I am now using Friday’s blog space to provide draft versions of the Modern Monetary Theory textbook that I am writing with my colleague and friend Randy Wray. We expect to publish the text sometime around mid-2014. Our (very incomplete) textbook home page – Modern Monetary Theory and Practice – has draft chapters and contents etc in varying states of completion. Comments are always welcome. Note also that the text I post here is not intended to be a blog-style narrative but constitutes the drafting work I am doing – that is, the material posted will not represent the complete text. Further it will change as the drafting process evolves.

Previous blogs in this series:

**Chapter X Growth and Inequality**

[CONTINUING – NOTE I REARRANGED THE PEDAGOGY A LITTLE IN THIS SECTION FROM THAT PRESENTED LAST WEEK – I HAD ACCESS TO HARROD’S ORIGINAL BOOK TODAY WHEREAS LAST WEEK IT WAS FROM MEMORY]

**X.2 Keynesian growth theories – Harrod-Domar**

An early growth theory came from Roy Harrod in 1939 when he attempted to provide a long-run reconstruction of Keynes’ *General Theory*. Eight years later, Evsey Domar introduced a model of long-run growth, which added to the approach taken by Harrod. It is conventional to refer to this work as the Harrod-Domar growth model.

Up until now we have only been considering the concept of short-run or static equilibrium. While, the Harrod-Domar approach to economic growth remains faithful to the fundamental principle of macroeconomics that the short-run analysis is based upon, their important contribution was the construction of the problem of economic growth in terms of the **dual** nature of investment expenditure.

We have seen that for an equilibrium to be maintained, the income that is distributed to productive inputs via production must flow back into the expenditure stream in some form (consumption, investment, net government spending, net exports). Otherwise, the aggregate demand expectations of firms that they used to determine their level of output and income generation would not be realised and the level of output and income would change.

A basic insight of the Harrod-Domar approach to economic dynamics is that this **static** concept of equilibrium is transitory because there is a **dual** characteristic of investment spending. While investment adds to aggregate demand in the current period it also adds to the productive capacity of the economy. To fully utilise the growing productive capacity the economy must also experience appropriate aggregate demand growth.

This means that the income level that might be consistent with full employment of capital and labour in the current period will be deficient over time as the productive capacity grows via investment expenditure

Thus, aggregate demand has to grow in the next period to ensure the extra capital is fully utilised. The expenditure side of the economy can be seen as always chasing the growth in capacity that it creates.

We expressed this idea earlier in the textbook in terms of the requirement that the leakages from the income-expenditure system in each period (saving, taxation and imports) have to be exactly offset by spending injections in the same period (from investment, government and export spending) for the current level of income to be stable.

The dual nature of investment raises the possibility of crises occurring where capital and labour resources lie idle as a result of aggregate demand failing to keep pace of the growth in productive capacity.

Harrod and Domar addressed the implications of the duality of investment in their work.

There are three components to their model:

1. The warranted rate of growth, which is the rate of growth that maintains full utilisation of capital.

2. The natural rate of growth, which is the rate of growth that ensures that all workers that desire work can find employment. Equilibrium occurs when the warranted rate of growth equals the natural rate of growth.

3. The razor’s edge or instability of growth, where the two rates of growth which define a full employment equilibrium are prone to deviate and cause crisis.

First, we need to consider how much productive capacity (the capital stock, K) increases as a result of investment expenditure in the current period. For simplicity we assume that there is no attrition of capital so that depreciation &delta is zero.

The simple Keynesian model of income determination suffices to develop the concepts.

(X.1) Y_{t} = C_{t} + I_{t}

where Y_{t} is total output/income in period t, C_{t} is total consumption in period t, and I_{t} is total investment in period t.

We are assuming a closed economy with no government spending or taxation to be faithful to the original work.

Income can be either consumed or saved and S_{t} = sY_{t}, where s is the marginal propensity to save (which is 1 – the marginal propensity to consume).

Thus:

(X.2) Y_{t} = C_{t} + I_{t} = C_{t} + S_{t}

In equilibrium:

(X.3) S_{t} = I_{t}

We will assume that this equilibrium is consistent with a full employment level of income (and hence the level of saving). We note that it is not an automatic tendency of the monetary system to ensure there is full employment. The level of income has to be managed through policy interventions to ensure there is full employment.

At the current full employment income flow equilibrium, there will be a particular capital stock K in place. We can thus define a full employment capital-output ratio, K/Y. The reciprocal (Y/K) is an indicator of the productivity of capital.

The ratio K/Y is measured in dollar terms and tells us what monetary value of capital needs to be in place to produce a given level of national income. A ratio of 4 would tell us that to produce $1000 worth of output, the economy would need to have a capital stock of $4000.

We can define the **marginal** capital-output ratio, which indicates how much extra income will be produced, ΔY for a given change in the capital stock, ΔK. So if ΔK/ΔY is equal to 4, then and for every dollar increase in the capital stock, total potential income rises by $0.25.

For simplicity we will retain the Harrod-Domar assumption that K/Y is equal to ΔK/ΔY. There is an extensive literature on this issue which we ignore in this treatment. The relationship between the two ratios indicates the nature of technical progress. An equality assumption basically means we are assuming that technical progress is neutral. A discussion of these issues is beyond the scope of this textbook.

Capital is accumulated over time according to the following expression (assuming depreciation of existing capital is zero):

(X.5) K_{t+1} = I_{t} + K_{t}

where K is the capital stock, and t indicates the period.

Equation (X.5) tells us that the capital stock increases as a result of the flow of investment in the current period.

This can be re-written as (suppressing the time subscripts):

(X.6) ΔK = I

To see how much the productive capacity of the economy (in dollars) increases as a result of investment, we define the inverse of the marginal capital-output ratio as:

(X.7) θ = ΔY/ΔK

Domar denoted θ the potential social average productivity of investment.

Using Equation (X.6), we can re-arrange Equation (X.7) as:

(X.8) ΔY_{p} = θI

This tells us what the change in potential output (ΔY_{p}) or productive capacity will for a given technology embodied in the marginal capital-output ratio, when net investment expenditure increases by $1. Our assumption of zero depreciation means that gross and net investment are equal.

This expression thus captures one of the dual characteristics of investment expenditure – the contribution to future productive capacity or aggregate supply potential.

Consider the following example. Assume that θ = 0.25 (that is, the ΔK/ΔY = 4) and the marginal propensity to consume is 0.8 (which means the marginal propensity to save is 0.2).

If the initial equilibrium national income is Y = C + I and C = 2000 and I = 100, which means that aggregate demand of $2,100 leads to national output or income of $2,100.

We can see that ΔY_{p} = 0.25 x 100 = $25. So at the end of the first period, potential output has increased by $25 to $2,125.

You can see that while aggregate demand of $2,100 was sufficient to generate full employment output in period 1, the required expenditure in period 2 to ensure full employment is retained (potential output is exhausted) has risen to $2,125 as a result of the investment in the first period adding to the productive capacity of the economy.

Harrod and Domar thus exposed the growth imperative if full employment is to sustained over time. While we should be concerned with the environmental impacts of growth, we cannot escape the conclusion that with aggregate spending growth, the economy will depart from full employment once it is attained and the misery of unemployment would become the norm.

Harrod and Domar then considered the other side of the investment duality which focuses on the implications of the supply-side insights for aggregate demand growth.

[NEXT WEEK I WILL TIE THIS BACK INTO THE WARRANTED AND NATURAL RATE OF GROWTH AND THE ANALYSIS OF INSTABILITY]

**References**

Domar, E. (1947) ‘Expansion and Employment’, *American Economic Review*, 37, 34-55.

Harrod, R. (1939) ‘An Essay in Dynamic Theory’, *Economic Journal*, 49, 14-33.

**Conclusion**

CONTINUING NEXT FRIDAY.

**Saturday Quiz**

The Saturday Quiz will be back again tomorrow. It will be of an appropriate order of difficulty (-:

That is enough for today!

(c) Copyright 2013 Bill Mitchell. All Rights Reserved.

Professor;

OT a little ….

I can’t quantify the amount of knowledge about macro economics that you have imparted to me. Thank you for that and your obvious dedication and astounding productivity to maintaining your blog.

There has been nothing that I have read in your blogs that is not empirically based. Your data, conclusions and summaries are beyond dispute. Yes, I understand that the Professor has a particular socioeconomic, political leaning etal yet that matters not.

Please keep up the cause Professor.