In this framework, the level of tariff indicates the extent to which our small economy is open to the external trade. The lower the tariff on imports, more open the economy is. Thus if a ceteris paribus tariff reduction can be shown to raise the share of the public good in total expenditure of the economy, we can conclude that more open economies have larger governments. This is the thought experiment we carry out in this section.
Since we have assumed in (8) that imports are capital intensive relative to exports, by the price-magnification effect, a tariff reduction raises the money wage and lowers the rate of return to capital
Wb ¼ hKX j jhXY
bT; ^r ¼ hLX j jhXY
bT ð9Þ
where hat over a variable denote the proportional change; hij is the cost share of input-i in sector-j;j jhXY¼ hLXhKY hLYhKX[ 0; and bT dð1 þ tÞð1 þ tÞ.
The price of the public good, on the other hand, rises with the tariff reduction by our assumption that the public good is the most labour intensive or least capital intensive (so thathKX[ hKG)
bPG¼ hKX hKG
j jhXY
bT ð10Þ
With the domestic price of imports falling, this means a switch in the domestic demand in favour of the import good Y and away from the public good. Since output of the public good is demand-determined by virtue of it being a non-traded good, its production must therefore contract on account of this demand effect of a tariff reduction. There will be, in addition, a market-clearing effect on the pro-duction of the public good necessitated by the homothetic taste assumption. As evident from (5),
bG ¼ hð KY hKGÞeG
j jhXY
bT þ bY ð11Þ
where, eG¼ bGbY
bPGbPY
is the (absolute) value of the price elasticity of (relative) demand for the non-traded good.
Thefirst term captures the demand effect induced by the change in the relative price of the public good following the tariff reduction. The second term captures the market-clearing effect under the assumption of homothetic taste. Under such an assumption, the relative demand for the public good can change only when the relative price of it changes. Thus, if a tariff reduction raises the output of the import-competing good, then for any given relative price, the production of the public good must also increase proportionately to match the (relative) demand for the public
good and, therefore, to clear the domestic market for the non-traded good.4Note by the assumption that our economy is small, any change in production of the traded goods cannot affect the world prices and hence the price of the public good relative to the (domestic) price of imports.
Thus, to determine how the production of the public good changes with a tariff reduction, we need to know how the outputs of the traded private goods change. First of all, these output levels would change by the factor substitution effect. The increase in the wage level relative the rate of return to capital makes all lines of production relatively more capital intensive. With labour and capital fully employed, the output levels must, therefore, adjust. Had all the three goods been traded, the output levels would have changed only by these factor substitution effects. But with the public good being non-traded, there will be further changes in the output of the traded private goods. As spelled out above, for any given output level of the import-competing good, the production of the public good falls with the tariff reduction. The consequent release of some labour and capital previously employed in the government sector, therefore, increases the net endowment of these factors for the private sub-economy. This triggers the output magnification effects. However, given the three goods, the direction of change in the output levels triggered by the factor substitution and net endowment changes will not follow directly from the particular intensity ranking assumed in (8). Formally, as shown in the appendix
bY ¼ gkKX kj jXGðhKY hKGÞeG
4In case of non-homothetic tastes, this market-clearing effect would not have been there. Instead, the output of the public good would have been determined, in addition to the demand effect induced by change in price, by the demand effect consequent upon change in the real income.
The first term in the parenthesis of each of the expressions in (12) and (13) captures the factor substitution effect whereas the second term captures the demand-induced net endowment-change effect as spelled out above. First of all, given the intensity ranking in (8), the output of the import-competing good rises, bY [ 0, if,
kKX[ kLX ð15Þ
Given (15), both the factor substitution effect and the net endowment-change effect are favourable for the import-competing sector.5 The factor substitution effect, on the other hand, lowers the output of the export good X whereas the other effect is ambiguous.
In rest of our analysis, we shall assume (15) in addition to (8) to reduce the multiplicity of cases. Thus, given these assumptions, a tariff reduction raises the production of the import-competing good whereas that of the export good may fall.6 Finally, since the output of the import-competing good increases, bY[ 0, from (11) it appears that the market-clearing effect raises the production of the public good and thus works in the opposite direction of the demand effect.
Therefore,
Proposition 1 By (8) and (15), a tariff reduction ðbT\0Þ raises the volume of production of good G and the absolute size of the government.
Proof By (8) and (15),j jhXY[ 0 and hKG\hKY. Then, it is immediate from (14) that bG[ 0. This completes the first part of the proof.
For the second part, define the expenditure on good G as the absolute size of the government
S¼ PGG ð16Þ
Thus, by (8), bS¼ bPGþ bG [ 0.
Hence the claim. □
A tariff reduction affects the absolute size of the government in three ways. First is the price effect. Tariff reduction raises the price of the public good and thus the expenditure on it for any given consumption and output levels. The other two effects
5This result that a fall in tariff actually raises the domestic production of the import-competing good is similar to the well-known Metzler Paradox in the literature on international trade. This, however, arises due to the existence of a non-traded good which uses the same factors of pro-duction as do the traded goods.
6The imports, M, can still increase. Since M = Yd– Y, we can writebYd¼ qbY þ ð1 qÞ bM , where q ¼YYd. Since Ydincreases, even if the output Y increases faster, by the property of weighted average, we have bY[ bYd[ bM . Thus bM may still be positive.
are the adverse demand effect and the favourable market-clearing effect on the output of the public good as spelled out earlier. By assumptions in (8) and (15), it turns out that the market-clearing effect outweighs the adverse demand effect. The output of the public good, therefore, rises at the end. Consequently, the initial expansion of the absolute size of the government through the price effect gets magnified.
How does a tariff reduction change the relative size of the government?
Following the literature, the relative size is defined as the expenditure on public good as a proportion of the national income (or total expenditure)
SR¼ PGG
PGGþ PXXþ PYY ð17Þ The (proportional) change in the relative size of the government thus can be written as,
bSR¼ ð1 hGÞ½bPGþ bG hXbX hYbY ð18Þ wherehj; j ¼ G; X; Y; is the share of sector-j in national income and P
jhj¼ 1.
Substitution of values from (10), (12)–(14) yields,
bSR ¼
½ ðhf Xþ hYÞðhKX hKGÞkKXþ hXkKXdgðkKX kLXÞ þ hXg
þ ðhKY hKGÞ½hYðkKX kLXÞ hXðkKY kLYeG
ðkKX kLXÞkKXj jhXY
bT ð19Þ
Following Lemma is useful in determining the change in the relative size of the government
Lemma 1 By (8) and (15),kKY[ kLY. Proof: By (8),
kLX kKX
[ kLY kKY
which by (15) implies,
kKY kLY
[kKX kLX
[ 1 ð20Þ
Hence the claim. □
It is immediate from (19) and Lemma 1 that since a tariff reduction means bT \ 0, so bSR[ 0 if,
hX
hY [kKX kLX
kKY kLY ð21Þ
But ifhhX
Y is smaller than the critical value defined in (21), then the coefficient of eGin (19) is positive, in which case bSR[ 0 if,
eG\fðhXþ hYÞðhKX hKGÞ þ hXgðkKX kLXÞkKXþ hXg
ðhKY hKGÞ½hYðkKX kLXÞ hXðkKY kLYÞ ð22Þ Therefore,
Proposition 2 Given (8) and (15), a more open economy will have a larger size of the government if the ratio of share of the export sector to that of the import-competing sector in national income is sufficiently large in the sense defined in (21).
Otherwise, a smaller value of the price elasticity of demand ensures that a more open economy will have a larger government.
Proof: Follows from the above discussion. □
The reason for these conditions is simple. As we have spelled out, given our factor intensity assumptions a more open economy in the sense of imposing a lower tariff on imports, will have a larger absolute size of import-competing sector whereas may have a smaller absolute size of the export sector. The relative size of the government as defined in (17) above therefore increases if the import-competing sector has a smaller share in the national income of our small open economy in the sense defined in (21). But if the share of the import-competing sector is larger, the size of the private sub-economy is expanding. Thus for the relative size of the government to increase, we need that this increase in the (absolute) size of the private sub-economy is small enough. Since, it is the fall in the demand for the public good and consequent increase in the net availability of labour and capital that cause expansion of the production of both the private traded goods, we must have this demand effect smaller. This is in fact captured by the condition in (22).
Figure1below illustrates the results summarized in Propositions1and2, where eGis the critical value of the price elasticity defined in the right-hand-side of (22).
Finally, it is interesting to note that the critical value of the price elasticity defined in (22) is greater than unity if (see Appendix),
hKX[ hX
hXþ hYhKGþ hY
hXþ hYhKY hKX ð23Þ
ˆ >0 S
ˆR>0
S Sˆ <R 0
O εG* εG
Fig. 1 Size of the government whenhhX
Yis small
By (8) and the property of a weighted average, with weights adding up to unity, hKG\hKX\hKY. Thus hKX can be larger than hKX and still satisfy the assumed intensity ranking in (8). Of course, since (23) is just a sufficient condition, the critical value ofeGmay exceed unity even when ishKXsmaller. This is illustrated in Fig.2.
Hence, the size of the government can be larger in a more open economy even if the non-traded public good that it produces is price elastic in demand.
All these results together imply the following
Corollary 1 Given (8), (15) and (23), a more open economy has a larger size of the Government regardless of the value ofhhX
Y, if the demand for the public good it produces is price inelastic.
This can be explained as follows. Note that ifhhX
Y is large in the sense defined in (21), the relative size of the government expands regardless of how large is the price elasticity of demand for the non-traded public good. On the other hand, if hhX
Y is smaller, under (23), the relative size of the government expands if the public good that it produces is not highly price elastic in demand. These two results together imply that if the capital cost share in the export sector satisfies (23), and the public good is price-inelastic in demand, the relative size of the government increases regardless of whetherhhX
Y is large or small.