We would like to thank an anonymous referee for helpful comments and suggestions.

Original Article

# Capital and Consumption Tax Reforms in a Small Open Economy

Article first published online: 13 JAN 2014

DOI: 10.1111/roie.12093

© 2014 John Wiley & Sons Ltd

Additional Information

#### How to Cite

Chao, C.-C. and Yu, E. S. H. (2014), Capital and Consumption Tax Reforms in a Small Open Economy. Review of International Economics, 22: 1–12. doi: 10.1111/roie.12093

#### Publication History

- Issue published online: 13 JAN 2014
- Article first published online: 13 JAN 2014

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### Abstract

The effects of a reform in capital and consumption taxes on private welfare and government tax revenue are examined for a small open, capital-importing economy. A trade-off between private welfare and tax revenue is encountered in maximizing social welfare. Nonetheless, lowering capital taxes and raising consumption taxes can increase both private welfare and tax revenue if the initial tax rates are not optimal. In addition, a tax reform by this fashion is a likely response to a rise in the foreign rate of return on capital.

### Introduction

Tax levies have been used as policy instruments to raise government revenue to finance public projects and to promote economic growth and welfare. The effects of various taxes on an economy can be intricately linked. The purpose of this paper is to examine such possible relationships and the effects, upon welfare and government revenue, of changes in two kinds of tax rates, namely capital and consumption taxes. This paper is motivated by tax reforms instituted by a number of countries, in particular Singapore. As a result of the 1997 financial crisis, most Asian nations suffered economic woes. To revive the economy, Singapore's government implemented sweeping changes to its tax system in 2002 by cutting the capital tax rate from 24.5% to 22% and then further down to 20% in 2003. In addition, to ensure that the cuts in the capital tax rate be revenue neutral, the government raised the consumption tax on goods and services from 3% to 5%.1

We consider a small open economy, e.g. Singapore, which produces both private and public goods. The provision of public goods is totally financed by tax revenue. Private sectors compete with the public sector for resources. It is notable that a tax reform to attract foreign capital by lowering the capital tax rate can result in a tightened government budget and hence less provision of public goods. In contrast, raising the consumption tax can boost government revenue, but this is possibly achieved at the expense of consumer surplus and private welfare. In designing tax reforms, governments often encounter the issue of a trade-off between gaining tax revenue and promoting the private welfare of the citizens. In this regard, several lines of inquiry can be pursued: What would be the optimal combination of tax rates on corporations and consumers to maximize private welfare? What would be the optimal tax rates to maximize tax revenue? How should a government adjust the tax rates if the existing tax rates are different from the optimal rates? What would be the changes in the optimal tax rates when external shocks occur? We will examine all these issues in the present paper.

It is instructive to have a succinct review of the literature. There have been substantial studies on tax coordination and reforms. For example, Michael et al. (1993) discuss the reform of consumption taxes and import tariffs for a small open economy. They argue that since a reduction in tariffs, which is welfare improving, renders a loss in revenue, increasing consumption taxes to make up the loss is desirable. This finding echoes the optimal taxation principle by Diamond and Mirrlees (1971), which states that a small open economy should eliminate all tariffs for the sake of production efficiency, while enacting destination-based consumption taxes for the sake of revenue.2 Hence, a tax reform, consisting of a precise way of replacing tariffs by consumption taxes, turns out to be not only welfare improving but also revenue-enhancing.

There are also extensive studies on corporate taxes and international capital flows. Using a general equilibrium framework, Neary (1993) shows that for a small open economy without any other distortions, higher capital taxes result in lower welfare by impeding foreign capital inflows. Thus, the first-best policy in the absence of any sorts of distortions is zero tariffs and zero capital taxes to facilitate free trade in commodities and free international mobility of factors. However, as indicated by Hatzipanayotou and Michael (1993), positive rather than zero capital taxes can be warranted if tax revenue is needed for financing the provision of public goods.

Analytically, both consumption and capital taxes can exert an impact on resource allocation and welfare by influencing the behavior of consumers and firms. The issues regarding the implications of the simultaneous imposition of the two taxes and the coordination of the tax instruments to improve welfare have been largely neglected. This paper thus aims to investigate the effects of overhauling these taxes on the economy. We will identify the adjustment paths for any given mix of tax rates to move toward the optimal rates, which maximize private welfare or government revenue. We then discuss the impacts of external shocks on the optimal tax rates for the economy.

The paper is organized as follows. Section 2 presents a three-sector, general-equilibrium model with inflows of foreign capital. Section 3 examines the welfare and tax-revenue effects of changes in capital and consumer taxes and derives the optimal tax combinations for attaining maximum welfare or tax revenue. Section 4 discusses the implications of the tax reform for welfare or revenue maximization. Section 5 concludes the paper with some remarks.

### The Model

Consider a small open, capital-importing economy that produces three kinds of goods: two private goods, manufacturing *X* and agricultural *Y*, and a public good, *g*. The production functions for the private goods are: *X* = *X*(*L _{X}*,

*S*,

*K*) and

*Y*=

*Y*(

*L*,

_{Y}*V*), where

*L*denotes workers employed in sector

_{i}*i*,

*S*and

*K*are factory space and foreign capital utilized specifically by sector

*X*, and

*V*is the land used in sector

*Y*only.3 The production of the public good requires labor only and we assume, for simplicity, that one unit of labor produces one unit of the public good, i.e.

*g*=

*L*. Note that in this simple framework, labor is the only factor used by all three sectors. The labor constraint for the economy is:

_{g}*L*+

_{X}*L*+

_{Y}*L*=

_{g}*L*, where

*L*is the endowment of labor. The wage received by workers in the private sectors is denoted by

*w*, but the public-sector wage rate is institutionally set at , which is higher than

*w*. The private sector is thus like a sponge that absorbs the remaining workers who are not employed in the public sector.4

We assume that private goods *X* and *Y* are traded internationally but the public good *g* is not. We choose agricultural good *Y* as the numeraire and denote the world price of good *X* by *p*. The value of the production of private goods in the economy can be represented by the revenue function: *R*(*p*, *K*, *g*) = max{*pX*(*L _{X}*,

*S*,

*K*) +

*Y*(

*L*,

_{Y}*V*):

*L*+

_{X}*L*=

_{Y}*L*−

*g*}, with respect to

*L*. Letting subscripts indicate partial derivatives, the gross rate of return on foreign capital is

_{i}*R*. A capital tax,

_{K}*τ*, is levied on firms in the manufacturing sector.5 Foreign capital flows in until its after-tax rate of return equals the rate in the world market (

*r**); that is,

*R*−

_{K}*τ*=

*r**. In addition, from the revenue function, −

*R*=

_{g}*w*> 0, the shadow price of the public good,6 and

*R*= −

_{gK}*∂w*/

*∂K*< 0, as a rise in

*K*promotes the production of good

*X*, inducing more demand for labor, e.g.

*∂w*/

*∂K*> 0.

Consider next the demand side. Consumers demand private goods *C _{X}* and

*C*and a public good

_{Y}*g*. For simplicity, a consumption tax

*t*is imposed on the purchase of good

*X*only.7 For a given amount of the public good

*g*, the minimum expenditure on the private goods to attain a given level of utility

*u*is:

*E*(

*p*+

*t*,

*g*,

*u*) = min{(

*p*+

*t*)

*C*+

_{X}*C*:

_{Y}*u*=

*U*(

*C*,

_{X}*C*,

_{Y}*g*)} with respect to

*C*, where

_{i}*U*(·) is the utility function. Note that −

*E*> 0, capturing the marginal willingness to pay for the public good.8

_{g}Using the foregoing demand and supply information, we obtain the following budget constraint for the domestic economy:

- (1)

Equation (1) states that total spending on the private goods equals total revenue from the production of private and public goods, minus the payments to foreign capital.

The government finances the production of the public good with the tax revenue collected from foreign capital and the consumption of good *X*:

- (2)

where *E _{p}* (=

*C*) denotes the compensated demand for good

_{X}*X*with

*E*(=

_{pp}*∂C*/

_{X}*∂p*) < 0.

Consider next the capital market. Since capital is internationally mobile, as stated earlier, equilibrium requires equality between the after-tax rates of return in the domestic and foreign countries:

- (3)

where *R _{KK}* (=

*∂R*/

_{K}*∂K*) < 0, expressing diminishing marginal product of capital. Note that

*R*(=

_{Kg}*∂R*/

_{K}*∂g*) is the effect of the change in the production of the public good on the rate of return to capital in the home country. An increase in the production of

*g*reduces the availability of labor in the private sectors, thereby lowering the productivity of capital (i.e.

*R*< 0). Furthermore, we have

_{Kg}*R*=

_{gK}*R*.9

_{Kg}The model, described in (1)–(3), possesses three unknowns, *u*, *K* and *g*, with two tax instrument variables, *τ* and *t*, and three parameters, *p*, and *r**. We will use this model to examine the effects of changes in each of the two taxes, capital tax and consumption tax, on the private welfare of the residents and the tax revenue of the government. We will also derive the welfare-maximizing and revenue-maximizing optimal tax rates, and then consider external shocks on these rates.

### Analysis

It is instructive to derive at the outset an expression for changes in welfare. Totally differentiating (1) yields the change in the private welfare of residents:

- (4)

where *E _{u}* (=

*∂E*/

*∂u*> 0) denotes the inverse of the marginal utility of income. Define

*B*=

_{g}*E*−

_{g}*R*as the gap between the marginal willingness to pay for the public good and its shadow price. Following Lahiri and Raimondos-Møller (1998), we assume

_{g}*B*< 0.

_{g}The welfare expression in (4) captures three domestic distortionary effects, arising from the consumption tax, the capital tax and the public good, and an external effect from the foreign rate of return on capital. Changes in taxes on consumption and capital affect consumer and producer surplus, thereby influencing private welfare. Furthermore, the consequent changes in tax revenue cause a secondary effect on the financing of the public good. This also affects private welfare. In addition, a rise in *r** lowers private welfare as a result of more payments to foreign capital.

As for the changes in the provision of the public good, we can differentiate (2) to obtain:

- (5)

where *E _{pg}* =

*∂E*/

_{p}*∂g*< 0, assuming that

*X*and

*g*are substitutes in consumption, and

*E*=

_{pu}*∂E*/

_{p}*∂u*> 0, indicating that good

*X*is normal in consumption. The first two terms on the right-hand side of (5) reveal the tax revenue effect on the supply of the public good, while the third term captures the demand effect via the income change.

Finally, the change in the inflow of foreign capital can be ascertained from (3) as

- (6)

where recalling that *R _{KK}* < 0 and

*R*< 0. Equation (6) is interpreted as follows: A hike in the capital tax raises the cost of capital, reducing demand. In addition, an increased production of the public good increases the demand for labor by the public good sector and hence less labor for the manufacturing sector. This leads to a lower demand for capital. Moreover, the inflow of foreign capital decreases when the foreign rate of return on capital rises. It is notable that the consumption tax has no direct effect on foreign capital.

_{Kg}#### Effects on Private Welfare

We examine fist the welfare effect of taxes on foreign capital. From (4)–(6), we obtain:

- (7)

where *D* > 0, as shown in the Appendix. As mentioned earlier in (5), a rise in the capital tax rate reduces the producer surplus, but the increase in the tax revenue can finance more provision of the public good. Hence, in general, the welfare effect of the capital tax rate in (7) is ambiguous. Nonetheless, by setting *du*/*dτ* = 0, we can solve for the optimal capital tax, denoted by *τ _{u}* as

- (8)

Clearly, the optimal capital tax depends on the consumption tax. Recall that *E _{pg}* < 0, the larger the consumption tax

*t*, the smaller the optimal capital tax

*τ*. This relation can be depicted in the (

_{u}*τ*,

*t*) space in Figure 1, where the

*τ*schedule is negatively sloped. Noted that

_{u}*τ*shows the set of

_{u}*τ*that gives maximum welfare for alternative values of

*t*. This can be verified by checking the curvature of the welfare function. Following the technique used by Neary (1993), we substitute

*τ*in (8) into (7) to yield:

_{u}- (7′)

Apparently, *du*/*dτ* < (>) 0 as *τ* > (<) *τ _{u}*. This implies that the utility function is concave with respect to the capital tax. That is, a reduction (rise) in the capital tax improves welfare when

*τ*> (<)

*τ*. In Figure 1, any horizontal adjustments in

_{u}*τ*toward

*τ*improve the private welfare of the residents.

_{u}Analogously, using (4)–(6), we can solve for the welfare effect of changes in the consumption tax as

- (9)

A rise in the consumption tax rate lowers consumer surplus but the higher tax rate may raise tax revenue for the public good provision. These two conflicting forces of a consumption tax again render an indeterminate effect on welfare. The optimal consumption tax *t _{u}* can be obtained by setting in (9)

*du*/

*dt*= 0 as

- (10)

Similar to the earlier case of *τ _{u}*,

*t*is dependent on

_{u}*τ*. Noting that −

*R*/

_{Kg}*R*< 0, (10) reveals a negative relation between

_{KK}*t*and

_{u}*τ*. We plot the

*t*schedule in Figure 1, in which any vertical movement of

_{u}*t*towards

*t*is welfare improving.10 This can be seen by substituting

_{u}*t*in (10) into (9) to obtain:

_{u}- (9′)

It is immediate that *du*/*dt* < (>) 0 as *t* > (<) *t _{u}*, and hence the welfare function is concave with respect to the optimal consumption tax

*t*.

_{u}Consider next the welfare effects if both capital and consumption tax rates are allowed to change simultaneously. Solving (8) and (10), the jointly optimal tax rates, denoted by *τ ^{u}* and

*t*are:

^{u}- (11)

- (12)

where . The optimal capital and consumption taxes, *τ ^{u}* and

*t*, in (11) and (12) are both positive for the purpose of financing the public good provision. Note that, from (11) and (12), the relationship between

^{u}*τ*and

^{u}*t*is:

^{u}- (13)

Converting this relation into the elasticity form, we have:

- (13′)

where *ε _{X}* = −(

*p*+

*t*)

^{u}*E*/

_{pp}*E*and

_{p}*ε*= −(

_{K}*∂K*/

*∂R*)[(

_{K}*r** +

*τ*)/

^{u}*K*] are respectively the price elasticities of demand for good

*X*and for foreign capital. It is notable that each optimal tax rate and its corresponding price elasticity have an inverse relationship; it is desirable for the government to levy a higher tax, if the corresponding demand function is less elastic.

Using (11) and (12), the welfare changes caused by adjustments in capital and consumption taxes in (7) and (9) can be expressed as

- (14)

This implies that altering the tax rates toward *τ ^{u}* and

*t*improve welfare, as illustrated in Figure 1. The jointly optimal tax rates,

^{u}*τ*and

^{u}*t*, are represented by point B, and an iso-welfare contour is depicted by the ellipse U. Adjustments from, say, point A to point B are welfare improving.

^{u}We summarize the above discussion in the following proposition:

Proposition 1. *For a small open economy providing a public good, maximizing private welfare requires positive rates of both capital and consumption taxes. In addition, the optimal capital and consumption tax rates depend inversely on the price elasticities of demand for capital and good X, respectively.*

#### Effects on Tax Revenue

Turn next to the effects of changes in tax rates on tax revenue. Since tax revenue is used only for the provision of the public good, we can focus on the tax impact on the production of the public good. Consider first the effect of the capital tax on public good production. From (4)–(6), we obtain:

- (15)

A hike in the capital tax rate directly increases tax revenue. However, the higher capital tax dampens the inflow of foreign capital, thereby lowering production and income and hence the demand for good *X*. This results in smaller consumption tax revenue for financing *g*. These conflicting forces render the production of the public good indeterminate in (15). However, the maximum level of *g* can be obtained by setting *dg*/*dτ* = 0 in (15) as

- (16)

Recalling *R _{KK}* < 0, the

*τ*schedule is plotted as negatively sloped in Figure 1. We can also show that the locus of the capital tax

_{g}*τ*yields a maximum amount of revenue (financing a maximum provision of public good). This can be seen by substituting

_{g}*τ*into (15):

_{g}- (17)

Since *D* > 0, we have *dg*/*dτ* > (<) 0 as *τ* < (>) *τ _{g}*. This implies that a horizontal movement of

*τ*toward

*τ*in Figure 1 increases tax revenue.

_{g}Similarly, from (4)–(6), we can obtain the revenue effect of the consumption tax:

- (18)

A consumption tax hike raises tax revenue, but the fall in the demand for good *X* reduces tax revenue. The maximum amount of revenue is obtained from (18) by setting *dg*/*dt* = 0:

- (19)

which is positive because *E _{pp}* < 0. Note that

*t*in (19) is independent of

_{g}*τ*and hence

*t*is plotted as a horizontal schedule in Figure 1, in which a vertical movement of

_{g}*t*toward

*t*raises tax revenue.

_{g}Solving (16) and (19), we obtain the jointly “optimal” capital and consumption tax rates in the sense of generating the maximum tax revenue for the government:

- (20)

- (21)

In Figure 1, the values of *τ ^{g}* and

*t*are depicted by point L and a representative iso-revenue contour is represented by the ellipse T. Note that both

^{g}*τ*and

^{g}*t*are positive in (20) and (21), and dividing (18) by (20), we obtain:

^{g}- (22)

It is interesting to note that this revenue maximizing tax relation is identical to that obtained in (13) for the case of welfare maximization, but evaluated at different values of the tax rates. The relation can be represented by the line OL in Figure 1, showing the ratios of revenue-maximizing tax rates and welfare-maximizing tax rates.

### Tax Overhauls

Two cases have been examined in section 3: (i) for attaining the maximum level of welfare, the jointly optimal taxes are *τ ^{u}* and

*t*; and (ii) for raising the maximum amount of tax revenue, the jointly optimal taxes are

^{u}*τ*and

^{g}*t*. However, because of different base points, these two sets of taxes are not directly comparable. Nevertheless, an indirect comparison of them can be made. In Figure 1, the welfare maximizing tax rates at point B gives

^{g}*du*= 0 in (4). An increase in either the capital or consumption tax rate from point B raises government revenue, i.e. or from (4). This implies that

*τ*>

^{g}*τ*and

^{u}*t*>

^{g}*t*; the revenue maximizing tax rates, denoted at point L, are larger than the welfare maximizing tax rates at point B.11 The following proposition is immediate:

^{u}Proposition 2. *For the economy using capital and consumption taxes to finance public good production, the tax rates for revenue maximization are larger than the tax rates for welfare maximization.*

Niskanen (1977) treats the government just like an individual agent that has its own utility. The government is concerned not only with private welfare but also its own tax revenue. In this vein, we specify social welfare *W* as a weighted average of both private welfare and government revenue:

- (23)

where *b*, between 0 and 1, represents the weight for private utility. The government is said to be benevolent (Leviathan), if *b* = 1 (0). Both are special cases of a Niskanen government.

Suppose the initial situation is denoted by point A in Figure 1, in which the capital tax rate is large and the consumption tax rate is small. The utility level at point A is U and tax revenue is T. Consider several scenarios: If the government simply wants to maintain the same amount of tax revenue, the tax rates can be adjusted in the direction of AG along the dashed ellipse. In this case, private welfare can be improved but cannot be maximized. Furthermore, if the government is benevolent, the tax rates will be adjusted from point A toward point B. This implies that not only private welfare will be improved but also tax revenue will be increased. On the contrary, if the government is Leviathan, the tax rates are adjusted solely for maximizing tax revenue along line AL, where private welfare cannot be maximized (compared with point B). For the general case of a Niskanen government, all the interior points between points B and L, say N, are possible targets for tax-rate adjustments. The exact location of point N is determined by the value of *b*. The tax rate will be adjusted from point A to N, thereby yielding an equilibrium consisting of optimal mix of capital and consumption tax rates located between the two boundary points B and L.

Next, what would happen to the optimal tax rates when an external shock, such as a rise in the foreign rate of return on capital, occurs? Differentiating (11), (12), (20) and (21) with respect to *r** and ignoring the third-order partials, we obtain:

- (24)

- (25)

- (26)

- (27)

where , , *M* = *E _{p}E_{pu}* −

*E*, and

_{pp}*S*> 0.12 The tax responses to

*r*

^{*}depend on the changes in private welfare, tax revenue and the inflow of capital, which can be solved from (4)–(6) as

- (28)

- (29)

- (30)

A rise in *r** immediately raises cost of capital and hence reduces the inflow of foreign capital. Both factors contribute to a lower private welfare in (28), causing less demand for good *X*. As tax revenue falls, the provision of the public good declines in (29). This increases the demand for *X* since *X* and *g* are substitutes in consumption (*E _{pg}* < 0). Because of these two conflicting forces, the expression of

*E*(

_{pu}*du*/

*dr**) +

*E*(

_{pg}*dg*/

*dr**) in (25)–(27) is ambiguous. However, for quasi-linear preference [

*U*(

*C*,

_{X}*C*,

_{Y}*g*) =

*v*(

*C*,

_{X}*g*) +

*C*], the income effect falls entirely on the demand for

_{Y}*Y*. This gives

*E*= 0 and hence

_{pu}*E*(

_{pu}*du*/

*dr**) +

*E*(

_{pg}*dg*/

*dr**) > 0.13 The higher demand for

*X*renders larger consumption taxes in (25) and (27).

In addition, the inflow of foreign capital affects the optimal tax rates in (24)–(27). As indicated in (6), a rise in *r** directly reduces the inflow of capital. This lowers tax revenue and hence the provision of the public good *g*. However, the lowered production of *g* releases workers to the private sector, which raises the productivity of capital. This attracts foreign capital to the home economy. These two opposite reasons make *dK*/*dr** in (30) indeterminate. Nonetheless, for *E _{pu}* = 0 under the quasi-linear preference, we have

*dK*/

*dr** < 0. This suggests that to allure foreign capital, smaller capital taxes in (24) and (26) are necessary.

In sum, if the income effect of the demand for *X* is small (i.e. *E _{pu}* − 0), it follows that

*dτ*/

^{u}*dr** < 0 and

*dt*/

^{u}*dr** > 0 in (24) and (25) and

*dτ*/

^{g}*dr** < 0 and

*dt*/

^{g}*dr** > 0 in (26) and (27). This result is illustrated in Figure 2, in which points B and L and hence point N shift counter-clockwise to points B′, L′ and N′. Therefore, lowering the capital tax and raising the consumption tax are a likely response to the rise in the foreign rate of return on capital.

We summarize the above discussion, as follows:

Proposition 3. *A Niskanen government takes into account a trade-off between private welfare and government tax revenue in maximizing social welfare. In face of a rise in the foreign rate of return on capital, a tax overhaul by lowering capital tax rates and raising consumption tax rates warrants for the economy with a small income effect on the taxable good.*

This finding provides a theoretical explanation for the tax reforms, implemented by some Asian nations, by moving the taxes from direct capital taxes to indirect consumption taxes.

### Conclusions

This paper has explored the implications of changes in capital and consumption taxes for a small open, capital-importing economy, in which the government collects tax revenue to provide public goods. It is shown that if the government has a Niskanen utility function, allowing a trade-off between private welfare and tax revenue in maximizing social welfare, lowering capital taxes and raising consumption taxes simultaneously can increase both private welfare and government revenue if the initial tax rates are not optimal. In addition, a tax overhaul by lowering the capital taxes and raising the consumption taxes is a likely response to the rise in the foreign rate of return on capital. This result sheds some light on the rationale behind the recent tax overhaul scheme adopted by Singapore.

### Appendix

#### Stability

We assume that the inflow of foreign capital to the domestic economy is based on:

where *α* is a positive coefficient and *δ* [*= R*_{K}(*p*, *K*, *g*) − (*r** + *τ*)] is the differential on capital return between domestic and foreign countries. By (2), *g* is function of *K* and hence *δ* is a function of *K*. Linearly approximating the adjustment equation gives:

where *K*^{e} denotes the equilibrium *K*. The stability condition requires *dδ*/*dK* < 0. From (4)–(6), we obtain:

where . Hence, stability needs *E*_{u}*E*_{pg} < *E*_{pu}(*B*_{g} + *KR*_{Kg}) < 0, so that *D* > 0 and hence *dδ*/*dK* < 0.

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- 1
Singapore's number 1 rival in the region is Hong Kong, which levies a corporate tax rate of 16% but does not tax goods and services. However, owing to the structural changes in the Hong Kong economy, its government budget has dipped into a deficit since 2001. To tackle the deficit, the government has considered broadening the current tax base. See

*South China Morning Post*(4 May 2002), a newspaper published in Hong Kong. - 2
See also in Michael and Hatzipanayotou (2001).

- 3
Domestic capital

*K*can be introduced in addition to foreign capital in the model. Then we can simply rewrite:_{d}*K*=*K*+_{d}*K*, where_{f}*K*is foreign capital. However, the appearance of_{f}*K*does not yield new results. So, we omit_{d}*K*for simplicity. See Khan (1982), Neary (1995), Oda (2008) and Oda and Shimomura (2012) for factor mobility and welfare. Also see Mah (2010) and Zhao and Zhang (2010) for empirical studies of foreign capital inflows on economic growth._{d} - 4
Beladi and Oladi (2007) consider another case that the public good is produced privately and then supplied free of charge by the government. Hence, the public–private wage differential does not exist.

- 5
See, for example, Batra and Beladi (1993).

- 6
Because the shadow price of the public good

*w*is smaller than the cost of producing it, , production inefficiency occurs in the public sector. - 7
It is assumed that the numeraire good

*Y*does not bear a consumption tax. - 8
See Michael and Hatzipanayotou (2001) for a discussion.

- 9
Using the revenue function, we have

*R*=_{g}*pX*(_{L}*dL*/_{X}*dg*) +*Y*(_{L}*dL*/_{Y}*dg*) = −*pX*and_{L}*R*=_{K}*pX*+_{K}*pX*(_{L}*dL*/_{X}*dK*) +*Y*(_{L}*dL*/_{Y}*dK*) =*pX*. These give:_{K}*R*= −_{gK}*pX*(_{LL}*dL*/d_{X}*K*) −*pX*and_{LK}*R*=_{Kg}*pX*(_{KL}*dL*/_{X}*dg*), where*dL*/_{X}*dK*=*pX*/(_{LK}*pX*+_{LL}*Y*) and_{LL}*dL*/_{X}*dg*= −*Y*/(_{LL}*pX*+_{LL}*Y*) obtained from the first-order conditions of the revenue function. Hence,_{LL}*R*=_{gK}*R*= −_{Kg}*pX*/(_{LK}Y_{LL}*pX*+_{LL}*Y*)._{LL} - 10
As shown in the Appendix, the slope of the

*τ*schedule is steeper than that of the_{u}*t*schedule._{u} - 11
- 12
- 13
The first-order conditions for maximizing the expenditure, (

*p*+*t*)*C*+_{X}*C*, subject to the utility constraint,_{Y}*u*=*v*(*C*,_{X}*g*) +*C*, are:_{Y}*p*+*t*=*v*(_{X}*C*,_{X}*g*) and*u*=*v*(*C*,_{X}*g*) +*C*. These yield:_{Y}*C*=_{X}*C*(_{X}*p*+*t*,*g*) and*C*=_{Y}*C*(_{Y}*p*+*t*,*g*,*u*).