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.

Wealth Concentration and Loanable Funds

Jo Michell has an interesting post on loanable funds.  This was prompted by the question of whether wealth concentration has led to rising asset prices and falling yields.  Whilst sharing Jo's views in many respects, my own analysis here is slightly different, so I thought it worth setting out.

First of all, I would agree that increasing wealth concentration is likely to increase the propensity to save out of income.  However, this in itself is not enough to change interest rates.

The diagram below is the standard model of loanable funds (borrowed from Jo, and in turn from Nick Rowe).  The idea here is that if there is an increase in the desired quantity of saving at any given interest rate, this is represented by a rightward shift in the Sd curve.  This leads to a fall in the equilibrium interest rate (even if the Id curve is vertical).

 

However, an increased propensity to save is not the same thing as a desire to save an increased quantity.  In fact, an increased desire to save leads to lower income.  The actual quantity people end up wanting to save is unchanged; it simply represents a higher proportion of income.  This is the paradox of thrift.  So there is no movement in the Sd curve and no change in the (partial) equilibrium interest rate.

There may be temporary movements, if people are mistaken in what they expect will happen.  If one person tries to save more without realising others are also doing so, then they may overestimate their own income and start to bid up asset prices.  But, assuming they eventually cotton on to what is happening, this will eventually reverse.

Having said all this, I would still say that increased wealth concentration has contributed to falling yields.  The reasons for saying so are as follows.

The argument above is based on what happens in a monetary economy.  In looking at how things pan out in a monetary economy, we can't get very far without asking how monetary policy is framed.  For example, if monetary policy takes the form of simply fixing or targeting a particular interest rate, then almost by definition changes in savings propensity are not going to change that rate.

However, for our purposes here we need to recognise that current monetary policy takes the form of inflation targeting.  This means that the central bank responds to perceived deflationary pressures by reducing the policy rate.  So an increased propensity to save does indeed lead to lower interest rates, but it does so because it depresses demand and because the central bank reacts to that.

One interpretation here is to suppose that implementation of monetary policy brings the economy back to its original level of demand, at whatever interest rate that requires.  Thus we can go back to our loanable funds diagram and say that the Sd curve has indeed shifted to the right, once we have taken into account the full operation of monetary policy.

This is one way of looking at it and I'm all in favour of looking at things in different ways, for the additional insight it provides.  However, I find it a little problematic.  We can easily conceive of a situation where there is an increased propensity to save but where reductions in interest rates are ineffective in restoring demand.  In this scenario, there is no solution compatible with the unchanged output interpretation and it fails as an explanation of interest rates.

Even if we think that interest rate manipulation can bring output back to its original level, there is no general reason to suppose that the terminal interest rate of this process is independent of the path taken to get there.  (In most models it is independent, but that doesn't mean it is in reality.)  There would then be no meaningful ceteris paribus solution to the loanable funds model. 

There is another important way in which wealth concentration has contributed to falling yields, one that was particularly important in the run up to the crisis.  This is not to do with increased propensity to save, but rather with portfolio preference.  (We might think of portfolio preference as a more general form of liquidity preference, when we are considering a range of assets rather than simply bonds and money.)

Wealth concentration makes investors more concerned about large exposures to single name risk.  Pre-crisis, managers of large cash pools already holding sizeable unsecured bank deposits increasingly sought alternative low risk short term investments.  This created a strong demand for traded high quality assets which could be used as collateral for short term secured instruments.  This in turn led to increased demand for the assets that could used to create such collateral, such as securitisable mortgages.  The effect of this demand pressure was to drive down the yields on such assets relative to policy rates and rates on unsecured bank deposits.

In short, wealth concentration is certainly an important part of the picture of what has happened to financial asset yields.  But whilst the loanable funds model might provide some kind of insight it is, in my view, an insufficient framework for understanding the mechanisms at play.