# Mathematical proof of dynamic stability of equilibrium in international trade, with quantity adjustment

I am studying dynamic stability of neoclassical international equilibrium given in Giancarlo Gandolfo Economic Dynamics, but I am failing to understand an intermediate step, namely linearization of the behavioural function. It assumes the followings behaviour of home and foreign (denoted by *):

\begin{align} d/dt (-\bar E_1) = f_1 [-E_1(\bar p) -(-\bar E_1)], && f_1(0) = 0, && \dot f_1(0) > 0\\ d/dt (-\bar E^*_2) = f_2 [-E^*_2(\bar p) -(-\bar E^*_1)], && f_2(0) = 0, && \dot f_2(0) > 0\\ \end{align} where $$E_i(\bar p)$$ is the desired export of $$i$$ (according to offer curves) at terms of trade $$\bar p = p_2/p_1$$, and $$\bar E_i$$ is the actual exports; $$f_1$$ and $$f_2$$ are sign preserving. From the equations, it is obvious home exports 1 and foreign 2. Each country adjusts their export towards that which it would desired at the terms of trade prevailing. It proceeds to linearise the functions at $$p_e$$, the equilibrium terms of trade, which I am failing to understand. I will only discuss linearisation of $$f_1$$ since understanding that will make the other understood. It linearises $$f_1$$ at $$p_e$$ as:

\begin{align} -d/dt(\bar E_1)=(1+\epsilon)[\bar E_1 - E_1(p_e)] - p_e \epsilon[\bar E^*_2 - E^*_2(p_e)] \end{align} where $$\epsilon$$ is the export elasticity of hom at $$p_e$$ given as: $$(dE_1/dp) (p_e/E_1(p_e))$$. The proof of this is what I do not understand. To prove this, firstly, it linearises the argument (of $$f_1$$) as: \begin{align} \bar E_1 - E_1(\bar p) = \bar E_1 - (dE_1/dp) (\bar p-p_e) - E_1(p_e) \tag{1} \end{align} This looks like a first-oder taylor expansion around $$p=p_e$$ with $$d\bar E_1/dp = 0$$, which does not makes sense; how can actual exports be independent of $$p$$? Moving on, the sum of value of excess demand in home must be zero; therefore: $$\bar E_1 + p\bar E_2 = 0$$. Import of one country, must equal exports of the other; therefore: $$\bar E_2 =-\bar E^*_2$$. Combining, it is obtained: $$\bar E_1 = p \bar E^*_2$$. The next step makes no sense to me. It proceeds to linearise this relation to obtain: \begin{align} \bar E_1 - E_1 (p_e) = p_e [\bar E^*_2 - E^*_2(p_e]+ E^*_2(p_e)(\bar p - p_e) \tag{2} \end{align} Where is this coming from, I cannot understand. It seems like a Taylor expansion, but there is no derivative. Next it manipulates to express $$(\bar p - p_e)$$ in terms of the other variables in the expression, which it substitutes in (1). From there, the linearisation of the behavioural function is obtained.

Alternate less vague proofs for the same behaviour is also welcome.

• Actual exports depend on $p$ only insofar as they approach (over time) desired exports, which depend on $p$. So there is no direct effect of $p$ on actual exports. Nov 25, 2021 at 11:15
• By definition, is terms of trade not the no. of units of 1 that home would have to actually export for 1 unit of 2? How can terms of trade change without changing the actual exports? In offer curve diagram, actual exports have to lie on the TOT line. Nov 25, 2021 at 12:11
• That's true in equilibrium. But a dynamic stability analysis asks what happens if you deviate from equilibrium and then slowly adjust. The tot is just the ratio of prices, and actual exports are not in equilibrium in the short run. Nov 26, 2021 at 12:36