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Saturday Quiz – April 30, 2011 – answers and discussion

Here are the answers with discussion for yesterday’s quiz. The information provided should help you work out why you missed a question or three! If you haven’t already done the Quiz from yesterday then have a go at it before you read the answers. I hope this helps you develop an understanding of modern monetary theory (MMT) and its application to macroeconomic thinking. Comments as usual welcome, especially if I have made an error.

Question 1:

National accounting shows us that a government surplus equals a non-government deficit. If the British government is successful in putting its budget back into surplus then the private domestic sector will become more indebted as a consequence which means that austerity amounts to swapping public for private debt.

The answer is False.

The point is that the non-government sector is not equivalent to the private domestic sector in the sectoral balance framework. We have to include the impact of the external sector.

So this is a question about the sectoral balances – the government budget balance, the external balance and the private domestic balance – that have to always add to zero because they are derived as an accounting identity from the national accounts. The balances reflect the underlying economic behaviour in each sector which is interdependent – given this is a macroeconomic system we are considering.

To refresh your memory the balances are derived as follows. The basic income-expenditure model in macroeconomics can be viewed in (at least) two ways: (a) from the perspective of the sources of spending; and (b) from the perspective of the uses of the income produced. Bringing these two perspectives (of the same thing) together generates the sectoral balances.

From the sources perspective we write:

GDP = C + I + G + (X – M)

which says that total national income (GDP) is the sum of total final consumption spending (C), total private investment (I), total government spending (G) and net exports (X – M).

From the uses perspective, national income (GDP) can be used for:

GDP = C + S + T

which says that GDP (income) ultimately comes back to households who consume (C), save (S) or pay taxes (T) with it once all the distributions are made.

Equating these two perspectives we get:

C + S + T = GDP = C + I + G + (X – M)

So after simplification (but obeying the equation) we get the sectoral balances view of the national accounts.

(I – S) + (G – T) + (X – M) = 0

That is the three balances have to sum to zero.

You can also write this as:

(S – I) + (T – G) = (X – M)

Which gives an easier interpretation (especially in relation to this question).

The sectoral balances derived are:

  • The private domestic balance (S – I) – positive if in surplus, negative if in deficit.
  • The Budget balance (T – G) – positive if in surplus, negative if in deficit.
  • The Current Account balance (X – M) – positive if in surplus, negative if in deficit.

These balances are usually expressed as a per cent of GDP but that doesn’t alter the accounting rules that they sum to zero, it just means the balance to GDP ratios sum to zero.

Using this version of the sectoral balance framework:

(S – I) + (T – G) = (X – M)

So the domestic balance (left-hand side) – which is the sum of the private domestic sector and the government sector equals the external balance.

For the left-hand side of the equation to be positive (that is, in surplus overall) and the individual sectoral components to be in surplus overall, the right-hand side has to be positive (that is, an external surplus) and of sufficient magnitude.

This is also a basic rule derived from the national accounts and has to apply at all times.

The following graph and accompanying table shows a 8-period sequence where for the first four years the nation is running an external deficit (2 per cent of GDP) and for the last four year the external sector is in surplus (2 per cent of GDP).

For the question to be true we should never see the government surplus (T – G > 0) and the private domestic surplus (S – I > 0) simultaneously occurring – which in the terms of the graph will be the green and navy bars being above the zero line together.

You see that in the first four periods that never occurs which tells you that when there is an external deficit (X – M < 0) the private domestic and government sectors cannot simultaneously run surpluses, no matter how hard they might try. The income adjustments will always force one or both of the sectors into deficit.

The sum of the private domestic surplus and government surplus has to equal the external surplus. So that condition defines the situations when the private domestic sector and the government sector can simultaneously pay back debt.

It is only in Period 5 that we see the condition satisfied (see red circle).

That is because the private and government balances (both surpluses) exactly equal the external surplus.

So if the British government was able to pursue an austerity program with a burgeoning external sector then the private domestic sector would be able to save overall and reduce its debt levels. The reality is that this situation is not occuring.

Going back to the sequence, if the private domestic sector tried to push for higher saving overall (say in Period 6), national income would fall (because overall spending fell) and the government surplus would vanish as the automatic stabilisers responded with lower tax revenue and higher welfare payments.

Periods 7 and 8 show what happens when the private domestic sector runs deficits with an external surplus. The combination of the external surplus and the private domestic deficit adding to demand drives the automatic stabilisers to push the government budget into further surplus as economic activity is high. But this growth scenario is unsustainable because it implies an increasing level of indebtedness overall for the private domestic sector which has finite limits. Eventually, that sector will seek to stabilise its balance sheet (which means households and firms will start to save overall). That would reduce domestic income and the budget would move back into deficit (or a smaller surplus) depending on the size of the external surplus.

So what is the economics that underpin these different situations?

If the nation is running an external deficit it means that the contribution to aggregate demand from the external sector is negative – that is net drain of spending – dragging output down.

The external deficit also means that foreigners are increasing financial claims denominated in the local currency. Given that exports represent a real cost and imports a real benefit, the motivation for a nation running a net exports surplus (the exporting nation in this case) must be to accumulate financial claims (assets) denominated in the currency of the nation running the external deficit.

A fiscal surplus also means the government is spending less than it is “earning” and that puts a drag on aggregate demand and constrains the ability of the economy to grow.

In these circumstances, for income to be stable, the private domestic sector has to spend more than they earn.

You can see this by going back to the aggregate demand relations above. For those who like simple algebra we can manipulate the aggregate demand model to see this more clearly.

Y = GDP = C + I + G + (X – M)

which says that the total national income (Y or GDP) is the sum of total final consumption spending (C), total private investment (I), total government spending (G) and net exports (X – M).

So if the G is spending less than it is “earning” and the external sector is adding less income (X) than it is absorbing spending (M), then the other spending components must be greater than total income.

Only when the government budget deficit supports aggregate demand at income levels which permit the private sector to save out of that income will the latter achieve its desired outcome. At this point, income and employment growth are maximised and private debt levels will be stable.

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Question 2:

The relentless push by neo-liberals to cut real wages growth has allowed the share of national income going to profits to expand over the last 30 years in many nations.

The answer is False.

A declining rate of real wages growth is not even a necessary condition for a falling wage share (rising profit share).

Abstracting from the share of national income going to government, we can divide national income into the proportion going to workers (the “wage share”) and the proportion going to capital (the “profits share”). For the profit share to rise, the wage share has to fall (given the proportion going to government is relatively constant over time).

The wage share in nominal GDP is expressed as the total wage bill as a percentage of nominal GDP. Economists differentiate between nominal GDP ($GDP), which is total output produced at market prices and real GDP (GDP), which is the actual physical equivalent of the nominal GDP. We will come back to that distinction soon.

To compute the wage share we need to consider total labour costs in production and the flow of production ($GDP) each period.

Employment (L) is a stock and is measured in persons (averaged over some period like a month or a quarter or a year.

The wage bill is a flow and is the product of total employment (L) and the average wage (w) prevailing at any point in time. Stocks (L) become flows if it is multiplied by a flow variable (W). So the wage bill is the total labour costs in production per period.

So the wage bill = W.L

The wage share is just the total labour costs expressed as a proportion of $GDP – (W.L)/$GDP in nominal terms, usually expressed as a percentage. We can actually break this down further.

Labour productivity (LP) is the units of real GDP per person employed per period. Using the symbols already defined this can be written as:

LP = GDP/L

so it tells us what real output (GDP) each labour unit that is added to production produces on average.

We can also define another term that is regularly used in the media – the real wage – which is the purchasing power equivalent on the nominal wage that workers get paid each period. To compute the real wage we need to consider two variables: (a) the nominal wage (W) and the aggregate price level (P).

We might consider the aggregate price level to be measured by the consumer price index (CPI) although there are huge debates about that. But in a sense, this macroeconomic price level doesn’t exist but represents some abstract measure of the general movement in all prices in the economy.

Macroeconomics is hard to learn because it involves these abstract variables that are never observed – like the price level, like “the interest rate” etc. They are just stylisations of the general tendency of all the different prices and interest rates.

Now the nominal wage (W) – that is paid by employers to workers is determined in the labour market – by the contract of employment between the worker and the employer. The price level (P) is determined in the goods market – by the interaction of total supply of output and aggregate demand for that output although there are complex models of firm price setting that use cost-plus mark-up formulas with demand just determining volume sold. We shouldn’t get into those debates here.

The inflation rate is just the continuous growth in the price level (P). A once-off adjustment in the price level is not considered by economists to constitute inflation.

So the real wage (w) tells us what volume of real goods and services the nominal wage (W) will be able to command and is obviously influenced by the level of W and the price level. For a given W, the lower is P the greater the purchasing power of the nominal wage and so the higher is the real wage (w).

We write the real wage (w) as W/P. So if W = 10 and P = 1, then the real wage (w) = 10 meaning that the current wage will buy 10 units of real output. If P rose to 2 then w = 5, meaning the real wage was now cut by one-half.

So the proposition in the question – that nominal wages grow faster than inflation – tells us that the real wage is rising.

Nominal GDP ($GDP) can be written as P.GDP, where the P values the real physical output.

Now if you put of these concepts together you get an interesting framework. To help you follow the logic here are the terms developed and be careful not to confuse $GDP (nominal) with GDP (real):

  • Wage share = (W.L)/$GDP
  • Nominal GDP: $GDP = P.GDP
  • Labour productivity: LP = GDP/L
  • Real wage: w = W/P

By substituting the expression for Nominal GDP into the wage share measure we get:

Wage share = (W.L)/P.GDP

In this area of economics, we often look for alternative way to write this expression – it maintains the equivalence (that is, obeys all the rules of algebra) but presents the expression (in this case the wage share) in a different “view”.

So we can write as an equivalent:

Wage share – (W/P).(L/GDP)

Now if you note that (L/GDP) is the inverse (reciprocal) of the labour productivity term (GDP/L). We can use another rule of algebra (reversing the invert and multiply rule) to rewrite this expression again in a more interpretable fashion.

So an equivalent but more convenient measure of the wage share is:

Wage share = (W/P)/(GDP/L) – that is, the real wage (W/P) divided by labour productivity (GDP/L).

I won’t show this but I could also express this in growth terms such that if the growth in the real wage equals labour productivity growth the wage share is constant. The algebra is simple but we have done enough of that already.

That journey might have seemed difficult to non-economists (or those not well-versed in algebra) but it produces a very easy to understand formula for the wage share.

Two other points to note. The wage share is also equivalent to the real unit labour cost (RULC) measures that Treasuries and central banks use to describe trends in costs within the economy. Please read my blog – Saturday Quiz – May 15, 2010 – answers and discussion – for more discussion on this point.

Now it becomes obvious that if the nominal wage (W) grows faster than the price level (P) then the real wage is growing. But that doesn’t automatically lead to a growing wage share. So the blanket proposition stated in the question is false.

If the real wage is growing at the same rate as labour productivity, then both terms in the wage share ratio are equal and so the wage share is constant.

If the real wage is growing but labour productivity is growing faster, then the wage share will fall.

Only if the real wage is growing faster than labour productivity , will the wage share rise.

The wage share was constant for a long time during the Post Second World period and this constancy was so marked that Kaldor (the Cambridge economist) termed it one of the great “stylised” facts. So real wages grew in line with productivity growth which was the source of increasing living standards for workers.

The productivity growth provided the “room” in the distribution system for workers to enjoy a greater command over real production and thus higher living standards without threatening inflation.

Since the mid-1980s, the neo-liberal assault on workers’ rights (trade union attacks; deregulation; privatisation; persistently high unemployment) has seen this nexus between real wages and labour productivity growth broken. So while real wages have been stagnant or growing modestly, this growth has been dwarfed by labour productivity growth.

So the question is false because you have to consider labour productivity growth in addition to real wages growth before you can make conclusions about the movements in factor shares in national income.

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Question 3:

If the stock of aggregate demand growth outstrips the capacity of the productive sector to respond by producing extra real goods and services then inflation is inevitable.

The answer is False.

Spending definitely equals income and too much spending relative to the real capacity of the economy to absorb it will create inflation. But those facts do not relate to the point of the question, which is, in fact, a very easy test of the difference between flows and stocks.

All expenditure aggregates – such as government spending and investment spending are flows. They add up to total expenditure or aggregate demand which is also a flow rather than a stock. Aggregate demand (a flow) in any period and it jointly determines the flow of income and output in the same period (that is, GDP) (in partnership with aggregate supply).

So while flows can add to stock – for example, the flow of saving adds to wealth or the flow of investment adds to the stock of capital – flows can also be added together to form a “larger” flow.

For example, if you wanted to work out annual GDP from the quarterly national accounts you would sum the individual quarterly observations for the 12-month period of interest. Conversely, employment is a stock so if you wanted to create an annual employment time series you would average the individual quarterly observations for the 12-month period of interest.

The question thus tests the precision of language as they relate to economic concepts. Too often the language is loose and the concepts become confused as a result.

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Question 4:

The Australian dollar is currently appreciating strongly against many of the key currencies and this has put pressure on our international competitiveness. Export competitiveness will be restored under these conditions if local workers accept a cut in nominal wages and the rate of inflation is contained.

The answer is False.

This question also applies to the EMU nations who cannot adjust their nominal exchange rate but are seeking export-led demand boosts as they cut government spending.

The temptation is to accept the dominant theme that is emerging from the public debate – that the appreciating dollar requires tighter domestic wage and price conditions.

However, deflating an economy under these circumstance will not guarantee that a nation’s external competitiveness will be increased.

We have to differentiate several concepts: (a) the nominal exchange rate; (b) domestic price levels; (c) unit labour costs; and (d) the real or effective exchange rate.

It is the last of these concepts that determines the “competitiveness” of a nation. This Bank of Japan explanation of the real effective exchange rate is informative. Their English-language services are becoming better by the year.

Nominal exchange rate (e)

The nominal exchange rate (e) is the number of units of one currency that can be purchased with one unit of another currency. There are two ways in which we can quote a bi-lateral exchange rate. Consider the relationship between the $A and the $US.

  • The amount of Australian currency that is necessary to purchase one unit of the US currency ($US1) can be expressed. In this case, the $US is the (one unit) reference currency and the other currency is expressed in terms of how much of it is required to buy one unit of the reference currency. So $A1.60 = $US1 means that it takes $1.60 Australian to buy one $US.
  • Alternatively, e can be defined as the amount of US dollars that one unit of Australian currency will buy ($A1). In this case, the $A is the reference currency. So, in the example above, this is written as $US0.625= $A1. Thus if it takes $1.60 Australian to buy one $US, then 62.5 cents US buys one $A. (i) is just the inverse of (ii), and vice-versa.

So to understand exchange rate quotations you must know which is the reference currency. In the remaining I use the first convention so e is the amount of $A which is required to buy one unit of the foreign currency.

International competitiveness

Are Australian goods and services becoming more or less competitive with respect to goods and services produced overseas? To answer the question we need to know about:

  • movements in the exchange rate, ee; and
  • relative inflation rates (domestic and foreign).

Clearly within the EMU, the nominal exchange rate is fixed between nations so the changes in competitiveness all come down to the second source and here foreign means other nations within the EMU as well as nations beyond the EMU.

There are also non-price dimensions to competitiveness, including quality and reliability of supply, which are assumed to be constant.

We can define the ratio of domestic prices (P) to the rest of the world (Pw) as Pw/P.

For a nation running a flexible exchange rate, and domestic prices of goods, say in the USA and Australia remaining unchanged, a depreciation in Australia’s exchange means that our goods have become relatively cheaper than US goods. So our imports should fall and exports rise. An exchange rate appreciation has the opposite effect which is what is occurring at present.

But this option is not available to an EMU nation so the only way goods in say Greece can become cheaper relative to goods in say, Germany is for the relative price ratio (Pw/P) to change:

  • If Pw is rising faster than P, then Greek goods are becoming relatively cheaper within the EMU; and
  • If Pw is rising slower than P, then Greek goods are becoming relatively more expensive within the EMU.

The inverse of the relative price ratio, namely (P/Pw) measures the ratio of export prices to import prices and is known as the terms of trade.

The real exchange rate

Movements in the nominal exchange rate and the relative price level (Pw/P) need to be combined to tell us about movements in relative competitiveness. The real exchange rate captures the overall impact of these variables and is used to measure our competitiveness in international trade.

The real exchange rate (R) is defined as:

R = (e.Pw/P) (2)

where P is the domestic price level specified in $A, and Pw is the foreign price level specified in foreign currency units, say $US.

The real exchange rate is the ratio of prices of goods abroad measured in $A (ePw) to the $A prices of goods at home (P). So the real exchange rate, R adjusts the nominal exchange rate, e for the relative price levels.

For example, assume P = $A10 and Pw = $US8, and e = 1.60. In this case R = (8×1.6)/10 = 1.28. The $US8 translates into $A12.80 and the US produced goods are more expensive than those in Australia by a ratio of 1.28, ie 28%.

A rise in the real exchange rate can occur if:

  • the nominal e depreciates; and/or
  • Pw rises more than P, other things equal.

A rise in the real exchange rate should increase our exports and reduce our imports.

A fall in the real exchange rate can occur if:

  • the nominal e appreciates; and/or
  • Pw rises less than P, other things equal.

A fall in the real exchange rate should reduce our exports and increase our imports.

In the case of the EMU nation we have to consider what factors will drive Pw/P up and increase the competitive of a particular nation.

If prices are set on unit labour costs, then the way to decrease the price level relative to the rest of the world is to reduce unit labour costs faster than everywhere else.

Unit labour costs are defined as cost per unit of output and are thus ratios of wage (and other costs) to output. If labour costs are dominant (we can ignore other costs for the moment) so total labour costs are the wage rate times total employment = w.L. Real output is Y.

So unit labour costs (ULC) = w.L/Y.

L/Y is the inverse of labour productivity(LP) so ULCs can be expressed as the w/(Y/L) = w/LP.

So if the rate of growth in wages is faster than labour productivity growth then ULCs rise and vice-versa. So one way of cutting ULCs is to cut wage levels which is what the austerity programs in the EMU nations (Ireland, Greece, Portugal etc) are attempting to do.

But LP is not constant. If morale falls, sabotage rises, absenteeism rises and overall investment falls in reaction to the extended period of recession and wage cuts then productivity is likely to fall as well. Thus there is no guarantee that ULCs will fall by any significant amount.

Further, the reduction in nominal wage levels threatens the contractual viability of workers (with mortgages etc). It is likely that the cuts in wages would have to be so severe that widespread mortgage defaults etc would result. The instability that this would lead to makes the final outcome uncertain.

Given all these qualifications, the answer is false. External competitive will not be automatically restored by deflating nominal wages and price levels.

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Premium Question 5:

The US Federal Reserve this week put out its updated projections which provide “central tendency” estimates of real GDP growth between 3.5 and 4.3 per cent in 2013. Their lowest estimate for 2013 was 3 per cent per annum. Assuming the current labour productivity growth continues (around 2 per cent per annum) and the labour force growth resumes more normal rates (around 1.4 per cent per annum) by 2013 and the average working week is constant in hours, then one consequence of the difference between the lower bound of the central tendency projections and their lowest estimate will be that the unemployment rate will fall more slowly in 2013 if the the latter projection is true.

The answer is False.

The facts were:

  • Real GDP growth projection 3.5 per cent compared to 3 per cent in 2013.
  • Labour productivity growth (that is, growth in real output per person employed) growing at 2 per cent per annum. So as this grows less employment in required per unit of output.
  • The labour force is growing by 1.4 per cent per annum. Growth in the labour force adds to the employment that has to be generated for unemployment to stay constant (or fall).
  • The average working week is constant in hours. So firms are not making hours adjustments up or down with their existing workforce. Hours adjustments alter the relationship between real GDP growth and persons employed.

We need a method of relating the projections of real GDP growth into labour market outcomes. The late Arthur Okun is famous (among other things) for estimating the relationship that links the percentage deviation in real GDP growth from potential to the percentage change in the unemployment rate – the so-called Okun’s Law.

The algebra underlying this law can be manipulated to estimate the evolution of the unemployment rate based on real output forecasts.

From Okun, we can relate the major output and labour-force aggregates to form expectations about changes in the aggregate unemployment rate based on output growth rates. A series of accounting identities underpins Okun’s Law and helps us, in part, to understand why unemployment rates have risen.

Take the following output accounting statement:

(1) Y = LP*(1-UR)LH

where Y is real GDP, LP is labour productivity in persons (that is, real output per unit of labour), H is the average number of hours worked per period, UR is the aggregate unemployment rate, and L is the labour-force. So (1-UR) is the employment rate, by definition.

Equation (1) just tells us the obvious – that total output produced in a period is equal to total labour input [(1-UR)LH] times the amount of output each unit of labour input produces (LP).

Using some simple calculus you can convert Equation (1) into an approximate dynamic equation expressing percentage growth rates, which in turn, provides a simple benchmark to estimate, for given labour-force and labour productivity growth rates, the increase in output required to achieve a desired unemployment rate.

Accordingly, with small letters indicating percentage growth rates and assuming that the average number of hours worked per period is more or less constant, we get:

(2) y = lp + (1 – ur) + lf

Re-arranging Equation (2) to express it in a way that allows us to achieve our aim (re-arranging just means taking and adding things to both sides of the equation):

(3) ur = 1 + lp + lf – y

Equation (3) provides the approximate rule of thumb – if the unemployment rate is to remain constant, the rate of real output growth must equal the rate of growth in the labour-force plus the growth rate in labour productivity.

It is an approximate relationship because cyclical movements in labour productivity (changes in hoarding) and the labour-force participation rates can modify the relationships in the short-run. But it provides reasonable estimates of what happens when real output changes.

The sum of labour force and productivity growth rates is referred to as the required real GDP growth rate – required to keep the unemployment rate constant.

Remember that labour productivity growth (real GDP per person employed) reduces the need for labour for a given real GDP growth rate while labour force growth adds workers that have to be accommodated for by the real GDP growth (for a given productivity growth rate).

So in the example, the required real GDP growth rate is 3.4 per cent per annum.

If the lowest bound of the central tendency estimates (3.5 per cent in 2013) are correct then the national unemployment rate (under our assumptions) would decline by a miserly 0.1 points in 2013.

If the lowest real GDP growth estimate overall (3 per cent in 2013) is correct then the rate of real GDP growth is below the required rate to keep the unemployment rate constant and the unemployment rate would rise by 0.4 per cent in 2013.

The rising unemployment rate would reflect the fact that real output growth was not strong enough to both absorb the new entrants into the labour market and offset the employment losses arising from labour productivity growth.

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    This Post Has 12 Comments
    1. Q1, “National accounting shows us that a government surplus equals a non-government deficit.”

      Technically true, but not specific enough.

      trade deficit equals gov’t deficit plus private deficit

      Better yet …

      savings of the rich equals dissavings of the gov’t plus dissavings of the lower and middle class

      Even better yet …

      savings of the rich plus savings of the lower and middle class equals dissavings of the currency printing entity with currency that has no bond attached plus the balanced budget(s) of the various levels of gov’t

      Also, “Y = GDP = C + I + G + (X – M)”

      Can there be imbalances that develop if the C, I, G, and/or (X-M) are coming from a time period outside of that measured by Y?

    2. I’d rather see Q3 like this:

      3. If the STOCK of AGGREGATE DEMAND GROWTH outstrips the capacity of the productive sector to respond by producing extra REAL goods and services then PRICE INFLATION is inevitable.

    3. Q2, “The relentless push by neo-liberals to cut real wages growth has allowed the share of national income going to profits to expand over the last 30 years in many nations.”

      Let’s add on to that. Assume over the last 30 years in many nations (mainly the high wage ones) that real wages have been flat or negative, productivity has been at least a positive 2%, and national income (GDP) has been growing most of the time. What has been going on?

    4. According to the rules of logic, the correct answer to question 3 is True because False implies whatever is True.

    5. “According to the rules of logic, the correct answer to question 3 is True because False implies whatever is True.”

      This is true of material implication but not formal implication (entailment). Most ordinary language arguments based on conditionals presume entailment.

    6. Bill –

      Question 2:

      The relentless push by neo-liberals to cut real wages growth has allowed the share of national income going to profits to expand over the last 30 years in many nations.

      Your solution seems to be strawmanning. The question did not ask whether it had caused the share of national income going to profits to expand over the last 30 years in many nations.

    7. @Tom, not sure what you mean, but a quick check on entailment with wikipedia to brush up some concepts shows that if S1 is inconsistent then S1 entails whatever.

      @Aidan, agreed. Yet another example of getting the “right” answer depends on how you interpret the question. Since in this case it refers to the past, and we know that in many countries cuts on real-wages has allowed productivity growth to outstrip real wages growth, the answer should be True.

    8. @MamMoTH,

      Wikipedia is not quite correct, based on your summary. In standard extensional (truth-functional) logic with the material conditional, the conditional is defined in such a way that A implies B is true whenever A is false. All standard extensional systems are like this. However, in a non-standard logical system, particularly with one with relevance conditions attached, this property is undesirable. Unfortunately, the term ‘entailment’ is loosely used by many, but the locus classicus for a logical system of entailment is Anderson and Belnap’s Entailment. There are also logical systems in which contradictions, under certain conditions, are acceptable, but these are non-standard, too.

      It seems to me that, in stating that ‘if S1 is inconsistent, then S1 implies whatever’, you are implicitly referring to Goedel’s incompleteness theorem. If so, then strictly speaking, the term ‘entailment’ should be replaced by ‘implies’ in the sense of the material conditional, particularly since entailment is distinct from (material) implication in those cases where you need to make the distinction. Goedel’s theorem is based on the system set out in Principia Mathematica and extensions of this system, which is pretty much all of standard mathematics. The theorem basically says that in any formalized system, it is possible to say more than you can prove utilizing the resources of the system itself, if the system is consistent. In other words, it can not prove all that it can say. That is, the system is incomplete.

      On this basis, I can agree with you and Tom Hickey, since properly interpreted, you are both saying much the same thing. It may be that Bill does not have the material conditional in mind. But any standard mathematization of economics will be based on the extensional predicate calculus which relies on the material conditional. Keynes was aware of this problem and it was one of the reasons that he felt that the mathematical tools of his time were inappropriate for formalizing an economic theory that included the ‘psychological states’ of the individual (such as animal spirits). This is still, pretty much, the case today. Except that we now have game theory, which is an advance on the mathematical tools available to Keynes – the minimax theorem that von Neumann had already proved a few years before the publication of The General Theory would not have been of much use to Keynes.

    9. Addendum:

      @MamMoTH,

      If the conditional in 3, A implies B, is false, this can only be because A is true and B is false, assuming the material conditional is in play. This is because with the material conditional, ‘F implies F’ is true. This situation is usually not the case in a system of entailment. The justification for allowing ‘F implies F’ to be true is that from a falsehood anything follows, the true and the false. If you begin with falsehoods, in an extensional calculus, you will no longer be able to distinguish between truths and falsehoods. Hence, it ought to be true that, since anything can follow from a falsehood, what follows can be either true or false. The rationale for a deductive system is that if you begin with truths, if you do not violate any rules of inference, you will end up only with truths. The material conditional satisfies this requirement. But in order for this to work, there can be no specification of a link between A and B in A implies B. This also means that the material conditional can be defined in terms of negation and disjunction – A implies B iff not-A or B. That is, A implies B just in case A is false or B is true. This doesn’t make much sense in the situation in which Bill is engaged.

      I take it that an implied link between A and B in Q 3 indicates that Bill is not intending that ‘implies’ stands for the implication of the material conditional, but rather some sort of relevance logic, say R. Unlike the standard extensional predicate calculus, there is more than one system, R. Until that has been specified, one must rely on the logical principles found in natural language and, hence, on some kind of informal reasoning with some math tacked on as an adjunct to the argument. There is nothing wrong in doing this, except that when this takes place in mathematics, what is not spelled out is clearly understood. This is not yet the case for logics of relevance and entailment.

    10. “It may be that Bill does not have the material conditional in mind.”

      I suspect he doesn’t. In ordinary language “if p then q” is true only if both “p” and “q” are semantically (descriptively) true.

      In modus ponens, if p implies q, and p, then q . In modus tollens, if not-q, then not-p, and not-q, then not-p, q is a necessary condition for p, but not a sufficient one.

      This is how modus ponens and modus tolens arguments are framed in ordinary discourse, which is semantically interpreted. For the argument to be sound in producing a semantically interpreted consequent, both p and q must be descriptive (factual) propositions. In modus ponens, if p is a tautology (syntactically true), anything follows from it because p is necessarily true, and similarly in modus tollens if q is a contradiction, hence, necessarily false, so this must be ruled out.

      For this reason, in ordinary language p and q are generally assumed to be semantic expressions (descriptive propositions) rather than syntactical expressions (symbols and operators). In ordinary language arguments, the concern is generally with descriptive propositions that assert or deny states of affairs, either particular or general. The truth and falsity of assertions and denials of states of affairs are tested against facts, ultimately through observation.

      Macro is about aggregates, i.e., general descriptions that can be traced to factual data. The meaning of macro expressions is grounded in facts summarized in data. Inference in macro is semantic. You can’t just make stuff up, at least not without risk of getting called out on it. Even when macro is uses syntactical identities like Y = C + I + G + (X-M), the identity depends on the semantic import of the terms. The identity follows from the semantic import, i.e., understanding what granular data is aggregated and the various relationships involved. For example, in order to understand C or I, you have to know what kind of data these symbols refer to.

      This is significant because try to dismiss as MMT as Just about accounting identities as tautologies that add nothing factual. The logic is syntactical, being based on accounting identities (tautologies), but the expression is semantically interpreted as a macro theory as a model. That is to say, 2 + 2 = 4 is logically true because it is derived from the axioms of arithmetic, but Y = C + I + G + (X-M) is true because of the meaning of the symbols with reference to data. It is an accounting identity because that is how data, e.g., transactions, get recorded. It is necessary to know what an economy is and how the kinds of expenditure relate to it. This requirement does not apply to purely logical or mathematical expressions, which are knowable a priori.

    11. @ jake, Tom: very interesting points. Not sure what is to be concluded but I noticed that you both refer to material implication and entailment. When I quickly looked for entailment in the wikipedia, I understood that if S1 is inconsistent then S1 entails S2. Did I get it wrong or do you have something else in mind when you talk about entailment?

    12. The terms “implication” and “entailment” do not have absolute meaning. This is part of the point I was making. Meaning is context-dependent. The ordinary language use and the technical use are different. Ordinary language is rather ambiguous, and one has to pick up from the context what meaning is intended. Generally speaking, in ordinary language the meaning is as I explained above.

      Moreover, the terms like “implication” and “entailment” are not used entirely consistently across technical discourse, which is a reason that mathematical and logical symbols defined in terms of other symbols, are used instead of words. For example, operators are defined by truth tables.

      Logical Operations and Truth Tables

      Arguments

      In ordinary language the the truth table is:

      p q p>q
      T T T
      T F F
      F T F
      F F F

      In formal languages it is:

      S1 S2 S1>S2
      T T T
      T F F
      F T T
      F F T

      where “P” and “q” are descriptive propositions, and S1 and S2 are either tautologies or contradictions.

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