UU Economics First Derivation Worksheet

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ECON 5080/6080 – Midterm Exam
Questions
The circuit of capital model uses C(t) to represent the flow of capital outlays at time t, P (t) to represent the flow of value of finished product (valued
at cost), S(t) to represent the flow of sales, N (t) to represent the stock of productive capital, X(t) to represent the stock of commercial capital (inventories
of finished goods awaiting sale valued at cost), F (t) to represent the stock of
financial capital, p to represent the capitalization rate, q the markup on costs,
TP the time delay in production, TR the time delay in selling the commodities,
and TF the time delay in reinvesting money capital.
The simplest case to analyze is when p = 0, that is, when no surplus value
is accumulated and the capitalists consume their whole income. Marx calls this
case simple reproduction.
P (t) = C(t ? Tp )
S(t) = [1 + q]P (t ? TR )
S 0 (t) = P (t ? TR ) = S(t)/[1 + q]
S 00 (t) = qP (t ? TR ) = qC(t)
0
C(t) = S (t ? TF ) = P (t ? [TR + TF ]) = C(t ? [TP + TR + TF ])
C(t) = P (t) = S(t)/[1 + q]
Question 1. When p = 0, in a simple reproduction model, above equations
are obtained from the circuit of capital model. What does above equations tell
us?
The profit rate r is r = S 00 (t)/[F (t) + N (t) + X(t)] We can also calculate
= [TF +TqP +TR ]
r = C(t)[TqC(t)
F +TP +TR ]
Question 2. What is the significance of the above formula? What is S 00 (t),
how is it different than S 0 (t)?
In Expanded Reproduction model we found the growth rate, g, as
g = ln(1 + pq)/(TF + TR + TP ) (5.25)
Question 3. According to equation (5.25) what determines the growth rate
in the system? What effect do p, q, TF , TR and TP have on the growth rate,
1
respectively (assume p > 0 and q > 0)?
While discussing proportionality and aggregate demand in simple reproduction we found SI (t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
Question 4. What does the above equation say? Carefully explain the
meanings of SI (t) , kI , C)I(t), kII and CII .
The social capital must be divided to allow for simple reproduction:
CI
CII
=
1?kII
qI +kI
Question 5. What is the significance of the above equation? Carefully
explain the meanings of CI , CII , kII , qI and kI . What happens if the equality
does not hold?
We can write the aggregate money demand for commodities in a simplified
expanded reproduction as:
D(t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
+kI CI (t) + kII CII (t)
+[1 ? pI ]SI00 (t ? TF ) + [1 ? pII ]SII (t ? TF ) (5.62)
Question 6. What do the first, second and third line represent?
Question 7. What is the relationship between Rosa Luxemburg’s argument
of imperialism and aggregate demand in expanded reproduction? What points
Duncan Foley made about it? What is Marx’s proposed solution to the demand
problem? What would be a modern-day adaption of this solution? What role
could the state play in solving this problem?
2
ECON 5080/6080 – Midterm Exam
Study Guide
The circuit of capital model uses C(t) to represent the flow of capital outlays at time t, P (t) to represent the flow of value of finished product (valued
at cost), S(t) to represent the flow of sales, N (t) to represent the stock of productive capital, X(t) to represent the stock of commercial capital (inventories
of finished goods awaiting sale valued at cost), F (t) to represent the stock of
financial capital, p to represent the capitalization rate, q the markup on costs,
TP the time delay in production, TR the time delay in selling the commodities,
and TF the time delay in reinvesting money capital.
The flow of value of finished product at time t, P (t), must be equal to the
flow of capital outlays TP periods earlier. We have: P (t) = C(t ? TP )
In a similar fashion, the flow of sales at time t corresponds to the flow of
finished product TR periods earlier, given the assumption that there is a fixed
time lag in sales. Of course, the value of sales is larger than the value of finished
product at cost because commodities are sold at prices that include surplus
value or, to put it another way, because commodities are sold at prices that are
marked up over costs: S(t) = [1 + q]P (t ? TR )
In a simple model that does not involve borrowing, new capital outlays
must be financed from past sales. If we write S 0 (t) for the part of sales that
represents the recovery of the costs of production, and S 00 (t) for the part of sales
that represents the realization of surplus value, we have:
S 0 (t) = P (t ? TR ) = [1/(1 + q)]S(t) =
S 00 (t) = qP (t ? TR ) = [q/(1 + q)]S(t) =
C(t) = S 0 (t ? TF ) + pS 00 (t ? TF )
S(t)
1+q
qS(t)
1+q
because only the fraction p of the surplus value is assumed to be recommitted to the circuit of capital. Productive capital is increased by capital outlays to
purchase means of production and labor-power and is decreased when finished
product emerges from the production process: dN (t)/dt = C(t) ? P (t)
In a similar fashion we can write down the laws governing the evolution of
the stocks of commercial capital and money capital.
1
dX(t)/dt = P (t) ? S(t)/[1 + q] = P (t) ? S 0 (t)
dF (t)/dt = S(t) ? [1 ? p]S 00 (t) ? C(t) = S 0 (t) + pS”(t) ? C(t)
1
Simple Reproduction
The simplest case to analyze is when p = 0, that is, when no surplus value is
accumulated and the capitalists consume their whole income. Marx calls this
case simple reproduction.
P (t) = C(t ? Tp )
S(t) = [1 + q]P (t ? TR )
S 0 (t) = P (t ? TR ) = S(t)/[1 + q]
S 00 (t) = qP (t ? TR ) = qC(t)
0
C(t) = S (t ? TF ) = P (t ? [TR + TF ]) = C(t ? [TP + TR + TF ])
C(t) = P (t) = S(t)/[1 + q]
Question 1. When p = 0, above equations are obtained from the circuit of
capital model. What does above equations tell us?
Answer to Q1. Above equations show (as our intuition tells us they should)
that if the capitalists consume all the surplus value, the flow of capital outlays
will be constant from one time to another and the flow of value of finished output
will also just be equal to this flow of capital outlays. The model cannot tell us
how large these flows are; they will stay at whatever level they have when the
process starts.
The profit rate r is the ratio of the surplus value (a flow) to the total capital
tied up in production, or r = S 00 (t)/[F (t) + N (t) + X(t)] We can also calculate
r = C(t)[TqC(t)
= [TF +TqP +TR ]
F +TP +TR ]
Question 2. What is the significance of the above formula?
Answer to Q2. It says that the profit rate is equal to the markup, which
determines how much each particle of value expands as it traverses the circuit
of capital, divided by the total turnover time of capital, which tells how long it
takes the particle to make the complete circuit.
2
Expanded Reproduction
To analyze expanded reproduction in the circuit of capital model, let us make the
assumption that all the flows and stocks are increasing at the same, unknown,
exponential rate g. Thus, for example, C(t) = C(0)exp(gt) = egt , where exp(.)
is the exponential function.
In what follows we shall use several properties of the exponential function,
especially
2
exp(gt ? gT ) = exp(g[t ? T ]) = exp(gt)exp(?gT ) (5.19)
d[exp(gt)]/dt = g[exp(gt)] (5.20)
Equation (5.19) shows that when we look backward T periods along a path of
exponential growth the size of the variable at that time, t ? T , is just exp(?gT )
times the size of the variable at time t.
Equation (5.20) shows that the increase in an exponentially growing variable
at any time is equal to the growth rate times the size of the variable at that time.
Substituting P (t) = C(t ? TP ) into S(t) = [1 + q]P (t ? TR ) we get S(t) =
[1 + q]C(t ? [TP + TR ])
Because current sales depend on past production and because past production depends on capital outlays even further back, current sales depend on
capital outlays in the past. The same reasoning leads to the conclusion that
capital outlays themselves depend on their own past values:
C(t) = [1 + pq]C(t ? [TF + TR + TP ]) (5.22)
If C(t) and all the other stock and flow variables are growing exponentially
at the same, as yet unknown, rate g, then C(t) = C(0)exp(gt), where C(0) is the
size of the flow of capital outlays at time 0. Then (5.22) becomes C(0)exp(gt) =
[1 + pq]C(0)exp(gt ? [TF + TR + TP ]) = C(0)exp(gt)[1 + pq]exp(?g[TF + TR +
TP ]). We can divide through by C(0)exp(gt) to get the characteristic equation
of this system: 1 = [1 + pq]exp(?g[TF + TR + TP ]) or, multiplying both sides
by exp(g[TF + TR + TP )) and taking natural logarithms, we get
g = ln(1 + pq)/(TF + TR + TP ) (5.25)
Equation (5.25) sums up a number of important insights about expanded
reproduction in capitalist economies. It shows that the rate of expansion does
indeed depend on the key parameters of the system.
Question 3. According to equation (5.25) what determines the growth rate
in the system? What effect do p, q, TF , TR and TP have on the growth rate,
respectively?
p
q
Answer to Q3. { 1+pq
, 1+pq
, ?, ?, ?}
Assuming C(0) = 1, we find r = q/(TP + TR + TF )
Question 4. Assuming C(0) = 1, are the equations defining the profit rate
any different in the simple and expanded reproduction?
Answer to Q4. No.
3
3
Proportionality and Aggregate Demand in Simple Reproduction
Marx raises, in Volume 2 of Capital the important question of what proportions
of social capital must be allocated to different functions to sustain smooth reproduction.
He proposes that for analytic purposes we divide the capitalist economy into
two departments: Department I consists of all those activities that produce
the elements of constant capital, that is, means of production; Department II
consists of all those activities that produce means of subsistence for the reproduction of labor-power.
The basic model for Department I (Department II is exactly the same, but
with the subscripts changed) is,
PI (t) = CI (t ? TP )
SI (t) = [1 + qI ]PI (t ? TR )
CI (t) = SI0 (t ? TF )
In simple reproduction there is a necessary link between the production of
each department and the social requirements for the output of that department.
This is Marx’s key insight into the requirements of proportionality in reproduction schemes.
The output of Department I is means of production, which must be bought
in order to meet the productive requirements of the two departments. Department I’s requirements for means of production in value terms are [1 ? kI ]CI (t)
because kI is the proportion of capital outlays of Department I spent on variable capital; hence the rest must go to purchase constant capital, or means of
production. Similarly, Department II’s requirements for means of production
are [1 ? kII ]CII (t). These purchases correspond to the replacement of means of
production used up in past cycles of production. In symbolic terms this relation
can be expressed as: SI (t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
Question 4. What does the above equation say?
Answer to Q4. Sales from all those activities that produce the elements of
constant capital (SI (t)) should be equal to the summation of requirements (in
value terms) for means of production in the constant capital producing department ([1 ? kI ]CI (t)) and the department that produces means of subsistence for
the reproduction of labor-power ([1 ? kII ]CII (t)).
A similar expression can be written relating the sales of Department II to the
wages paid in the two departments and to the surplus value, which by the assumption of simple reproduction is all consumed: SII (t) = kI CI (t)+kII CII (t)+
4
00
SI0 (t) + SII
(t)
In simple reproduction, we know that CI remains constant through time.
Thus we can write SI (t) = [1 + qI ]CI (t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t). We
can solve this equation for the proportion CI /CII , in which the social capital
must be divided to allow for simple reproduction:
CI
CII
=
1?kII
qI +kI
Question 5. What is the significance of the above equation?
Answer to Q5. Above equation says in a simple reproduction model if the
system begins with the capitals in the two departments in the proportions defined
as above, it can continue smoothly with no change in capital outlays or outputs.
There will be just the right output from each department to allow production in
the future to occur in the same quantities.
Intertwined with the analysis of the necessary proportions for reproduction
in Capital we find an investigation of the problem of aggregate demand.
The demand for produced commodities can be divided into three broad and
exhaustive categories: the demand of capitals for means of production, the demand of workers for means of subsistence, and the demand of capitalist households (or of other households whose incomes arise from surplus value, or of the
State) for means of subsistence or luxuries.
To simplify the mathematical examples, we shall assume that there is no
time delay in spending of wages by workers and that capitalist households have
the same time delay, TF , as capitalist firms have in spending their share of
the surplus value. We also shall assume that luxury production is a part of
Department II-to avoid the unnecessary multiplication of departments. Then
we can write the aggregate money demand for commodities as
D(t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
+kI CI (t) + kII CII (t)
00
+SI00 (t ? TF ) + SII
(t ? TF )
where the first line represents the demand of capitalist firms for means of
production, the second the spending of wages by worker households, and the
third the spending of surplus value by capitalist households.
The second important point in Marx’s analysis is that the capital outlays of
capitalist firms are themselves financed from past sales. This idea is expressed
in the general circuit of capital model. Using this model and the fact that the
time delay in capitalist household spending is assumed to be the same as the
time delay in capitalist firm spending, we find
5
D(t) = SI (t ? TF ) + SII (t ? TF ) = S(t ? TF )
Question 6. What is the significance of the above equation?
Answer to Q6. It shows that current demand depends on past sales, as
long as capital outlays are assumed to be financed solely out of past sales.
Question 7. Sum up Marx’s analysis of simple reproduction in two statements.
Answer to Q7. First, social capital must be allocated between the two
departments in the appropriate proportions to allow reproduction to continue
smoothly. Second, as long as the capitalists hold a sufficiently large fund of
money balances, there is no difficulty in financing the aggregate demand required
to realize all the commodities produced.
4
Proportionality in Expanded Reproduction
We now have to take account of the fact that the capitalization rates in the two
departments will be positive rather than zero. The basic model for Department
I (Department II is exactly the same, but with the subscripts changed) is, just
as in the case of simple reproduction,
PI (t) = CI (t ? TP ) SI (t) = [1 + qI ]PI (t ? TR )
CI (t) = S 0 I(t ? TF ) + pI SI00 (t ? TF )
where CI (t) is the flow of capital outlays for Department I, PI (t) is the
flow of finished product, and so on. Exactly as in the case of simple reproduction, as SI (t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t). A similar relation can
be written down relating the sales of Department II to the wages paid in
the two departments and the part of the surplus value consumed: SII (t) =
00
(t)
kI CI (t) + kII CII (t) + [1 ? pI ]SI00 (t) + [1 ? pII ]SII
This relation assumes that there is no time delay in spending of wages by
workers or spending of their share of surplus value by capitalist households.
The general proportionality condition for expanded reproduction, which is the
foundation of Marx’s results in Capital (1893, chap. 21) becomes:
CI (0)
CII (0)
=
[1–kII ]e(g[TP +TR ])
[1+qI ]?[1–kI ]e(g[TP +TR ])
(5.56)
Marx implicitly treats the problem of expanded reproduction as a period
model, with periods that he refers to as “years.” Capital outlays take place
at the beginning of a year, and production is completed within the year. The
product is realized at the beginning of the next year by the sale of the output. In
the notation that we have been using, for Department I CI (t) is capital outlays
at the beginning of year t, PI (t) is the flow of finished product at the end of year
6
t, and SI (t) is the sales at the beginning of year t; similar notation is used for
Department II. Marx’s assumptions in working out his schema of reproduction
are
P (t) = C(t)
S(t) = [1 + q]P (t ? 1) C(t) = S 0 (t) + pS 00 (t)
for each department. The balance condition he proposes is that the output
of Department I be realized through the capital outlays for constant capital in
the two departments. SI (t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
Marx’s system is a set of difference equations, but a comparison shows that
they are exactly the same as the circuit of capital equations, with TP = TF = 0
and TR = 1. Thus (5.56) also gives the necessary initial conditions for balanced
growth in Marx’s schemes. Because in Marx’s model [1 + pq] = exp(g) (for
either department), we can write (5.56) as
CI (0)
CII (0)
=
[1?kII ][1+pI qI ]
[1+qI ]?[1?kI ][1+pI qI ]
(5.61)
Question 8. What is the significance of above equation?
Answer to Q8. The significance of (5.61) in terms of Marx’s schemes is
simple. If we start with the capitals in the two departments in the proportions
indicated by (5.61), it is possible for the system to continue smoothly along a
path of balanced expanded reproduction. If we start with any other proportions,
it is impossible to meet all the conditions for expanded reproduction.
5
Aggregate Demand in Expanded Reproduction
Assume that there is no time delay in spending of wages by workers and that
capitalist households have the same time delay, TF , as capitalist firms in spending their share of the surplus value. We also shall assume that luxury production
is a part of Department II, to avoid the unnecessary multiplication of departments. Then we can write the aggregate money demand for commodities as
D(t) = [1 ? kI ]CI (t) + [1 ? kII ]CII (t)
+kI CI (t) + kII CII (t)
+[1 ? pI ]SI00 (t ? TF ) + [1 ? pII ]SII (t ? TF ) (5.62)
where the first line represents the demand of capitalist firms for means of
production, the second the spending of wages by worker households, and the
third the spending of surplus values by capitalist households. As in the case of
simple reproduction, capital outlays of capitalist firms are themselves financed
from past sales, thus D(t) = SI (t ? TF ) + SII (t ? TF ) = S(t ? TF ) Which
shows that current demand depends on past sales, as long as capital outlays are
7
assumed to be financed solely out of past sales. If we assume that the system
is on a path of expanded reproduction at rate g, then we find that
D(t) = D(0)exp(gt) = S(t ? TF )
= S(0)exp(gt)exp(?gTF )
= S(t)exp(?gTF ) (5.64)
In the case of simple reproduction, when g = 0, equation (5.64) creates
no puzzles. It says that the current aggregate demand is just large enough to
realize all the produced commodities at the appropriate rate of sales to enable
the reproduction process to continue. As we have seen, the lag in capitalist firm
and household spending means that firms and households must hold reserves of
money equal to S(0)TF to finance their spending streams. But in the case of
expanded reproduction, when g > 0, equation (5.64) seems to create a paradox
because it shows that the aggregate money demand for produced commodities is
smaller than the amount required to maintain smooth expanded reproduction.
Any finite initial reserve of money would be exhausted at some point on the
path of expanded reproduction.
Question 9. What is the relationship between Rosa Luxemburg’s argument
of imperialism and aggregate demand in expanded reproduction? What points
Duncan Foley made about it?
Answer to Q9. Rosa Luxemburg argued that a closed capitalist system undergoing accumulation would always run into inadequacies of aggregate demand
and as a consequence would be forced to seek external markets to realize its surplus production.
Duncan Foley claims, although Luxemburg is correct in pointing to the difference between aggregate demand and necessary realization that is inherent in
Marx’s setup, her solution is not convincing. He asks “Where do the external
markets get the money to buy the surplus product of the capitalist system?” and
points out “If they are supposed to hold large stocks of gold, which they spend on
capitalist commodities, the same problem arises as in the closed system, namely,
that with continuous expanded reproduction any finite stock of gold will be exhausted in a finite time.”
Question 10. What is Marx’s proposed solution to the demand problem?
What would be a modern-day adaption of this solution?
Answer to Q10. Marx himself, at the very end of Volume 2 of Capital, proposes one solution, which is taken up by Bukharin (1972) in his critique of Rosa
Luxemburg. Marx points out that it is not true that all the commodities produced
have to be realized by being exchanged against money. The money-commodity
gold, once produced, is already value in the money form and thus does not need
to be sold. If one posits a gold-producing sector of exactly the right size, namely,
with production exactly equal to the difference between money demand and realization expressed in equation (5.64), then the problem of realization is solved.
8
The money demand on the right-hand side of (5.64) is sufficient to realize all the
non-money commodities, and the rest of commodity production is gold, which
does not need to be realized. The gold-producing sector must grow at the same
rate as the rest of the system in order to maintain this balance. Of course, any
improvements in financial methods that shorten TF will reduce the required size
of the gold-producing sector.
The other way to resolve the paradox of equation (5.64) is to relax the assumption that current capital outlays are financed entirely from past sales. If
some capital outlays are financed by borrowing against the prospect of future
sales, then the realization gap can be closed.
This requires an alteration: C(t) = S 0 (t ? TF ) + pS 00 (t ? TF ) + B(t). Where
B(t) is new capitalist borrowing. If we use (5.65) in calculating the aggregate
demand for the system, we get D(t) = S(t ? TF ) + B(t). We see that new
borrowing by the capitalists to finance capital outlays can close the gap between
demand financed by past sales and the level of demand required to maintain
reproduction. In fact, borrowing by capitalist households, or by the State, can
also close the gap. The sustainable rate of growth of the system obviously depends
on the level of such new borrowing: the higher the total borrowing, the faster the
rate of expanded reproduction that can be achieved by the system.
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