Linking the main soil degradation problems to current management practices and possible

The goods and services produced by the government often generate positive externalities for other sectors or agents. Better roads or physical infrastructure, for example, improves factor productivity and reduce production costs for goods produced in the rest of the economy. Similarly, health care services are likely to improve productivity of workers. One might wonder then how the results derived above would have changed if such positive production externalities had been taken into account. To examine this in the simplest possible manner, suppose marginal productivities of labour and capital improves with larger production (and con-sumption) of good G, though not uniformly. That is, for any given wage-rental ratio, less number of workers and smaller amount of capital will be required to produce one unit of each good when the production of good G increases. The

*

ε > or < 1 G ε > 1 ^*G

θKG

*

θKX θKY θKX

Fig. 2 Capital cost share and the critical value of price elasticity

least-cost choice of input as specified will still be relevant but now that will reflect factor substitution for any given production (and consumption) of good G.

However, we assume that the productivity of workers and capital employed in sector G does not change. This may not necessarily be the case, but this assumption makes the algebra less cumbersome. To capture these external effects, we rewrite the input requirement condition (4) as

aij¼ aijðW=r; GÞ; i ¼ L; K; j ¼ X; Y ð4aÞ

a_iG¼ aiGðW=rÞ; i ¼ L; K ð4bÞ

The change in per unit requirement of labour in sector X, for example, then comes from two sources: change in the wage-rental ratio, and change in the volume of production of good G:

^aLX¼ gLXð bW  ^rÞ  eLXbG ð24Þ where,gLX  ^ðw=rÞ_{aLX @ðW=rÞ}^@aLX is the absolute factor price elasticity of labour demand per unit of output, and eLX _a^G_LX^@a_@G^LX [ 0 is the percentage decline in per unit labour requirement following one percent increase in the production of good G, which captures the positive externality in terms of gain in factor productivities. The changes in other factor coefficients can similarly be specified. To simplify matters further, we assume that productivity increase is uniform for all workers in which-ever private sector they are employed such that eLX ¼ eLY¼ eL. Similarly, we assume eKX ¼ eKY ¼ eK.

The positive externality captured this way produces two additional effects that are important for examining its impact on the size of the government. First is the scale effect. Since per unit labour and capital requirement now declines, at initial output levels of good X and Y, more labour and capital will now be available. Such increases in the effective supply of labour and capital lead to changes in the pro-duction of the three goods. At the initial wage-rental ratio, we have scale expansion or contraction. However, the direction of changes in the output levels depend on the magnitude of increases in labour productivity vis-à-vis that in capital productivity.

If labour productivity increases less than the capital productivity, for example, the effective relative supply of labour declines. For any given output level of good G, in the private sector of the economy we can then expect the production of relatively labour intensive good X to fall, similar to the output magnification effect at initial wage-rental ratio. Such fall in production of good X makes labour available in excess of what may be required to expand the production of good Y. This excess supply of labour in turn creates scope for an expansion of the production of good G.

The second additional effect is the technique effect induced by the positive externality generated by the production of good G. Productivity increase raises the money wage and the rate of return to capital proportionately. The wage-rental ratio

thus increases if labour productivity rises more than proportionately than the increase in capital productivity. Algebraically, as shown in the appendix,

Wb  ^r ¼ ðeL eKÞ bG  1

j jh_XY bT ð25Þ

The change in the wage-rental ration induces a technique effect and brings in further changes in the output levels. An increase in the wage-rental ratio, for example, substitutes labour by capital in all lines of production. The consequent availability of labour per unit of output expands the output of the relatively labour intensive good X vis-à-vis good Y, again similar to the output magnification effect.

The net availability of labour for production of good G thus falls and accordingly this good being the most labour intensive by assumption, the production of it declines. But if capital productivity increased more than the labour productivity, the production of good G would have increased.

What appears from the above discussion is that the positive externality generated by the production of good G as captured by the productivity effects causes an increase in the production of good G and an unambiguous expansion of the size of the government sector if labour productivity increase is not larger than the capital productivity increase. Note that the price of good G rises unambiguously since both the labour cost and the capital cost increase due to the externality effect.

Algebraically all these can be verified from the following expression (see appendix) bPG¼ hð LGeLþ hKGeKÞ bG hKX hKG

j jh_XY bT ð26Þ

bG ¼  g/kKXþ hð KY hKGÞ kj j_XYeG

ðkKX kLXÞ hj j_XY gðe L eKÞkKXþ wkKXþ ð/  1Þ kj j_XY j jh_XY

" #

bT ð27Þ where,w ¼ ðkLXþ kLYÞeL^{ðkKXþ k}_kKX^KYÞkLXeK and/ ¼ 1 þ ðhLGeLþ hKGeKÞ [ 1.

As shown in the appendix,w\0 for eL eK. Accordingly, given the assumption in (15),j jkXY[ 0 and hj jXY[ 0, an increase in the production of good G after tariff reduction is more likely when eL eK. Since the productivity increase raises the price of good G unambiguously, so in such a case the absolute size of the gov-ernment sector (as measured by PGG) rises as well.

The changes in the conditions for an increase in the relative size of the gov-ernment can similarly be worked out.

In sum, increased trade causing a larger size of the government sector is plau-sible even when the good produced by the government generates positive pro-duction externality by raising factor productivities. However, size expansion will be less if there had been no positive externality.

5 Conclusion

What we have established in this paper is that under homothetic taste, and thus even without the real income effect, a more open (small) economy may indeed have a larger size of the government even when it produces a non-traded public good. The absolute size of the government expands under a reasonable set of assumptions regarding factor intensity and employment shares of the goods. For a larger relative size of the government, we require an additional condition of either a higher share of the export sector in national income or a sufficiently small, although not nec-essary less than unity, value of the price elasticity of demand for the public good.

Appendix

1. Changes in the output levels.

From the full employment conditions (6) and (7) in the text, we can write:

k^LXðbX þ ^aLXÞ þ kLYðbY þ ^aLYÞ þ kLGð bG þ ^aLGÞ ¼ 0 ð28Þ

Solving for bX from (29) yields, bX ¼ ^a^KXk^KY

kKXbY þ ^a^KY

k^KG

kKXbG þ ^a^KG

ð30Þ

Substituting (30) in (28) we get:

kLX^aLXþ kLX ^aKXkKY

Using bG¼^ðhKY_{j j}^h_h ^KGÞeG

hKGÞrGand substituting values from (9) in the text in (33) yields the solution of bY : bY ¼  gk^KX kj jXGðhKY hKGÞeG

ðkKX kLXÞ hj jXY

 

bT

Proceeding as before, from (30) we get, bX ¼hLXð^aLX ^aKXÞ kKY

which after substitution of values from (9) and (12) in the text boils down to the solution of bX .

2. The condition fore^G[ 1.

By (22) in the text, the critical value of the price elasticity of demand is,

e^G¼ðhXþ hYÞðhKX hKGÞkKXd þ hX/kKXd þ hXc ðhKY hKGÞ½hYðkKX kLXÞ  hXðkKY kLYÞ Subtracting one from each side we get,

hXðhKX hKGÞðkKX kLXÞ þ hX/kKXd þ hXc þ

e^G 1 ¼hXðhKY hKGÞðkKY kLYÞ þ hYðhKX hKG hKYþ hKGÞðkKX kLXÞ ðhKY hKGÞ½hYðkKX kLXÞ  hXðkKY kLYÞ

¼kLXd½hX/ þ fðhXþ hYÞhKX hXhKG hYhKYg þ hXðhKY hKGÞðkKY kLYÞ þ hXc ðhKY hKGÞ½hYðkKX kLXÞ  hXðkKY kLYÞ

Therefore, a sufficient condition for e^G[ 1 is that,

hKX[ hX

hXþ hYhKGþ hY

hXþ hYhKY  h^KX

as stated in (23) in the text.

3. Price and Output changes under productivity effect.

Using^aLX ¼ gLXð bW ^rÞ  eLXbG as in (24) and similar expressions for other input coefficients, (31) now boils down to,

 kLXðgLXþ gKXÞ þ kLXkKY On the other hand, from the zero profit conditions we get,

bP^_X ¼ 0 ¼ hLXWb þ hKX^rþ h½ LXe_Lþ hKXe_K bG bPY ¼ bT ¼ hLYWb þ hKY^rþ h½ ^LYeLþ hKYeK bG From these two easy to solve the changes in factor prices as,

Wb ¼ eLbG  h^KX

Then from the zero profit condition for good Z we obtain, bPG¼ hLGWb þ hKG^r

which upon substitution of values from (35) boils down to, bPG¼ hð LGeLþ hKGeKÞ bG hLGhKX hKGhLX

From the market-clearing condition, on the other hand, we get,

eGðbPG bPYÞ ¼ bG  bY which upon substitution of values from (37) reduces to,

 eGðhLGe_Lþ hKGe_KÞ bG þe^Gðh^KX hKGÞ

Substitution of (36) and (38) in (34) yields,

 g ðeL eKÞ bG  1

Substitution of (39) in (38) then yields,

bG ¼  hg/ þ eGðhKY hKGÞ^{j j}_k^k_KX^XYi gðeL eKÞ þ w þ^{j j}_k^k_KX^XG þ^{/ k}_k^{j j}_KX^XY

h i

j jh_XY bT ) bG ¼  g/kKXþ eGðhKY hKGÞ kj j_XY gðeL eKÞkKXþ wkKXþ kj jXGþ / kj jXY

j jhXY

bT

) bG ¼  g/kKXþ eGðhKY hKGÞ kj j_XY j jk_XGþ kj j_XY

j jh_XYþ gðe L eKÞkKXþ wkKXþ þ ð/  1Þ kj j_XY j jh_XY bT bG ¼  gkKXþ hð KY hKGÞ kj jXYeG

ðkKX kLXÞ hj j_XY gðe L eKÞkKXþ wkKXþ ð/  1Þ kj j_XY j jh_XY

" #

bT ð40Þ which is as specified in (27).

Finally, to check the sign ofw, note that by definition,

w ¼ ðkLXþ kLYÞeLðkKXþ kKYÞkLX

k^KX eK

¼ ðkLXþ kLYÞ eLðkKXþ kKYÞkLX

ðkLXþ kLYÞkKX

eK

  ð41Þ

Now,^ðk_ðk^{KXþ kKYÞkLX}

LXþ kLYÞkKX 1 ¼^{kKXkLXþ k}_ðk^KY_LX^kLX_{þ k}^k_LY^KX_Þk^k_KX^LXkKXkLY

¼k^KYk^LX kKXk^LY ðkLXþ kLYÞkKX

which is positive by the factor intensity assumption. That is, the coefficient of eKin (41) is larger than unity so that w\0 for eL eK. For eL[ eK also w may be negative if e_L is not too large.

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