Two-Sector Growth: Analytical Solutions

Contents

(A) Analytical Solution I: Classical Hypothesis
(B) Analytical Solution II: Proportional Savings

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The solution to the two-sector model depends crucially on the kind of savings assumptions we make. In his first version, Uzawa (1961) considered the "Classical hypothesis" that workers consume all their income and capitalists save all their profits. In other words, wL is the demand for consumer goods and rK the demand for new capital goods. In equilibrium, demand equals supply for the consumer goods and investment goods markets, i.e.

wL = Y_c

rK = pY_i

or, in per capita form:

w = y_c

rk = py_i

Recalling that y_c = l _cｦ_c(k_c) and y_i = l _iｦ_i(k_i) by definition and that w = ｦ_c｢ (k_c) and r = pｦ_i｢ (k_i) by competition, these conditions become:

ｦ_c｢ (k_c) = l_c ｦ_c(k_c)

pｦ_i｢ (k_i)ｷk = pl_i ｦ_i(k_i)

or, solving for l _c and l _i respectively:

l _c = ｦ_c｢(k_c)/ｦ_c(k_c)

l _i = kｷｦ_i｢(k_i)/ｦ_i(k_i)

But we also know from the wage-profit ratio equation that w + k_c = (ｦ _c(k_c)/ｦ_c｢(k_c)) and w + k_i = (ｦ_i(k_i)/ｦ_i｢(k_i)), so:

l _c = 1/(w + k_c)

l _i = k/(w + k_i)

Now, recall that the capital-market clearing condition was k = l_ck_c + l_ik_i. Consequently, rewriting this for l _c:

l_c = (k - l_ik_i)/k_c

so, substituting in our expression for l _i:

l_c = [k - k_ik/(w + k_i)]/k_c

or:

l_c = [kw /(w + k_i)]/k_c

Now, by the labor-market clearing condition, l _c + l _i = 1. Thus adding this l _c to our previous l _i::

l _c + l _i = [kw /(w + k_i)]/k_c + k/(w + k_i) = 1

or multiplying through by k_c and rearranging:

k(w + k_c)/(w + k_i) = k_c

So, we can express k as:

k = k_cｷ(w + k_i)/(w + k_c)

As k_c = k_c(w ) and k_i = k_i(w ), then this expresses k as a function of w . This is the locus k(w) of factor market equilibria.

We can decipher the slope of this function as:

dk/dw = {[(dk_c/dw )(w + k_i)(w +k_c) + k_c(1+dk_i/dw )ｷ(w +k_c) - (1 + dk_c/dw ) k_cｷ(w + k_i)}/(w + k_c)²

so, multiplying by 1/k = (w + k_c)/k_cｷ(w + k_i):

(dk/dw )ｷ(1/k) = {[(dk_c/dw )(w + k_i)(w +k_c) + k_c(1+dk_i/dw )ｷ(w +k_c) - (1 + dk_c/dw ) k_cｷ(w + k_i)}

/ k_cｷ(w + k_i)ｷ(w + k_c)

so canceling terms:

(dk/dw )ｷ(1/k) = (dk_c/dw )/k_c + (1+dk_i/dw )/(w +k_i) - (1 + dk_c/dw )/(w + k_c)

and rearranging:

(dk/dw )ｷ(1/k) = (dk_c/dw )ｷ[1/k_c - 1/(w +k_c)] + (dk_i/dw )/(w +k_i) + [1/(w +k_i) - 1/(w +k_c)]

Now, as dk_c/dw , dk_i/dw > 0 and as [1/(w +k_i) - 1/(w +k_c)] = w /(w +k_c) > 0, then the sign of this equation depends crucially on the sign of [1/(w +k_i) - 1/(w +k_c)] = (k_c - k_i)/[(w +k_i)ｷ(w +k_c)]. So, if we assume that consumer goods are more capital-intensive than investment goods, so that k_c > k_i, then we are guaranteed that dk/dw > 0. If, in contrast, k_i > k_c, so that investment goods are more capital intensive, then dk/dw is ambiguous.

We shall refer to the assumption that k_c > k_i for all w , i.e. that consumer goods are more capital-intensive that investment goods, as the "Uzawa capital-intensity condition". As we have seen, this is a sufficient condition for uniqueness of factor market equilibria, i.e. only by adopting the Uzawa capital-intensity condition can we conclude with certainty that dk/dw > 0 for all w ﾎ [w _min, w _max]. In other words, the Uzawa capital-intensity assumption makes the factor market equilibrium locus, k(w), monotonically upward-sloping wherever it lies between the boundaries k_c(w ) and k_i(w ).

[Note: Edwin Burmeister (1968) characterizes the determinacy of factor market equilibria in terms of the Jacobian determinant J(w ) of the system, which is particularly useful. See Burmeister and Dobell, 1970: Ch. 4.]

All this is for factor market equilibria -- or what the literature calls "momentary equilibria". We still have not touched upon the existence, uniqueness or stability of the a steady-state growth path. To prove stability of steady-state growth, Uzawa appeals to a theorem of Arrow, Block and Hurwicz (1959) regarding the limits of a differentiable equation. We shall skip the formal stability proof and refer to his article. Intuitively, the idea behind stability comes from recalling the basic underlying differential equation:

dk/dt = y_i - nk

As we have made the "Classical hypothesis" that all profits are saved and all wages are spent, remember that this implies that rk/p = y_i, and by the marginal productivity assumption, rk/p = ｦ _i｢ (k_i), so this reduces to:

dk/dt = ｦ ｢ (k_i) - nk

Is this stable? Stability of the steady-state capital-labor ratio, k*, requires that d(dk/dt)/dk < 0 around k*. But the capital-labor ratio of the investment goods sector, recall, is a function of equilibrium factor prices, w , and these, in turn, are determined by the aggregate capital-labor ratio, k. So by intuitive chain rule logic:

d(dk/dt)/dk = (dｦ｢(k_i)/dk_i)ｷ(dk_i/dw )ｷ(dw /dk) - n

Determining the sign of this is the crucial step. Obviously, ｦ｢(k_i) is negatively related to k_i by simple diminishing marginal productivity, i.e. dｦ｢(k_i)/dk_i < 0. We know already that dk_i/dw > 0. So the question boils down to dw/dk. We have proved that if Uzawa's capital-intensity condition holds, then dw /dk > 0, and thus we are home free because this implies that for values of k above n, d(dk/dt)/dk < 0 and thus our system is stable.

This is interesting. The Uzawa capital-intensity condition was imposed to guarantee uniqueness of factor-market equilibrium for every k. But it also implies uniqueness and stability of the steady-state growth path. Why does relative capital-intensity matter for stability of growth equilibrium? Because of the infamous "Wicksell Effects". To see why, recall that by the Classical hypothesis, wL = Y_c and rK = pY_i, then rK/wL = pｷ(Y_i/Y_c), or:

K/L = (w/r)ｷpｷ(Y_i/Y_c)

or, in per capita terms:

k = w ｷpｷ(y_i/y_c)

So if w rises, everything else constant, k rises. But everything else is not constant. Specifically, the price of investment goods, p, is a function of w and relative outputs, y_i/y_c = ｦ_c(k_i)/ｦ_c(k_c), are also functions of sectoral allocation k_i and k_c which are also functions of w . So we should rewrite this as:

k(w ) = w ｷp(w)ｷ(y_i(w)/y_c(w))

Evidently, lots of things can happen now. So let us try it again: suppose w rises, then k(w ) must rise unless p(w) falls and/or y_i(w)/y_c(w) falls sufficiently, in which case k(w ) falls. So, what rules this possibility out? Uzawa's capital-intensity assumption. To see this, let us begin by proving that p cannot fall in response to a rise in w if the consumer goods sector is more capital-intensive. As we know the price of investment goods p can be expressed as:

p = ｦ_c｢ /ｦ_i｢

But as the marginal products are themselves functions of k_c and k_i (which are, in turn, functions of w ), then we can write p as a function of w :

p(w) = ｦ_c｢ (k_c(w)) /ｦ_i｢(k_i(w))

To decipher the relationship between p and w , just differentiate:

dp/dw = {ｦ_c｢｢ ｷ(dk_c/dw )ｷｦ_i｢ - ｦ_i｢｢ ｷ(dk_i/dw )ｷｦ_c｢ )/[ｦ_i｢ ]²

which seems pretty ugly. But recall that, from before, dk_c/dw = -ｦ_c｢ ²/(ｦ_c｢｢ ｷｦ _c) and dk_i/dw = -ｦ_i｢ ²/(ｦ_i｢｢ ｷｦ _i). So, plugging in:

dp/dw = {-ｦ_c｢｢ ｷ(ｦ_c｢ ²/(ｦ_c｢｢ ｷｦ _c))ｷｦ_i｢ + ｦ_i｢｢ ｷ(ｦ_i｢ ²/(ｦ_i｢｢ ｷｦ_i))ｷｦ_c｢ }/[ｦ_i｢ ]²

and canceling terms:

dp/dw = {(ｦ_i｢ ²/ｦ_i)ｷｦ_c｢ - (ｦ_c｢ ²/ｦ _c)ｷｦ_i｢ }/[ｦ_i｢ ]²

or:

dp/dw = {(ｦ_i｢ ²/ｦ_i)ｷｦ_c｢ - (ｦ_c｢ ²/ｦ _c)ｷｦ_i｢ }/[ｦ_i｢ ]²

dp/dw = ｦ_c｢ /ｦ_i - (ｦ_c｢ ²)/(ｦ_cｷｦ_i｢ )

Recalling that p = ｦ_c｢ /ｦ_i｢ , then multiplying through by 1/p = (ｦ_i｢ /ｦ_c｢ ):

(dp/dw )ｷ(1/p) = (ｦ_i｢ /ｦ_c｢ )ｷｦ_c｢ /ｦ _i - (ｦ_i｢ /ｦ_c｢ )ｷ(ｦ_c｢ ²)/(ｦ_cｷｦ_i｢ )

or:

(dp/dw )ｷ(1/p) = ｦ_i｢ /ｦ_i - ｦ_c｢ ²/ｦ_c

so, finally, remembering from before that w + k_i = ｦ_i/ｦ_i｢ and w + k_c = ｦ_c/ｦ_c｢ , then:

(dp/dw )ｷ(1/p) = 1/(w + k_i) - 1/(w + k_c)

So, if k_c > k_i (consumer goods more capital-intensive), then dp/dw > 0, while if k_i > k_c (investment sector more capital-intensive), then dp/dw < 0. This should not be a surprising result. It is reminiscent of the famous Stolper-Samuelson theorem: specifically, that a rise in the price of a good is positively related with a rise in the return to the factor in which that good in intensive. So, if the investment-goods industry is capital-intensive, a relative rise in the return to capital (a fall in w) will be associated with a rise in the price of investment goods (p). Conversely, if the investment-goods industry is labor-intensive, then a relative rise in the wage (a rise in w ) will be associated with a rise in p.

So, Uzawa's capital-intensity assumption implies that dp/dw > 0. That's half the problem solved. Now, we only need to make sure that y_i/y_c cannot fall sufficiently in response to a rise in w to make k fall as well. We resort to basic intuition: a rise in w is automatically related to an increase in the capital-intensity of both sectors, i.e. k_i and k_c rise and thus y_i and y_c rise. Furthermore, by Uzawa's capital-intensity assumption, then if y_i/y_c falls, we are releasing factors from a labor-intensive investment goods industry into a capital-intensive consumer goods industry, so we are increasing the average capital-intensity of the economy. Thus k simply cannot fall in response to a rise in w because not only do both sectors become more capital-intensive, but we are transferring factors from labor-intensive to more capital-intensive industries. The aggregate capital-labor ratio k must rise.

However, if we violate Uzawa's capital-intensity condition, and assume that investment goods are more capital intensive, then note that dp/dw < 0 and if y_i/y_c falls, we are releasing factors from a capital-intensive sector into a labor-intensive one. Consequently, there is a strong countervailing tendency: it is quite possible that the fall in py_i/y_c more than outweighs the rise in w so that k falls. In other words, an increase in the wage-profit rate can lead to a fall in the capital-labor ratio, i.e. we are employing more of the factor which has become relatively more expensive! In geometric terms, the market demand for capital is not everywhere negatively related to profit and/or the market demand for labor is not negatively related to wages. The market factor demand curves will have curious shapes with upward-sloping portions. This is the kind of thing that yields multiplicity of factor market equilibria and make our system indeterminate. In other words, at k, we do not know which factor prices will result, and thus we may end up at a higher or lower capital-labor ratio tomorrow, which means that we cannot tell whether we will be moving towards or away from the steady-state capital-labor ratio.

Of course, the Uzawa capital-intensity condition is sufficient for stability, but not necessary. One can construct many examples where the investment goods sector is more capital-intensive than consumer goods and still have stability of the steady-state. But, and this is more important, without it, we can also construct many reasonable examples with instability. And these are more reasonable.

[Note: This intuition is derived largely from Solow (1961) and Hahn (1965). Uzawa (1963) traces the source of his capital-intensity condition to Knut Wicksell's discussion of "ﾅkerman's problem" (Wicksell, 1923). Morishima (1969: p.45) calls this the "Shinkai-Uzawa" condition, in recognition of Y. Shinkai's (1960) work on two-sector models with fixed coefficients of production, where he obtained the result that growth equilibrium is stable if and only if the consumer goods industry is more capital intensive than the investment goods sector. When we have flexible production functions, as in the Uzawa model, this is merely a sufficient, but not necessary, condition for stability. John Hicks (1965) finds this condition, but does not dwell much over it. He finds it again and gives it more attention in his Neo-Austrian model (Hicks, 1973).]

Robert Solow (1961) criticized Uzawa's (1961) result on the basis that it seemed to depend to heavily on the "Classical hypothesis" that all wages are spent and all profits are saved. As a result, Hirofumi Uzawa (1963) hit the drawing board again and introduced a more flexible savings hypothesis.

For flexibility, let us assume that s_w is the average propensity to save out of wages and s_r is the average propensity to save out of profits. [Note: this actually follows Drandakis (1963) -- in his original presentation, Uzawa (1963) assumes that s_w = s_r = s.] So total demand for investment goods is s_wwL + s_rrK, while total demand for consumer goods is (1-s_w)wL + (1-s_r)rK. In equilibrium, demand equals supply for the consumer goods and investment goods markets, i.e.

(1-s_w)wL + (1-s_r)rK = Y_c

s_wwL + s_rrK = pY_i

so, in per capita form:

(1-s_w)w + (1-s_r)rk = y_c

s_ww + s_rrk = py_i

We shall concentrate on the second equation. Note that s_ww = s_wwr/r = s_wwr as w = w/r by definition. Thus substituting in:

(s_ww + s_rk)r = py_i

Dividing through by p and recalling that y_i = l _iｦ_i(k_i) by definition and r/p = ｦ_i｢ (k_i) by competition then:

(s_ww + s_rk)ｦ_i｢ (k_i) = l _iｦ_i(k_i)

Now, recall that our capital-accumulation equation is:

dk/dt = y_i - nk

which, in view of our expression above, we can write as:

dk/dt = (s_ww + s_rk)ｦ_i｢ (k_i) - nk

which is our fundamental differential equation.

Let us now concentrate on the factor market equilibrium locus. In factor market equilibrium:

l _i + l _c = 1

l_ck_c + l_ik_i = k

which are the conditions for labor market clearing and capital market clearing respectively. Combining:

k = (1-l_i)k_c + l_ik_i

So recalling our original macroeconomic equilibrium equation, we now have a system of simultaneous equations in l _i and k, specifically:

l _i(k_i - k_c) - k = -k_c

l _iｦ_i(k_i) - s_rkｦ _i｢ (k_i) = s_wwｦ_i｢ (k_i)

or, in matrix form:

・/font>	(ki - kc)	-1	・/font>	・/font>	l i	・/font>		・/font>	-kc	・/font>
・/font>			・/font>	・/font>		・/font>	=	・/font>		・/font>
・/font>	ｦi(ki)	-srｦi｢ (ki)	・/font>	・/font>	k	・/font>		・/font>	swwｦi｢(ki)	・/font>

By Cramer's Rule, the solution k is:

k = |A₂|/|A|

where A is the determinant of the matrix on the left, while A₂ is the determinant of that matrix with the solution vector in the second column. Thus note that:

|A| = -s_rｦ_i｢ (k_i)(k_i - k_c) + ｦ _i(k_i)

and:

|A₂| = s_wwｦ_i｢ (k_i)(k_i - k_c) + ｦ_i(k_i)k_c

so:

k = [s_wwｦ_i｢ (k_i)(k_i - k_c) + ｦ _i(k_i)k_c]/[ｦ_i(k_i) -s_rｦ_i｢ (k_i)(k_i - k_c)]

so, dividing both numerator and denominator by ｦ_i｢ :

k = [s_ww(ｦ_i｢ /ｦ_i｢)(k_i - k_c) + (ｦ_i/ｦ_i｢ )k_c]/[(ｦ_i/ｦ_i｢ ) - s_r(ｦ_i｢ /ｦ_i｢ )(k_i - k_c)]

or:

k = [s_ww(k_i - k_c) + (ｦ_i/ｦ_i｢ )k_c]/[(ｦ_i/ｦ_i｢ ) -s_r(k_i - k_c)]

Finally, recalling from our competition condition that ｦ_i/ｦ_i｢ = k_i + w , then:

k = [s_ww(k_i - k_c) + (k_i + w )k_c]/[(k_i + w ) - s_r(k_i - k_c)]

Now, adding w to both sides:

k + w = [s_ww(k_i - k_c) + (k_i + w )k_c]/[(k_i + w ) - s_r(k_i - k_c)] + w

= [s_ww(k_i - k_c) + (k_i + w )k_c + w (k_i + w ) - w s_r(k_i - k_c)]/[(k_i + w ) - s_r(k_i - k_c)]

so:

k + w = [w(s_w - s_r)(k_i - k_c) + (k_c + w )(k_i + w )]/[(k_i + w ) - s_r(k_i - k_c)]

Which is, effectively, the form of our factor market equilibrium locus k(w). Notice that if we take Uzawa's (1963) special case that s_w = s_r = s (which we shall call the Uzawa savings assumption), then this reduces to:

k + w = [(k_c + w )(k_i + w )]/[(k_i + w ) - s(k_i - k_c)]

or, noting that the denominator can be written as:

(k_i+w ) - s(k_i-k_c) = s(k_i+w ) + (1-s)(k_i+w ) - sk_i + sk_c

= s(k_c+w ) + (1-s)(k_i+w )

then we can rewrite our whole equation as:

k + w = [(k_c + w )(k_i + w )]/[s(k_c+w ) + (1-s)(k_i+w )]

which is exactly Uzawa's (1963) equation (21).

Enough of algebra. The important question that emerges from all this is the following: is there a positive monotonic relationship between w and k? In other words, is it true that there is a unique equilibrium factor-price ratio w for any given k? We shall not bother with existence (although that also ought to be proven) and concentrate on uniqueness of w . Adopting Uzawa's savings assumption (s_w = s_r = s), then let us define z_c = k_c + w and z_i = k_i + w and g(w) = k, so:

g(w)= z_cz_i/[sz_c + (1-s)z_i] - w

Then, differentiating with respect to w :

dg(w)/dw = [(z_c｢z_i + z_cz_i｢ )ｷ(sz_c + (1-s)z_i) - (sz_c｢ + (1-s)z_i｢ )ｷz_cz_i] /[sz_c + (1-s)z_i]² - 1

or:

dg(w )/dw = [sz_c²z_i｢ + (1-s)z_i²z_c｢ ]/[sz_c + (1-s)z_i]² - 1

Now, recall that z_c = k_c + w and z_i = k_i + w , which implies that z_c｢ = k_c｢ + 1 and z_i｢ = k_i｢ + 1 where, as k_c｢ > 0 and k_i｢ > 0, implies then that z_c｢ > 1 and z_i｢ > 1. Thus:

dg(w)/dw > [sz_c² + (1-s)z_i²]/[sz_c + (1-s)z_i]² - 1

But examine the fraction in this expression. Now, we know that (z_c - z_i)² ｳ 0, so:

z_c² + z_i² - 2z_cz_i ｳ 0

thus, multiplying by s(1-s):

s(1-s)z_c² + s(1-s)z_i² - 2s(1-s) z_cz_i ｳ 0

or:

s(1-s)z_c² + s(1-s)z_i² ｳ 2s(1-s)z_cz_i

Thus, adding s²z_c² + (1-s)²z_i² to both sides:

s(1-s)z_c² + s²z_c²+ s(1-s)z_i² + (1-s)²z_i^{2 ｳ} s²z_c² + (1-s)²z_i² + 2s(1-s)z_cz_i

Or as s(1-s) + s² = s and s(1-s) + (1-s)² = (1-s), and noticing that the term on the right is merely [sz_c + (1-s)z_i]², then:

sz_c² + (1-s)z_i^{2 ｳ} [sz_c + (1-s)z_i]²

which implies that

dg(w)/dw > [sz_c² + (1-s)z_i²]/[sz_c + (1-s)z_i]² - 1 > 0

so dg(w)/dw is positive and the factor equilibrium locus is everywhere upward-sloping, i.e. w is a monotonically increasing function of k and there is a unique factor-market clearing w associated with every k.

This is intriguing. When we made the "Classical" savings hypothesis, we needed Uzawa's capital-intensity assumption to guarantee uniqueness of factor market equilibrium. But now no capital-intensity assumptions are needed. The Uzawa savings hypothesis, that s_w = s_r = s, is by itself sufficient to obtain uniqueness.

What if we drop this special case? If s_w ｹ s_r, Drandakis (1963) has shown that either of the following are sufficient for uniqueness:

(i) s_r > s_w and k_c > k_i

(ii) s_r < s_w and k_c < k_i

We shall call these the Drandakis mixed conditions. Effectively, it states that either configuration (i) or configuration (ii) is sufficient to guarantee there is a unique w for every k. This is striking. Note that in the Uzawa savings case, no capital-intensity hypothesis was needed to guarantee uniqueness. But by allowing s_w ｹ s_r, then not all configurations work: we must append a capital-intensity hypothesis to the savings propensity hypothesis to ensure uniqueness. Notice that the Drandakis mixed conditions (i) and (ii) are not necessary but rather sufficient conditions, so other configurations might work, but it is not guaranteed.

So much for factor market equilibrium. Let us now turn to steady-state growth. Our fundamental differential equation is:

dk/dt = (s_ww + s_rk)ｦ_i｢ (k_i) - nk

so, assuming steady-state equilibrium, dk/dt = 0 and so at the solution k (and associated factor-market clearing w and sectoral allocation k_i, k_c):

(s_ww+ s_rk)ｦ_i｢(k_i)/k = n

Of course, there is nothing, in principle, that rules out multiple steady-states, i.e. there could be several solutions to this. How can we rule this out? Drandakis (1963) showed that if the first of the Drandakis mixed conditions are met (i.e. if s_r ｳ s_w and k_c ｳ k_i), then the steady-state k is unique and stable. So, for sufficiency, not only do we require the old Uzawa capital-intensity condition (consumer goods are more capital-intensive than investment goods), we also need to assume that proportionally more is saved out of profits than out of wages. To prove this, note that from before, condition (i) states that if s_r > s_w, and k_c > k_w, then dk(w)/dw > 0. Now:

d(dk/dt)/dk = (s_w(dw /dk) + s_r)ｦ_i｢ + (s_ww + s_rk)ｷｦ_i｢｢ ｷ(dk_i/dw )ｷ(dw /dk) - n

Now, recall that dk_i/dw = -(ｦ_i｢)²/ｦ _iｷｦ_i｢｢ , so ｦ_i｢｢ｷ(dk/dw ) = -(ｦ_i｢ )²/ｦ_i, so:

d(dk/dt)/dk = (s_w(dw /dk) + s_r)ｦ_i｢ - (s_ww + s_rk)ｷ(ｦ_i｢)²ｷ(dw /dk)/ｦ_i - n

or, rearranging a bit:

d(dk/dt)/dk = s_w (dw /dk)ｦ_i｢ + s_rｦ_i｢- (s_wwｦ_i｢ + s_rkｦ_i｢ )ｷｦ_i｢ ｷ(dw /dk)/ｦ_i - nk

Now, from the macroeconomic equilibrium condition (s_ww + s_rk)ｦ_i｢(k_i) = l _iｦ_i(k_i), so:

s_rkｦ_i｢ = l_iｦ_i - s_wwｦ_i｢

and:

s_rｦ_i｢ = (l_iｦ_i - s_wwｦ_i｢)/k

so:

d(dk/dt)/dk = s_w (dw /dk)ｦ_i｢ + (l_iｦ_i - s_wwｦ_i｢)/k - (s_wwｦ_i｢ + l_iｦ_i - s_wwｦ_i｢)ｷｦ_i｢ ｷ(dw /dk)/ｦ _i - n

rearranging:

d(dk/dt)/dk = s_w (dw /dk)ｦ_i｢ + l_iｦ_i/k - s_wｦ_i｢(w /k) - l_iｷｦ_i｢ｷ(dw /dk) - n

or, factoring ｦ_i｢:

d(dk/dt)/dk = [(s_w - l _i)ｷ(dw /dk) - s_w(w /k)]ｦ_i｢ + l_iｦ _i/k - n

So the sign of d(dk/dt)/dk depends crucially on the sign of [(s_w - l_i)ｷ(dw /dk) - s_w(w /k)]. If:

(s_w - l _i)(dw /dk) < s_w(w /k)

then d(dk/dt)/dk < 0 around equilibrium and we will have local stability.

Let us now deduce the implications of this. We know that dk/dt = y_i - nk, or simply:

dk/dt = l_iｦ_i(k_i) - nk

So, if we have steady-state k*, then dk/dt = 0, so:

l_i = nk*/ｦ_i(k_i*)

Now, we know that d(dk/dt)/dk < 0 if (s_w - l_i)(dw /dk) < s_w(w /k). So, plugging in, we see that around steady-state

(s_w - nk*/ｦ_i(k_i*))(dw /dk) < s_w(w /k*)

So, if s_w < nk*/ｦ_i(k_i*), then this inequality holds for certain, so we will be assured that dk/dt < 0 around the steady-state k*.

Alternatively, consider the following. As (s_ww + s_rk)ｦ_i｢(k_i) = l_iｦ_i(k_i) by macroeconomic equilibrium, then:

s_w = l_iｦ_i(k_i)/(wｦ_i｢(k_i)) - s_rk/w

and as ｦ_i/ｦ_i｢= k_i + w by competition, then:

s_w = l_i(k_i + w )/w - s_rk/w

or:

s_w - l_i = (l_i k_i - s_rk)/w

Thus, if l_ik_i < s_rk, then it must be that s_w - l_i < 0. But we know that (s_w - l_i)(dw /dk) < s_w(w /k) is sufficient condition for (dk/dt)/k < 0. So, if l_ik_i < s_rk, then (dk/dt)/k < 0. Now, as we know, at steady-state, l_i = nk*/ｦ_i(k_i*), thus substituting in, if:

(nk*/ｦ_i(k_i*))k_i* < s_rk*

then k* is stable. Rearranging, this condition becomes:

s_r > nk_i*/ｦ_i(k_i*)

So, in sum if:

(i) s_w ｣ nk*/ｦ_i(k_i*)

or:

(ii) s_r ｳ nk_i*/ｦ_i(k_i*)

then we are guaranteed that (dk*/dt)/k* < 0, so the steady-state capital-labor ratio, k* is (at least locally) stable.

Drandakis's (1963) sufficiency conditions may be examined more closely now. More precisely, if the Drandakis's mixed condition that s_r > s_w and k_c < k_i is met, then both (i) and (ii) will hold. To see this, note that if s_r > s_w, then s_rk* > s_wk*. Also, if k_c > k_i, then s_wk_c* > s_wk* > s_wk_i*. So combining:

s_rk* > s_wk_i*

so, dividing through by w * + k_i*

s_rk*/(w * + k_i*) > s_wk_i*/(w * + k_i*)

But as k_i*/(w * + k_i*) = 1 - w */(w *+k_i*), then this can be written:

s_rk*/(w * + k_i*) > s_w [1- w */(w * + k_i*)]

Now, recall by competition that ｦ_i(k_i*)/ｦ_i｢ (k_i*) = w * + k_i*. So:

s_rk*ｦ_i｢ (k_i*)/ｦ_i(k_i*) > s_w [1- w*ｦ_i｢(k_i*)/ｦ_i(k_i*)]

rearranging:

[s_rk* + s_w w*]ｷｦ_i｢(k_i*)/ｦ_i(k_i*) > s_w

Now, in steady-state, we know that (s_ww * + s_rk*)ｦ_i｢(k_i*) = nk*. Thus, substituting in:

nk*/ｦ_i(k_i*) > s_w

which is precisely the condition (i) for stability. Condition (ii) follows by extension. Thus, the first Drandakis mixed condition (s_r > s_w and k_c > k_i) is indeed sufficient for stability.

An interesting observation is to realize that the sufficient condition for stability, (s_w - l_i)(dw /dk) < s_w(w /k), can be rewritten as:

s_w [(dw /dk) - (w /k)] - l _i(dw /dk) < 0

But recall from our discussion of production theory the elasticity of substitution between factors is s = (dk/dw)ｷ(w /k). Notice, then that if s ｳ 1, then this inequality is guaranteed. Thus, s ｳ 1 is also a sufficient condition for stability.

Finally, an even more famous sufficiency condition derived by Drandakis (1963) is that the elasticity of substitution of the consumer-goods sector be greater than 1, i.e. that

s_c = (dk_c/dw )ｷ(w /k_c) ｳ 1.

which we shall not attempt to prove here.

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