Friday, 21 October 2016

Productivity Growth and Trade: A Model

I did a post back in May about manufacturing in the UK and its role in productivity growth and in foreign trade.  My purpose was to stress that, for an economy as open as the UK, industry concentration was more about trade than about productivity growth.  

The fact is that simply securing productivity growth can actually be detrimental for a country.  The reasons for this are not immediately obvious, so I drew up a little model to illustrate it. 

There are two countries: Country A and Country B.  Each country produces haircuts and one type of fruit - Country A produces apples and Country B produces bananas.  Haircuts are not traded internationally; fruit is.  So households in each country consume two types of fruit and domestic haircuts.

Wages are fixed in the currency of each country.  All prices are set at the same fixed mark-up to unit labour costs.  We'll call Country A's currency the $, and Country B's £.

The elasticity of substitution in demand is the same for each product and in each country.  We start by assuming that in both countries, households spend an equal amount on each of the three products they consume and that 1/3rd of the workforce is employed in producing haircuts and 2/3rds in producing fruit.

The £ / $ exchange rate floats to ensure that the value of exports equals the value of imports for each country.  The labour supply is fixed and demand is managed to ensure continual full employment.

So far, each country is identical.  The difference we want to introduce is to suppose that there is a 3% per period growth in labour productivity in the production of bananas.  There is no change in labour productivity in the production of apples or haircuts.

The charts below are based on an elasticity of substitution in demand of 0.75 and are normalised to give opening values of unity.

The first thing to notice is that the banana producing Country B has GDP growth and Country A does not.  (GDP is calculated here as a chained volume measure at opening year prices.) This is hardly surprising.  The GDP growth rate is less than the rate of growth in banana productivity, because there is no change in productivity in haircuts.

Rising banana productivity means falling unit labour costs and falling banana prices in the domestic currency, £.
At the prevailing exchange rate, a fall in the £ banana price would lead to a drop in the value of exports for Country B, even though the volume of exports rises, given that the demand elasticity is less than 1.  The exchange rate therefore has to change leading to a fall in the value of the £ against the $.  This means that the $ price of bananas falls by even more than the £ price.  It also means that £ price of apples rises, even though the $ price of apples is unchanged.

These further price changes alter trade volumes until the values of trade flows balance.  The chart below shows that this involves a big increase in the banana exports of Country B, whilst there is a slight decline in Country A's apple exports.  This is consistent with Thirlwall's Law and what is happening here to GDP.

The exchange rate movement also means that consumer prices fall by more in Country A than in Country B.  This means that real wages (based on a consumption price index - not the GDP deflator) rise more slowly in Country B than in Country A, notwithstanding that Country B is generating all of the growth in production.

In this model, Country A wages rise faster than those in Country B whenever the elasticity of substitution in demand is less than 1.  In fact, if the elasticity is less than about 0.61, then real wages in Country B actually fall, because the £ price of apples rises faster than the £ price of bananas falls.  This result is somewhat counter-intuitive.

These elasticity levels are not at all unrealistic for international trade flows. 

As a further point it is worth noting that Country B can mitigate the reduction in its own real wages by depressing domestic demand.  This reduces employment and GDP in Country B.  It raises real wages in Country B, but reduces them in Country A.  Imposing tariffs (whether on exports or on imports) will also raise real wages in Country B at the expense of those in Country A, but does not involve reduced employment.

The purpose of this post is simply to highlight two points:

1. GDP growth is not the same as growth in living standards.  A country that has a high proportion of activity in industries with strong productivity growth is likely to have high GDP growth.  But this, in itself, is not a good reason to concentrate on such industries.

2. Elasticities in traded goods are crucial.

However, it is not the purpose of this post to suggest that it is a bad thing to have industries with high potential productivity growth.  In practice growth in productivity is not mainly about producing more of the same for given inputs; it is about producing new and better products.  This innovation is itself important in developing and sustaining export demand.  We cannot separate developments in trade from what is happening with productivity growth.  The important point though is that trade is a critical part of the picture; productivity growth alone tells us very little.

Equation Listing

Consumption of each good in each country is based on a consumption index and the price relative to a consumption price index.
1.            CAa = CA / 3 . (p$a / pA)
2.            CAb = CA / 3 . (p$b / pA)
3.            CAh = CA / 3 . (p$h / pA)
4.            CBa = CB / 3 . ( p£a / pB)
5.            CBb = CB / 3 . (p£b / pB)
6.            CBh = CB / 3 . (p£h / pB)

With the price indices calculated as:
7.            pA = ( p$a . CAa + p$b . CAb + p$h . CAh) / CA
8.            pB = ( p£a . CBa + p£b . CBb + p£h . CBh) / CB

All domestic prices are set at the same mark-up to unit labour costs.
9.            p£b = λ . wB / σb
10.          p£h = λ . wB / σh
11.          p$a = λ . wA / σa
12.          p$h = λ . wA / σh

Import prices reflect the exchange rate.
13.          p£a = e . p$a
14.          p$b = p£b / e

The value of exports equals the value of imports.  (This equation is used to find the market clearing exchange rate.)
15.          CAb . p$b = CBa . p$a

Employment is based on consumption and productivity.  (In the basic scenario described, the levels of the consumption indices CA and CB are set so that all available labour is employed in both countries.)
16.          LB = CBh / σh + ( CAb + CBb ) / σb
17.          LA = CAh / σh + ( CAa + CBa ) / σa


CXy          Consumption of y in country X
CX            Consumption index in country X
pzy          Price of y denominated in z
pX            Price index in country X, denominated in domestic currency
wX           Nominal wages in country X, denominated in domestic currency
σy                  Labour productivity in production of y
LX            Employment in country X
e             Exchange rate ( £ per $ )

σ is given the same value for each good, in the first period.


  1. One of my current projects is to better understand foreign exchange. When I found this post, I read it with great interest. Hindering my progress, I thought I needed a much better understanding of the equations behind the graphs but I could not find a definition for -ε and λ. I also thought that equations 13 and 14 would give different values if the exchange rate was used to manage demand. And then I noticed that I needed to study "elasticity of substitution in demand" which I have only begun to do.

    Well, if both of these economies are "demand managed", it seems like there should be more factors, one of which would be controlled by a (presently) unnamed player(s).

    It also seems that if there are productivity gains, they should be shared by achieving greater production and consumption, which you seem to show. Another potential gain should be more leisure time but this option has been ruled out by assumption.

    To conclude, I like the model but find it incomplete in the sense that I cannot quickly replicate the graphs independently. Apparently, I just don't yet understand all the assumptions.

    1. Epsilon is the elasticity of demand. This gives the percentage change in quantity demanded divided by the percentage change in price, other things beng equal. Lambda is the pricing mark-up. Since only relative prices matter here and it applies to all prices, its value is irrelevant.

      I could add extra bits into the model to explain how domestic demand is determined, but it wouldn't add anything to the point here, so it's clearer if I leave it out and just assume that C is set at the required level.

  2. Thanks Nick. I will study the post further with these comments in mind.

  3. Nick,

    I wrote a long comment after spending a lot of time studying your post -- and managed to vanish the text during posting. Not enough time to recreate that comment but here is an abbreviated version:

    It seems to me that the definition of "full employment" drives results. There seems to be no valid standard that relates time UNIVERSALLY to products produced. There seems to be only local standards.

    Without a definition of "full employment", we cannot define the effects of a productivity change. While we know full well that a productivity change does bring changes in the mix of employment distribution, we are at a loss to know exactly what those changes will be. As a result, we can only achieve assumption driven estimations.

    Thanks for sharing this model and thereby posing an intellectual tease. Macroeconomics is a difficult study.

    1. I'm not sure why your post is not showing, but I received notification of it, so I'm copying it as follows:

      "Well, I have indeed thought a lot more about your post. Thanks for sharing it, thanks for the provocative model.

      Because I usually think in a mechanical way, I like to break down problems into components. Here I am finding several definition components that have more than one solution or choice.

      1. Define "full employment". How for the haircut person? How between before and after banana productivity change?

      2. Define "domestic prices". In the post we assume a price relationship to unit labor cost. As you indicate, this would change as we varied productivity. It would also vary as we redefined "full employment".

      3. Define "price in productivity change". Here we have two choices: More output at same price or increased price per individual output unit. You used a combination of both (in the post) when you added the concept of "elasticity of substitution in demand". The choices we make here seem to affect our definition of "full employment".

      And 4. Define "exchange rate". You write "The £ / $ exchange rate floats to ensure that the value of exports equals the value of imports for each country. The labour supply is fixed and demand is managed to ensure continual full employment." We see that "exchange rate" is affected by our definition of "full employment".

      If you have read this far into my comment, you can see that I think the definition of "full employment" is critical to the discussion. Defining "full employment" is difficult, particularly if productivity changes. The case of the haircut person is a good example; the real world product is only needed (perhaps) once a month (requiring a few minutes of time) contrasted to apples which might be consumed once a day. What kind of labor units are we comparing?

      This (your post) is a great model but the choices required (seem to me) make a solution very assumption driven. Not to say that your result is wrong, but to say that other solutions are possible.

      One alternate option is to break the model into sub-sectors. Both apple and banana sectors can be broken into pound and dollar sectors. Then each sector can be modeled with the country of currency. Since these are the only portions of either economy that are traded using foreign exchange, the foreign exchange rate has a primary effect on these sectors. This sector division makes clear that consumer substitution-of-demand effects (and productivity changes) produce results that are sector concentrated.

      Thanks again for this post. Macroeconomics is a complicated study."

    2. And my reply:

      1. Actually, it's probably unhelpful to refer to full employment, because what actually matters here is that labour utilisation is constant, rather than that it is equal to available labour. In terms if the equations, that simply means that I am holding LA and LB constant.

      2. By domestic prices, I mean the sale prices expressed in the currency of the seller.

      3. I'm not sure where I used this term and it doesn't mean much to me in isolation I'm afraid.

      4. I don't know how I can further define exchange rate. Here, it's the number of £ you would get in exchange for $1.

      I think I have divided the apple and banana markets into each country in that a distinct quantity of sales is determined for each country. I have assumed however that the same price applies in both countries (after translating at the exchange rate), because i have assumed simply mark-up pricing. I could instead have had sellers charge a different mark-up in each market. I chose not to do so to limit the number of equations and because I don't think it makes any difference to the point here.