}^{\left(1/\rho \right)}+\lambda \left[I-\sum \limits_{i=1}^N{p}_i{c}_i\right] $$

Differentiating ℒ with respect to c _j and equating it to zero gives us

$$ {\left[\sum \limits_{i=1}^N{c}_i^r\right]}^{\left(1/r\right)-1}{c}_j^{r-1}=\lambda {p}_j\kern3.75em \mathrm{for}\ j=1,2,\dots, N $$

Take the ratio of these first-order conditions with respect to variety 1, and define ε = 1/(1 − ρ) as discussed in the main text. Then:

$$ \frac{{\left[{\sum}_{i=1}^N{c}_i^{\rho}\right]}^{\left(1/\rho \right)-1}{c}_j^{\rho -1}}{{\left[{\sum}_{i=1}^N{c}_i^{\rho}\right]}^{\left(1/\rho \right)-1}{c}_1^{\rho -1}}=\frac{\lambda {p}_j}{\lambda {p}_1} $$

$$ \frac{c_j^{\rho -1}}{c_1^{\rho -1}}=\frac{p_j}{p_1} $$

$$ {c}_j^{\rho -1}={p}_j{\left({p}_1\right)}^{-1}{c}_1^{1-\rho } $$

$$ {c}_j={p_j}^{\left(\rho -1\right)}{\left({p}_1\right)}^{-1\left(\rho -1\right)}{c}_1^{\frac{\rho -1}{\left(\rho -1\right)}} $$

$$ {c}_j={p_j}^{-\left(1-\rho \right)}{\left({p}_1\right)}^{\left(1-\rho \right)}{c}_1^1\kern1.75em or\kern1.25em {c}_j={p_j}^{-\varepsilon }{p_1}^{\varepsilon }{c}_1 $$

Substituting these relations in the budget equation gives:

$$ I=\sum \limits_{i=1}^N{p}_j{c}_j\kern0.75em \Rightarrow \kern0.5em I=\sum \limits_{i=1}^N{p}_j{p_j}^{-\varepsilon }{p_1}^{\varepsilon }{c}_1\kern0.5em \Rightarrow \kern0.5em I={p_1}^{\varepsilon }{c}_1\sum \limits_{i=1}^N{p_j}^{1-\varepsilon } $$

$$ I={p_1}^{\varepsilon }{c}_1{P}^{1-\varepsilon}\kern1em or\kern1em {c}_1={Ip_1}^{-\varepsilon }{P}^{\varepsilon -1} $$

$$ P\equiv {\left[\sum \limits_{i=1}^N{p_j}^{1-\varepsilon}\right]}^{\frac{1}{\left(1-\varepsilon \right)}} $$

In the above equation, c ₁ represents the demand for variety 1. Similarly, we can derive the demand for other varieties. To answer the question of why we defined a P type of price index in the above equation, we need to substitute the derived demand for all the varieties in the utility function along with: $\varepsilon =1/\left(1-\rho \right)\kern0.75em \Rightarrow \kern0.75em 1-\varepsilon =-\varepsilon \rho \kern0.75em \Rightarrow \kern1em \frac{1-\varepsilon }{\varepsilon }-=\rho \kern0.5em \Rightarrow \kern0.75em \frac{1}{\rho }=-\frac{\varepsilon }{1-\varepsilon }$

$$ U={\left(\sum \limits_{i=1}^N{c}_i^{\rho}\right)}^{1/\rho }={\left(\sum \limits_{i=1}^N{\left({Ip_i}^{-\varepsilon }{P}^{\varepsilon -1}\right)}^{\rho}\right)}^{1/\rho } $$

$$ ={IP}^{\varepsilon -1}{\left(\sum \limits_{i=1}^N{p_i}^{-\varepsilon \rho}\right)}^{1/\rho }={IP}^{\varepsilon -1}{\left(\sum \limits_{i=1}^N{p_i}^{1-\varepsilon}\right)}^{-\frac{\varepsilon }{1-\varepsilon }} $$

Using the price index again¹⁰;

$$ U={IP}^{\varepsilon -1}{\left({P}^{1-\varepsilon}\right)}^{-\frac{\varepsilon }{1-\varepsilon }}={IP}^{\varepsilon -1}{P}^{-\varepsilon }={IP}^{-1} $$

$$ U=\frac{I}{P} $$