Equilibrium Condition

6.3 A Model of the Rural Industrial Sector

6.3.3 Output in the Short-Run

6.3.3.2 Equilibrium Condition

separately. It provides us a handle to examine, how a change in income of this class impacts output of this segment⁸⁴. It could possibly be explored further in the future. The short-run total demand 𝐷 for the output of the rural industrial sector is given as,

𝐷 = 𝐷_𝑜+ 𝐷_𝑎𝑤 + 𝐷_𝑖𝑤+ 𝐷_𝑐

𝐷 = 𝐷_𝑜+ 𝑎_𝑎𝑤+ 𝑎_𝑖𝑤+ 𝑎_𝑐 +𝑘_𝑎𝑤. 𝑊_𝑎+ 𝑘_𝑖𝑤.𝑊 + 𝑘_𝑐 . 𝜋 𝐷 = 𝐷_𝑜^′+ 𝑘_𝑖𝑤. 𝑌. 𝑎.^𝑤𝑛

𝑃 + 𝑘_𝑎𝑤. 𝐿_𝑎.^𝑤𝑎

𝑃_𝑎𝑤 + 𝑘_𝑐.^𝑌.𝜆

𝑃 . (𝑎. 𝑤_𝑛 + 𝑏. 𝑐) where 𝐷_𝑜^′= 𝐷_𝑜+ 𝑎_𝑖𝑤+ 𝑎_𝑐+ 𝑎_𝑎𝑤

[6 − 10𝑒] 𝐷 = 𝐷_𝑜^′+ { 𝑘_𝑖𝑤. 𝑎.^𝑤𝑛

𝑃 + 𝑘_𝑐.^𝜆

𝑃. (𝑎. 𝑤_𝑛+ 𝑏. 𝑐)}. 𝑌 + 𝑘_𝑎𝑤. 𝐿_𝑎.^𝑤𝑎

𝑃𝑎𝑤

𝐶𝑎𝑠𝑒 1: Change in 𝑌^∗ with 𝑤_𝑛, for 𝜆=constant, 𝑎 =constant

We partially differentiate [6 − 11𝑎] with respect to 𝑤_𝑛 to get the comparative static result,

[6 − 12] ^𝜕𝑌∗

85

𝜕𝑤𝑛 = −^𝐷0

′.^𝜕∝

𝜕𝑤𝑛

∝²

,

where ∝= {1 − ¹

(𝜆+1)(𝑎.𝑤𝑛+𝑏.𝑐)(𝑘_𝑖𝑤.𝑎. 𝑤_𝑛+ 𝑘_𝑐. 𝜆(𝑎. 𝑤_𝑛 + 𝑏. 𝑐))}

We obtain, [6 − 12𝑎] 𝜕𝑌^∗

𝜕𝑤_𝑛 = 𝐷_𝑜^′

∝². (𝜆 + 1). { 𝑘_𝑖𝑤. 𝑎𝑏𝑐 (𝑎. 𝑤_𝑛+ 𝑏. 𝑐)²} Thus, ^𝜕𝑌

∗

𝜕𝑤_𝑛 > 0 for 𝜆 =constant.

The result implies that an increase in 𝑤_𝑛 brings about an increase in 𝑌^∗ . Increase in 𝑤_𝑛

Figure 6-2 Change in Output in the Short-Run with Change in 𝐰_𝐧

𝑌

₂

𝑌

^∗

𝑌

₁

D2

P1 S1

Ysr P3 S3

P2 S2

Quantity of Output Price of

Rural Industrial Goods

D1

increases both the demand for industrial goods (𝐷_𝑖𝑤 i.e. demand-side effect) and the price of output (𝑃 i.e. supply-side effect).

The demand curve moves from 𝐷₁ to 𝐷₂ and price moves from 𝑃₁ to 𝑃₂ The new equilibrium point is 𝑌₁ and it satisfies the condition ^𝜕𝑌

∗

𝜕𝑤𝑛 > 0. However, if the price moves to 𝑃₃ the new equilibrium point is 𝑌_2. This is not a valid equilibrium point in our model as the condition ^𝜕𝑌

∗

𝜕𝑤𝑛 > 0 is violated. The intuition underlying the finding ^𝜕𝑌∗

𝜕𝑤𝑛 > 0 is that a rise in wn increases the industrial worker’s purchasing power. This results in a substantial increase in the industrial workers’ consumption demand (𝐷_𝑖𝑤). A rise in wn also increases the input cost and hence the price 𝑝 of the industrial produce, which is likely to pull down the demand. Its impact on 𝐷_𝑖𝑤 overshadows the impact on 𝑝 viz. ∆𝐷_𝑖𝑤 ≫ ∆𝑝 . Hence, 𝑌^∗increases with increase in 𝑤 and vice-versa

Goodwin (1983) had also observed that the impact of a rise in money wage rate on rise in price is small, which would imply that the dampening effect on demand of a wage rise is likely to be low.

𝐶𝑎𝑠𝑒 2: Change in 𝑌^∗ with 𝑤_𝑛, for 𝜆= 𝜆(𝑤_𝑛), 𝑎 =constant

We partially differentiate [6 − 11𝑎] with respect to 𝑤_𝑛 for 𝜆= 𝜆(𝑤_𝑛), to get the comparative static result,

𝜕𝑌^∗86

𝜕𝑤_𝑛 = −^𝐷0

′.^𝜕∝

𝜕𝑤𝑛

∝²

Differentiating [6 − 11𝑎] we obtain [6-12b] ^𝜕𝑌∗

𝜕𝑤𝑛 = ^.𝐷𝑜′

∝².[(1+𝜆)𝑎.𝑏.𝑐.𝑘𝑖𝑤−𝑎.𝑤_𝑛.𝑘_𝑖𝑤.(𝑎.𝑤_𝑛+𝑏.𝑐).𝜆^′(𝑤_𝑛)

{(𝜆+1)(𝑎𝑤𝑛+𝑏.𝑐)}² + ^{𝑘𝑐.𝜆′(𝑤𝑛)}

(𝜆+1)² ]

Since, ^𝜕𝜆

𝜕𝑤𝑛 < 0, 𝜆^′(𝑤_𝑛) has a negative bearing on the second and third R.H.S terms of [6-12b].

𝜕𝑌^∗

𝜕𝑤𝑛 > 0, for 𝜆 = 𝜆(𝑤_𝑛), if the following condition is satisfied:

(1+𝜆)𝑎.𝑏.𝑐.𝑘𝑖𝑤.

(𝜆+1)²(𝑎𝑤𝑛+𝑏.𝑐)² - ^{𝑎.𝑤.𝑘𝑖𝑤.(𝑎.𝑤𝑛+𝑏.𝑐).−|𝜆′(𝑤𝑛)|}

(𝜆+1)²(𝑎𝑤𝑛+𝑏.𝑐)² + ^{𝑘𝑐.−|.𝜆′(𝑤𝑛)|}

(𝜆+1)² > 0

1

(𝜆+1)²[(1+𝜆)𝑎.𝑏.𝑐.𝑘𝑖𝑤.

(𝑎𝑤𝑛+𝑏.𝑐)² + ^{𝑎.𝑤.𝑘𝑖𝑤.(𝑎.𝑤𝑛+𝑏.𝑐).|𝜆′(𝑤𝑛)|}

(𝑎𝑤𝑛+𝑏.𝑐)² ] > ^{𝑘𝑐|.𝜆′(𝑤𝑛)|}

(𝜆+1)²

The above condition is satisfied when i) the marginal propensity of consumption of the capitalist (𝑘_𝑐) = 0 and the marginal propensity of consumption of the industrial worker (𝑘_𝑖𝑤) = 1. This is an extreme case. It guarantees that there is at least one case where this condition is satisfied. On the other hand, if ii) 1 > 𝑘_𝑐 > 0 and 1 > 𝑘_𝑖𝑤 > 0 and the marginal propensity of the industrial workers (𝑘_𝑖𝑤) is sufficiently greater than 0 then also the effect of wage rate on output is positive. This is a likely case since workers being more poor would have a high marginal propensity of consumption for industrial goods, many of which are mass production goods.

Given the aforementioned assumptions two different cases follow from [6 − 11]. They are,

[6 − 13𝑎] 𝑌 = ℎ(𝑤_𝑛, 𝑎) for 𝜆 =constant [6 − 13𝑏] 𝑌 = 𝑘(𝑤_𝑛, 𝑎, 𝜆) for 𝜆 = 𝜆(𝑤_𝑛)

[6 − 14] ∆𝑌 = 𝜕𝑌

𝜕𝑤_𝑛 . ∆𝑤_𝑛+𝜕𝑌

𝜕𝑎 . ∆𝑎

𝐶𝑎𝑠𝑒 3: Change in 𝑌^∗ with 𝑎 and 𝑤_𝑛, for 𝜆 =constant

𝜕𝑌

𝜕𝑤𝑛 has been calculated for 𝜆 =constant above.

We partially differentiate [6 − 11𝑎] with respect to 𝑎 to get the comparative static result

[6 − 12𝑐] 𝜕𝑌⁸⁷

𝜕𝑎 = 𝐷_𝑜^′

∝². (𝜆 + 1). { 𝑘_𝑖𝑤. 𝑏𝑐𝑤_𝑛 (𝑎. 𝑤 + 𝑏. 𝑐)²} Thus, ^𝜕𝑌

𝜕𝑎> 0. This implies that when 𝑎 declines, that is when labour per unit of output falls or in other words labour productivity rises 𝑌 declines. The underlying intuition is that when 𝑎 declines, it impacts both 𝐷_𝑖𝑤 (demand side condition) and 𝑃 (supply side condition). 𝐷_𝑖𝑤 falls (demand curve moves from 𝐷₁ to 𝐷₂ ) as the size of the industrial workforce declines with declining 𝑎 since less labour is required per unit of output.

𝑃 (the supply curve shifts from 𝑆₁ to 𝑆₂ ) falls due to rising labour productivity (¹

𝑎) and the resultant declining wage cost per unit of output. A fall in price boosts real demand but that gets eclipsed with fall in 𝐷_𝑖𝑤 due to fall in size of the industrial workforce. The equilibrium output therefore falls from 𝑌^∗ to 𝑌₁, where 𝑌₁ < 𝑌^∗. As the impact of change in real demand due to change in size of the industrial workforce prevails over that that due to change in price with change in labour productivity we obtain ^𝜕𝑌

𝜕𝑎> 0.

For 𝜆 =constant, substituting [6 − 12𝑎] and [6 − 12𝑐] in [6 − 14] we get ∆𝑌 > 0, [6 − 14𝑎] ∆𝑌 = ^𝐷𝑜′

∝².(𝜆+1). { ^{𝑘𝑖𝑤.𝑎𝑏𝑐}

(𝑎.𝑤+𝑏.𝑐)²}. ∆𝑤 + ^𝐷𝑜′

∝².(𝜆+1). { ^{𝑘𝑖𝑤.𝑏𝑐𝑤}

(𝑎.𝑤+𝑏.𝑐)²}. ∆𝑎

as it follows from [6 − 12𝑎] and [6 − 12𝑐] respectively that the coefficients of ∆𝑤 and

∆𝑎 are positive. In other words, the short-run equilibrium output of the rural industrial sector increases with increase in the rural industrial wage rate and increase in labour per unit of output.

𝐶𝑎𝑠𝑒 4: Change in 𝑌^∗ with 𝑎 and 𝑤_𝑛, for 𝜆 = 𝜆(𝑤)

𝜕𝑌

𝜕𝑤𝑛 has been calculated for 𝜆 = 𝜆(𝑤_𝑛) above.

For 𝜆 = 𝜆(𝑤), substituting [6 − 12𝑏] and [6 − 12𝑐] in [6 − 14] we get,

[6 − 14𝑏]∆𝑌 = ^𝐷𝑜′

∝².{(1+𝜆)𝑎.𝑏.𝑐.𝑘𝑖𝑤−𝑎.𝑤.𝑘𝑖𝑤.(𝑎.𝑤+𝑏.𝑐).𝜆^′(𝑤).

{(𝜆+1)(𝑎𝑤+𝑏.𝑐)}² + ^{𝑘𝑐.𝜆}

′(𝑤) (𝜆+1)²}. ∆𝑤 + ^𝐷𝑜′

∝².(𝜆+1). { ^{𝑘𝑖𝑤.𝑏𝑐𝑤}

(𝑎.𝑤+𝑏.𝑐)²}. ∆𝑎

The coefficient of ∆𝑎 is positive refer ([6 − 12𝑐]) and the coefficient of ∆𝑤 is positive for the scenarios stated in case 2. Hence, it follows that ∆𝑌 > 0 i) when both the coefficients of ∆𝑤 and ∆𝑎 are positive. ii) when the coefficient of ∆𝑎 overshadows a negative coefficient of ∆𝑤⁸⁸. A summary of the results obtained so far are given in the Table 6-1 below.

A summary of the results obtained so far are given in Table 6-1 below.

Table 6-1 Results of Comparative Static Analysis: Short-Run Equilibrium Output

Case No. Case Description Result

1 Case 1:

Change in 𝑌^∗ with the change in 𝑤𝑛 for 𝜆=constant, 𝑎=constant

The short-run equilibrium output (𝑌^∗) increases with an increase in the wage rate of rural industrial workers(𝑤_𝑛) . Thus, ^𝜕𝑌∗

𝜕𝑤_𝑛> 0 𝜆 = constant, 𝑎 = constant

1 Case 2:

Change in 𝑌^∗ with change in 𝑤_𝑛 for 𝜆= 𝜆(𝑤),𝑎= constant

The short-run equilibrium output (𝑌^∗) increases with an increase in the wage rate of rural industrial workers (𝑤𝑛) for 𝜆= 𝜆(𝑤), if marginal propensity of the industrial worker (𝑘_𝑖𝑤) is sufficiently greater than 0.

2 Case 1:

Change in 𝑌^∗ with change in 𝑎 and change in 𝑤_𝑛 for 𝜆=constant

The short-run equilibrium output increases with increases in labour per unit of output (𝑎) and wage rate of rural industrial workers(𝑤_𝑛) , for 𝜆 = constant.

2 Case 2:

Change in 𝑌^∗ with change in 𝑎 and change in 𝑤𝑛 for 𝜆= 𝜆(𝑤)

The short-run equilibrium output increases with increases in labour per unit of output (𝑎) and wage rate of rural industrial workers(𝑤_𝑛) , for 𝜆= 𝜆(𝑤) for two conditions. i) when both ^𝜕𝑌

𝜕𝑤𝑛

and ^𝜕𝑌

𝜕𝑎 are positive ii)when ^𝜕𝑌

𝜕𝑤_𝑛< 0 but the positive effect of ^𝜕𝑌_𝜕𝑎 overshadows the negative effect of ^𝜕𝑌

𝜕𝑤_𝑛.

Empirically speaking we have found that short-run equilibrium output ( 𝑌^∗) of the rural industrial sector increased with a simultaneous i) increase in wage rate and ii) increase in labour productivity of rural industrial workers that is a decrease in labour per unit of

output. The empirical observations can be expressed as ∆𝑌 > 0 when ∆𝑤 > 0 and

∆ (¹

𝑎) > 0. To reconcile the empirical observations with the theoretical results we take a relook at [6 − 14𝑎]. The second empirical finding that the labour productivity is rising implies that the second term in the equation [6 − 14𝑎] has a negative bearing. We observe rising rural industrial output (𝑌^∗) with rising rural industrial wage rate (𝑤_𝑛) and falling labour per unit of output (𝑎), as the positive bearing of 𝑤_𝑛 (first term on R.H.S. of [6 − 14𝑎] ) surpasses the impact of 𝑎 (second term on R.H.S. of [6 − 14𝑎] )