An SFC Version of the Diamond Growth Model

A number of my posts have attempted to compare and draw parallels between very different approaches in economics.  It is an exercise I find very interesting and informative.  In this vein, I have recently gone about constructing a Stock-Flow Consistent (SFC) version of the Diamond growth model set out in the paper National Debt in a Neoclassical Growth Model.

The first thing to make clear is that the stocks and flows in Diamond's model are perfectly consistent, so I don't mean that I am somehow correcting accounting errors.  Rather, what I have attempted to do is to restructure the model into the format typically used in post-Keynesian stock-flow models.  This looks like the models that appear in Godley & Lavoie's Monetary Economics.  With appropriate choice of parameter values, however, it is structurally equivalent to Diamond's neoclassical model and produces the same results.

The purpose of this exercise was to focus on the behavioural assumptions of the model and see where post-Keynesian and neoclassical views might diverge.  Doing so requires "padding out" the original model a bit.  Most notably, Diamond's model involves only real values - there is a real wage level, but no separate price and nominal wage levels.  This doesn't fit well with a post-Keynesian worldview, so a major innovation is to turn this into a monetary model with nominal values, as well as real. 

As it turns out, this is quite a useful exercise.  One of Diamond's main themes is the impact of national debt, and including prices and inflation allow us to examine some dynamics in a richer way than what is covered in that paper.  This was also part of my purpose in undertaking this exercise and I intend to devote a separate post to examining the role of national debt within this model and saying something about Diamond's conclusions.

As usual I have relegated the equations and variable listing to the end of the post, but the schematic below illustrates the structure of this model.  The policy tools are shown by the dotted boxes and the arrows show the direction of causation.  The dotted arrows show causation through expected future values.

There are two assets in this model: government bonds and physical capital.  Firms hold capital and households own the firms.  I have assumed that households fund firms' capital investment by making capital contributions.  Households hold bonds directly.  I have not include banks and bank money.  I could easily do so but it would add complexity, and it would only make a difference if I wanted to start making additional assumptions about how banks behave.  If you worry about the lack of money, imagine there's a bank that holds all the government bonds, funded by interest bearing deposits held by households.

The table below then shows the flow of funds matrix for the model.

	Households	Firms	Government
Consumer spending	-C	C
Investment expenditure		-I, I
Government spending		G	-G
Wage bill	WB	-WB
Operating surplus	OS	-OS
Taxes	-T		T
Bond interest	r.B(-1)		r.B(-1)
Capital contribution	-ΔK	ΔK
Bond issuance	-ΔB		ΔB

Although the causal structure illustrated in the schematic is essentially post-Keynesian, the model can still produce the same results as those of Diamond's neoclassical model.  However, this does depend on what we assume about certain parameters.  There are also a few particular points about the way the model is drawn up.  I want to look at these now.

Expectations

Where expected future values are used in SFC models, these are usually based on some form of adaptive expectation mechanism.  The Diamond model requires forward looking expectations, i.e. expected values have to be equal to the actual outcome.  This makes the solving process for an SFC model rather more complicated, but it is perfectly doable (see note on Solution Technique below)

Adjustment Periods

SFC models will typically involve a number of functions where values adjust incrementally towards target values.  I have included two here: the capital stock and nominal wages.  However, to conform with Diamond's model, we have to assume that the lag on adjustment is zero, i.e. these values adjust fully in the current period.  Because this applies to wages, it implies that there is no wage or price stickiness.  This must be the case, of course, in Diamond's model as nominal values do not figure.

As Diamond's time periods each represent half a lifetime, there is something to said for allowing full adjustment.

Production Functions

Diamond uses a Cobb-Douglas production function.  In my version, if the parameters are chosen appropriately, this is implicit in the labour demand function.  This function allows employment to follow from the level of aggregate demand and also provides that greater capital investment improves labour productivity.  Whilst this is generally OK, I think post-Keynesians would (rightly) be suspicious of this type of relationship in a model.  Whilst it is nice to be able to tie output, employment and capital investment together, I think that use of artificial production functions can throw up spurious results, which may then be taken for granted.

Pricing in this model is based on a mark-up to unit labour costs.  Diamond's version uses marginal cost pricing.  With Cobb-Douglas production and suitable choice of parameter, these are equivalent.

Consumption Coefficients

Consumption in this model is based on labour income, investment income and wealth.  Diamond's model is an overlapping generations model with two period lives, where generations work in the first period only and spend their savings in the second.  To conform with his model, the parameters for investment income and wealth therefore need to equal one.  A high parameter on wealth is not unreasonable given the length of the time period and this would reduce if we were to adapt the model to three period generations, for example.  However, many post-Keynesians would not like having a higher parameter on investment income than on labour income, pointing to the importance of recognising household heterogeneity beyond generational differences.

Taxation

As is common in neo-classical models, Diamond assumes that taxes are lump sum.  Usually in SFC models, the policy tool is the rate of tax with the amount of tax determined endogenously.  This is much better, in my view, but I have used lump sum taxes here to conform with Diamond.  It actually makes little qualitative difference to the results.

Risk Premium

Diamond assumes that, in equilibrium, the same return applies on physical capital as on government bonds.  Typically SFC models would look to include some form of risk premium on the former.  Again, I don't think it qualitatively impacts much on the results.

Labour Supply

Diamond assumes labour is supplied inelastically, i.e. the amount people wish to work is fixed. I have allowed employment to vary, but used employment levels to affect the real wage target, as is conventional in SFC models.  In fact, as we allow the elasticity of the real wage target with respect to employment to tend to infinity, we approach the same position as in Diamond's model.  I could rearrange my model to achieve this exactly, but to keep it closer to the normal SFC structure, I have simply used a very high parameter, i.e. made target real wages very sensitive to employment variation.

Technical details follow.  I will do a post (probably my next), looking at some results of this model and showing how it gives some additional perspective on Diamond's conclusions on the impact of national debt.

Equation Listing

GDP is the sum of consumer expenditure, private investment and government expenditure.

(1)          y_t = c_t + i_t + g_t

Real consumer expenditure is based on real values of  income and carried forward wealth.  Different coefficients are applied to post-tax wage income and to investment income.

(2)          c_t  = α₁ ( WB_t / p_t - t_t ) + α₂ ( OS_t + r_t B_t-1 ) / p_t +  α₃ ( k_t-1 + B_t-1 / p_t)                  0 <= α₁, α₂, α₃ <= 1

Investment is some proportion of the difference between the target capital stock and the existing capital stock,

(3)          i_t = ε₁ ( kt_t - k_t-1)                                                                                                         0<=ε₁<=1

where the target level of the capital stock is based on expected future demand levels and the expected real rate of interest.

(4)          ln( kt_t ) = β₁ + β₂ ln[ E( y_t+1 ) ] + β₃ ln[ E( rr_t+1 ) ]                                                      β₂ > 0 ; β₃ < 0

The wage bill is the product of the wage rate and employment.

(5)          WB_t = w_t e_t

Operating surplus (profit) is nominal GDP less the wage bill.

(6)          OS_t = y_t p_t - WB_t

Employment is a (positive) function of output and a (negative) function of the capital stock.

(7)          ln( e_t ) = ψ₁ + ψ₂ ln( y_t ) + ψ₃ ln( k_t-1 )                                                                         ψ₂ > 0 ; ψ₃ < 0

Nominal wages adjust incrementally towards the level that would achieve the target real wage, given the price level.

(8)          w_t = ε₂ wrt_t p_t  + ( 1- ε₂ ) w_t-1                                                                                                              0<=ε₂<=1

where the target real wage is a function of the level of employment. 

(9)          ln( wrt_t ) = φ₁ + φ₂ [ ln( e_t ) - ln( n_t ) ]                                                                                              φ₂ > 0

Prices are a fixed mark-up to unit labour costs.

(10)        p_t = λ w_t n_t / y_t                                                                                                                      λ>1

The expected real rate of interest is based on the nominal rate on government bonds and the expected price level for the next period.

(11)        E( rr_t+1 ) = ( 1 + r_t+1 ) p_t / E( p_t+1 ) - 1

Finally, two equations govern the development of the stock of government bonds and the stock of capital.

(12)        B_t = (1 + r_t ) B_t-1 + g_t p_t - t_t p_t

(13)        k_t = k_t-1 + i_t                                                                                                                          

Variables

Real quantities are represented by lowercase letters and nominal quantities by uppercase letters.  Subscripts denote the time period.  The values for stocks (capital and bonds) are measured at the end of the period denoted by the subscript.  Interest rates are the rate paid in the period denoted by the subscript.  E(  ) denotes an expected value.

Name	Description
c	Consumer spending
e	Employment
g	Real government expenditure
i	Investment
k	Capital stock
kt	Target capital stock
n	Workforce
p	General price level
r	Nominal interest rate on government debt
rr	Real rate of return on government debt
t	Taxes
w	Nominal wage level
wrt	Target real wage
B	Government bonds
OS	Operating surplus
WB	Wage bill

Exogenous variables are the policy tools g, t and r and the workforce n.  In general n is assumed to grow at a fixed rate and g and t are assumed to grow at the same rate as n, apart from any shocks.

Solution Technique

The first step in solution was to determine per capita steady state values.  A simulation was then run over a long period by solving each period sequentially, using the steady state values to form expectations.  Subsequent simulations were then run using values from the previous simulation to derive expected values, until the difference between expectations and actual outcomes was negligible.

Pinning down the price level expectation can be quite tricky.  The technique used was to rely on the government budget constraint, using a form of Fiscal Theory of the Price Level.  Values for future real rates of return and government surpluses were taken from the previous simulation and compared with the nominal bond stock from the current period to determine an implied price level for the next period.  This technique proved very effective in facilitating rapid convergence.

[EDIT - equation 12 modified for typo spotted by Anton.]