HET: Von Neumann System with consumption

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Von Neumann System
with Consumption

Michio Morishima (1960) expanded the original von Neumann model by adding consumption to the system. We follow the basic form of the original von Neumann system, but then add the following additional data: w = given real wage and a₀ = vector of labor input coefficients (all positive - thus labor necessary input to all processes). So prices are governed by:

pB £ (1+r)(pA + wa₀)

postmultiplying by z:

pBz = (1+r)(pAz + wa₀z)

which forces equality due to the same assumption of free disposibility (excess cost)

Three sources of demand:

(1) industrial demand (as before = (1+g)Az)
(2) capitalists' demand (from profit income, R = r(pAz + wa₀z), they have propensity to consume of c_r, so demand = c_rR. Let d_c be a vector of goods capitalists desire, then budget constraint of capitalist implies: pd_c = c_rR)
(3) workers' demand (from W = wa₀z, workers consume all wages so if d_w is vector of goods workers desire, then budget constraint of workers implies: pd_w = W).

As a result, the output relations become:

Bz³ (1+g)Az + (1+g)d_w + d_c

(note: workers are paid at the same time as other factor costs, whereas capitalists take profits from previous period's output). Assume that d_w and d_c are also functions of price and that there is unit income elasticity of demand. Let m_c(p) be a vector of the capital-owner's Engel coefficients and let m_w(p) be a vector of laborers' Engel coefficients, both assumed to be related to the price vector. By the assumptions imposed, then, c_r = pm_c(p) and 1 = pm_w(p). Thus d_w = (W/p) = Wm_w(p). Similarly, d_c = c_rR/p = d_c = Rm_c(p). Thus, plugging these terms into our equation for d_w and d_c:

Bz ³ (1+g)Az + (1+g)Wm_w(p) + Rm_c(p)

or, substituting for W and R:

Bz ³ (1+g)(Az + wa₀zm_w(p)) + r(pAz + wa₀z)m_c(p)

Now, multiplying through by p, we obtain:

pBz = (1+g)(pAz + wa₀pm_w(p)z) + r(pA + wa₀)pm_c(p)z

which forces the equality as per the excess supply rule. As c_r = pm_c(p) and 1 = pm_w(p), then:

pBz = (1+g + c_rr)(pAz + wa₀z)

By the assumptions of Kemeny, Morgenstern and Thompson (1956), we add the condition that the total value of output must be positive (pBz > 0). Now, the two strict equalities for the price and quantity side imply that:

(1+g + c_rr)(pAz + wa₀z) = (1+r)(pAz + wa₀z)

or simply:

1+g + c_rr = 1+r

so that the Golden Rule (g = r) is restored if and only if c_r = 0, the propensity to consume out of profits reduces to zero. Alternatively:

g/(1 - c_r) = r

so the equilibrium is where the rate of interest equals the rate of growth divided by the propensity to save of capitalists (1 - c_r is the propensity to save). Perhaps unsurprisingly, this replicates precisely a form of the Kaldor-Pasinetti rule of Cambridge growth models.

Other References:

Michio Morishima (1960) "Economic Expansion and the Interest Rate in Generalized von Neumann Models", Econometrica, vol. 28 (2), p.352-63.

M. Morishima (1964) Equilibrium, Stability and Growth: A multi-sectoral analysis. Oxford: Oxford University Press.

M. Morishima (1969) Theory of Economic Growth, Oxford: Oxford University Press.

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