In the standard models of trade, such as Ricardo or Hecksher-Ohlin, I know that openness to trade will lead to increased consumption as (roughly) the consumption possibility frontier expands beyond the PPF. However, does this mean that the countries will grow?

Technically, consumption expands because the relative price of the goods in which you won't specialize will decrease. Therefore, I don't see how I could infer from this increased consumption that there has been economic growth.

I am asking because I have seen many lectures jump from the classic diagrams of international trade to the idea that the baseline theory suggests that trade is benefitial to economic growth.

• In a very basic sense, increased consumption resulting from balanced trade should be equivalent to real economic growth, because the real expenditure measure of GDP (equal to the other measures) should reflect the change in consumption after adjusting for changes in prices – Henry Feb 5 '18 at 21:05

It depends on how you define "growth". Trade models have multiple goods in them, so there's no unique way to define "growth" - you are just moving along a fixed production possibility frontier. You can calculate nominal GDP, but then you have to come up with a measure of inflation so that you can deflate the nominal figure into something real.

One of the results here is that if your economy which produces $n$ goods has a production possibility frontier given by the zero set of a $C^2$ function $Q : \mathbb R^n \to \mathbb R$ (with the usual properties, i.e increasing in each coordinate and with positive definite Hessian at every point), then a simple application of the chain rule gives (here the $dx$ notation denotes derivatives with respect to time, so we're assuming that the economy shifts continuously along the ppf instead of a unit step jump from one point to another - this assumption is important, otherwise we can't lose the higher order terms in the Taylor expansion of $Q$...)

$$0 = dQ = \sum_{k=1}^{n} \frac{\partial Q}{\partial x_k} dx_k$$

if you're moving along the ppf, so that the value of $Q$ is conserved. However, we also know from the first order conditions of producers in a competitive market that

$$\frac{\partial Q/\partial x_i}{\partial Q / \partial x_j} = \frac{p_i}{p_j}$$

where $p_i, p_j$ are the nominal prices of good $i$ and good $j$ respectively. Substituting this into the above identity gives

$$0 = \sum_{k=1}^n p_k dx_k$$

This is an important result: it says that if you calculate growth by looking at changes in production and hold the prices fixed, then you will detect no growth if the economy moves along a fixed production possibility frontier. Furthermore, this identity combined with the identity

$$NGDP = \sum_{k=1}^{n} p_k x_k$$

gives, using the product rule, that

$$d(\log(NGDP)) = \frac{1}{NGDP} \sum_{k=1}^n x_k d p_k = \textrm{GDP deflator}$$

In other words, if we deflate GDP by using the GDP deflator which is defined as the percent change in the price of a basket of goods weighted by the share of total production they represented in a given time period, we will measure zero RGDP growth so long as the economy remains along the same production possibility frontier $Q(x_1, x_2, \ldots, x_n) = 0$. The results hold true in autarky as well as in an environment of free trade, so in this sense free trade does not lead to "more growth", at least not due to the effects present in Heckscher-Ohlin.

We can get away from this result if we choose to deflate nominal GDP by an alternative measure, but I think this should be enough illustration that you should be thinking about the welfare effects of trade and not about the "growth" effects. If you do come up with an appropriate notion of CPI inflation in a model, it will most likely be a "cost of living index" of the kind you find in Dixit-Stiglitz type models, so you will still indirectly be measuring welfare. The basic proof of the first welfare theorem carries over to the case of international free trade vs autarky, so under the usual assumptions (locally nonsatiated preferences and such) free trade outcomes are Pareto optimal, whereas autarky outcomes are in general not Pareto optimal.