Trade, the Exchange Rate and Real Wages in the UK

Simon-Wren Lewis wrote an interesting post recently, which among other things referred to the impact of sterling depreciation on UK real wage levels.

Any increase in domestic demand in the UK will lead to a greater demand for imports and potential pressure on the exchange rate.  A weaker exchange rate means a higher level of demand can be sustained with the same level of trade balance, but this has implications for real wages.

What I thought would be useful was to make a rough estimate of what exchange rate would be needed if UK GDP for 2016 were to be 5% higher, but with a comparable trade deficit and what this would imply for real wage levels.  To do this, I have set up a simple model using estimated parameters.  These are based on a combination of my own estimates and third party estimates.  Exact specification of the model used is given at the end of the post.

It should be stressed that all of the parameters used here are for long-term elasticities.  Most transactions are based on the use of established suppliers.  Volumes and, to some extent, prices do not respond quickly to exchange rate movements.  If an exchange rate movement is subsequently reversed, there may be no noticeable effect at all.  However, companies do choose where to supply from and long-term differences in cost will effect this.

The point here is that this is not indicative of how the economy will respond in the immediate period following a depreciation.  This is an exercise in counterfactuals or comparative statics.

The scenario I wanted to consider here was what variation in the exchange rate would be required to maintain the trade balance at a constant percentage of GDP, were GDP to be 5% higher, based on 2016 figures.  It turns out that this requires an exchange rate that is 13.4% lower.  It also requires domestic expenditure to be 4.5% higher.

The table below shows the percentage difference in each of the variables:

Variable	Variation
Exchange rate	-13.4%
Export price index	+7.8%
Import price index	+9.2%
Domestic expenditure price index	+2.5%
GDP deflator	+2.1%
Export volume	+1.5%
Import volume	+0.1%
Domestic expenditure volume	+4.5%
GDP	+5.0%

Here, the potential increase in import volume arising from the higher expenditure is substantially offset by the fall in the exchange rate.  Export volume grows slightly, as although export prices rise, they do not rise as much as world prices in sterling terms.

A key assumption here is that domestic unit labour costs are unchanged, so that the only thing impacting on these price indices is the change in the sterling equivalent of world prices.  Given the same level of productivity, the 2.5% higher domestic price index implies 2.5% lower real wages.  (Treating the consumer price index as being the same as that for all domestic expenditure - a simplification.)

Sustaining the change in real exchange rate necessary to achieve this result therefore requires that the reduction in real wage is fully absorbed and is not eroded by increased wage inflation.  (Of course the impact could be alternatively absorbed by a change in production taxes or a reduction in profitability).

Model Specification

Equations

Real GDP is the sum of domestic expenditure and exports less imports.

(1)          gdp = dx + ex - im

(2)          GDP = dx . pd + ex . px - im . pm

Exports vary based on the relative price with an elasticity of -0.3.

(3)          ex = 545 . ( px / pw )^-0.3

Imports are based on relative price and both domestic expenditure and exports.  Exports have a much greater concentration of import content than domestic expenditure, and this is reflected by inclusion of a term for the share of exports in expenditure.  The price elasticity used is -0.33.

(4)          im = 0.3885 . ( dx + ex ) . [ ex / ( dx + ex ) ]^0.34 . ( pm / pd )^-0.33

The balance of trade is based on export and import volumes and prices.

(5)          BT = ex . px - im . pm

Price indices for imports, exports and domestic expenditure prices are a weighted average of world prices and domestic unit labour costs.  World prices here means some appropriate measure of prices in the UK's main trading partners, translated into sterling.  General price levels in the rest of the world is assumed unchanged so the only change is due to the exchange rate.  Domestic unit labour costs are also assumed unchanged.

(6)          pm = pw^0.7 . ulc^0.3

(7)          px = pw^0.6 . ulc^0.4 

(8)          pd = pw^0.2 . ulc^0.8

Variables

Name	Description
BT	Nominal trade balance
dx	Real domestic expenditure
ex	Real exports
gdp	Real GDP
GDP	Nominal GDP
im	Real imports
pd	Price of domestic expenditure
pm	Price of imports
pw	World prices (translated into sterling)
px	Price of exports
ulc	UK unit labour costs

Unit labour costs and all prices are indexed at 1 for 2016 and volume is measured in 2016 prices.  dx and pw are set to give the required level of gdp and ratio of BT to GDP.

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.            C_Aa = C_A / 3 . (p_$a / p_A)^-ε

2.            C_Ab = C_A / 3 . (p_$b / p_A)^-ε

3.            C_Ah = C_A / 3 . (p_$h / p_A)^-ε

4.            C_Ba = C_B / 3 . ( p_£a / p_B)^-ε

5.            C_Bb = C_B / 3 . (p_£b / p_B)^-ε

6.            C_Bh = C_B / 3 . (p_£h / p_B)^-ε

With the price indices calculated as:

7.            p_A = ( p_$a . C_Aa + p_$b . C_Ab + p_$h . C_Ah) / C_A

8.            p_B = ( p_£a . C_Ba + p_£b . C_Bb + p_£h . C_Bh) / C_B

All domestic prices are set at the same mark-up to unit labour costs.

9.            p_£b = λ . w_B / σ_b

10.          p_£h = λ . w_B / σ_h

11.          p_$a = λ . w_A / σ_a

12.          p_$h = λ . w_A / σ_h

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.          C_Ab . p_$b = C_Ba . p_$a

Employment is based on consumption and productivity.  (In the basic scenario described, the levels of the consumption indices C_A and C_B are set so that all available labour is employed in both countries.)

16.          L_B = C_Bh / σ_h + ( C_Ab + C_Bb ) / σ_b

17.          L_A = C_Ah / σ_h + ( C_Aa + C_Ba ) / σ_a

Variables

C_Xy          Consumption of y in country X

C_X            Consumption index in country X

p_zy          Price of y denominated in z

p_X            Price index in country X, denominated in domestic currency

w_X           Nominal wages in country X, denominated in domestic currency

σ_y                  Labour productivity in production of y

L_X            Employment in country X

e             Exchange rate ( £ per $ )

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