Indirect tax reform and the specification of demand: the case of Ireland

Abstract

This paper examines the sensitivity of marginal tax reform analysis to changes in the underlying demand system. In particular, we analyse the sensitivity of results from Ahmad and Stern’s (J Publ Econ 25(3):259–298, 1984) marginal tax reform model to different specifications of Deaton and Muellbauer’s (Am Econ Rev 70(3):312–326, 1980) Almost Ideal Demand System (AIDS) and Banks et al.’s (Rev Econ Stat 79(4):527–539, 1997) Quadratic AIDS. Using Irish Household Budget Survey data, we show that tax reform results exhibit a low degree of sensitivity to changes in the underlying demand system. An adjustment for a mass of observed zero-expenditures in the data for certain goods produces most sensitivity in the tax reform results. Even in these cases, many of the tax reform recommendations remain constant. Including demerit good arguments in the tax reform model can substantially alter the tax reform recommendations relating to demerit goods. Notably though, when we include these arguments in the tax reform model, the results are particularly insensitive to changes in the underlying demand system.

Appendices

Appendices

1.1 Appendix A: Derivation of marginal revenue cost expression

The production side of the model is straightforward. Assuming fixed producer prices and constant returns to scale, we have:

$$\begin{aligned} q=p + t \end{aligned}$$

(20)

where $q$ is the vector of consumer prices, $p$ is the fixed vector of producer prices, and $t$ is a vector of specific taxes. Given the simplifying assumptions about the production side of the economy, Eq. 20 shows that any tax increases will be fully reflected in consumer prices.³²

With consumer prices $q$, the demand of household $h, x^{h}(q)$, maximises utility $u_{h}(x^{h})$, subject to the household budget constraint. The indirect utility function, $v^{h}(q)$, then gives the maximum utility possible at prices $q$. Assuming incomes are fixed, we have a Bergson–Samuelson social welfare function which can be written as a function of prices:

$$\begin{aligned} V(q)=W\left( v^{1}(q),v^{2} (q),\ldots ,v^{H} (q)\right) \end{aligned}$$

(21)

We have aggregate demand vector $X(q)$ given by:

$$\begin{aligned} X(q)= \sum \limits _{h}x^{h}(q) \end{aligned}$$

(22)

and government tax revenue given by:

$$\begin{aligned} R=tX= \sum \limits _{i}t_{i}x_{i} \end{aligned}$$

(23)

Then, from Eq. 23 we have:

$$\begin{aligned} \frac{\partial {R}}{\partial {t}_{i}}=X_{i} +\sum \limits _{k}t_{k}\frac{\partial {X}_{k}}{\partial {t}_{i}} \end{aligned}$$

(24)

By Roy’s identity, we have:

$$\begin{aligned} \frac{\partial {v}^{h}}{\partial {q}_{i}}=-\alpha ^{h}x_{i}^{h} \end{aligned}$$

(25)

where $\alpha ^{h}$ is the private marginal utility of income. We can then say that:

$$\begin{aligned} \frac{\partial {V}}{\partial {t}_{i}}=-\sum \limits _{h}\beta ^{h}x_{i}^{h} \end{aligned}$$

(26)

where $\beta ^{h}=\alpha ^{h}\frac{\partial {W}}{\partial {u}^{h}}$ is the social marginal utility of income, or the welfare weight of household $h$.

In the original Ahmad and Stern methodology, tax reform recommendations were made using the $MSC$ rather than $MRC$ of taxation of a good, where the $MSC$ is defined as:

$$\begin{aligned} MSC_{i}=-\frac{{{\partial {V}}}/{\partial {t_{i}}}}{{\partial {R}}/{\partial {t_{i}}}} \end{aligned}$$

(27)

From Eqs. 26 and 24, we can estimate the $MSC$ of a euro raised through a change in the indirect tax on good i by:

$$\begin{aligned} MSC_{i}=\frac{\sum \nolimits _{h}\beta ^{h}x_{i}^{h}}{X_{i} +\sum \nolimits _{k}t_{k}(\frac{\partial {X}_{k}}{\partial {t}_{i}})} \end{aligned}$$

(28)

In order to deal with issues in this methodology caused by commodity-specific Laffer effects, where $\frac{\partial {R}}{\partial {t}_{i}}<0$ resulting in difficulties ranking $MSCs$, Madden (1995a) recommends using the marginal revenue cost ($MRC$). $MRC_{i}={1}/{MSC_{i}}$ is simply the inverse of Eq. 28. With some rearranging:

$$\begin{aligned} MRC_{i}=\frac{q_{i}X_{i}}{\sum \nolimits _{h}\beta ^{h}q_{i}x_{i}^{h}} +\frac{\sum \nolimits _{k}\tau _{k}q_{k}X_{k}\epsilon _{ki}}{\sum \nolimits _{h}\beta ^{h}q_{i}x_{i}^{h}} \end{aligned}$$

(29)

which is the expression in the paper.

Note that the same expression can be used to estimate the $MRCs$ in the case of proportional taxes($t^p$), rather than specific taxes ($t$). With proportional taxes, we have:

$$\begin{aligned} q=p(1+t^p) \end{aligned}$$

(30)

Government tax revenue then becomes:

$$\begin{aligned} R=t^ppX= \sum \limits _{i}t^{p}_{i}p_{i}x_{i} \end{aligned}$$

(31)

The expression for the $MRC$ can then be written as:

$$\begin{aligned} MRC_{i}=\frac{p_{i}X_{i}+\sum \nolimits _{k}t^{p}_{k}p_{k}(\frac{\partial {X}_{k}}{\partial {q}_{i}})p_{i}}{\sum \nolimits _{h}\beta ^{h}p_{i}x_{i}^{h}} \end{aligned}$$

(32)

Letting the $p_i$s cancel, and multiplying above and below by $q_i$, we have:

$$\begin{aligned} MRC_{i}=\frac{q_{i}X_{i}}{\sum \nolimits _{h}\beta ^{h}q_{i}x_{i}^{h}} +\frac{\sum \nolimits _{k}\tau _{k}q_{k}X_{k}\epsilon _{ki}}{\sum \nolimits _{h}\beta ^{h}q_{i}x_{i}^{h}} \end{aligned}$$

(33)

which, again, is the same expression as used in the paper. Madden (1995b) specifies the Ahmad and Stern tax reform model using proportional taxes when extending the model to include labour supply. He found that the ranking of $MRCs$ in Ireland showed little sensitivity to the inclusion of labour supply.

1.2 Appendix B: Summary statistics

The graph below shows how the price of each good varies across the sample period. Most price variation occurs between years rather than within years (Fig. 2).

Fig. 2

Price data used in the analysis, base Dec 2011 = 100

Table 9 shows the budget share of each good in each year of the analysis. Between 1987 and 2005, services and other goods increased its budget share by over 12 percentage points. The budget share of the other five goods decreased over the same time period. In 2009/2010, however, which was during a deep recession in Ireland, this trend is reversed. The share of services and other goods decreases for the first time in the sample period, while the share of food, tobacco, clothing and transport and fuel increases. The share of alcohol continues to decline. The pattern of budget shares for food is reflective of Engel’s law. The budget share for food was decreasing as average incomes were rising in Ireland. The trend was reversed in the 2009/2010, however, when Ireland was in deep recession.

Table 9 Budget shares (%) 1987–2009

In Sect. 2.3.2, we highlighted the importance of taking account of observed zero-expenditures when estimating elasticities. Table 10 shows this issue is prevalent in three of the six goods used in the analysis. There are also negligible levels of observed zero-expenditures for food and transport and fuel. However, these are less than half a percentage of total observations in each case, so we do not apply the ZA- correction for these goods. As mentioned above, we use Shonkwiler and Yen’s approach to account for the zero-expenditures in the data.

Table 10 Frequency of zero budget shares (%)

1.3 Appendix C: MRC rankings

	1987						1999						2009
	AIDS			QUAIDS			AIDS			QUAIDS			AIDS			QUAIDS
	NA	PS	ZA	NA	PS	ZA	NA	PS	ZA	NA	PS	ZA	NA	PS	ZA	NA	PS	ZA
$e=0$
Food	5	6	3	4	5	3	4	6	1	4	5	1	4	6	1	4	5	1
Alcohol	2	1	1	2	2	1	2	2	2	2	2	3	2	2	3	2	2	3
Tobacco	1	2	6	1	1	6	1	1	6	1	1	6	1	1	6	1	1	6
Clothing	4	4	5	5	4	5	5	4	5	5	4	4	5	4	4	5	4	4
T&F	3	3	4	3	3	4	3	3	3	3	3	2	3	3	2	3	3	2
Serv&Oth	6	5	2	6	6	2	6	5	4	6	6	5	6	5	5	6	6	5
$e=1$
Food	3	5	2	3	5	2	3	5	1	3	5	1	3	5	1	3	5	1
Alcohol	2	1	1	2	2	1	2	2	2	2	2	3	2	2	3	2	2	3
Tobacco	1	2	5	1	1	5	1	1	6	1	1	6	1	1	6	1	1	6
Clothing	5	4	6	5	4	6	5	4	4	5	4	4	5	4	4	5	4	4
T&F	4	3	4	4	3	4	4	3	3	4	3	2	4	3	2	4	3	2
Serv&Oth	6	6	3	6	6	3	6	6	5	6	6	5	6	6	5	6	6	5
$e=2$
Food	3	5	2	3	4	2	3	5	1	2	3	1	3	5	1	3	4	1
Alcohol	2	1	1	2	2	1	2	2	3	3	2	3	2	2	3	2	2	3
Tobacco	1	2	5	1	1	4	1	1	6	1	1	6	1	1	6	1	1	6
Clothing	5	4	6	5	5	6	5	4	5	5	5	4	5	4	4	5	5	4
T&F	4	3	4	4	3	5	4	3	2	4	4	2	4	3	2	4	3	2
Serv&Oth	6	6	3	6	6	3	6	6	4	6	6	5	6	6	5	6	6	5
$e=5$
Food	2	4	1	2	2	1	2	4	1	2	2	1	2	4	1	2	3	1
Alcohol	3	1	2	3	3	2	4	2	4	4	4	4	4	2	6	4	5	6
Tobacco	1	2	3	1	1	3	1	1	3	1	1	3	1	1	4	1	1	4
Clothing	5	5	6	5	5	6	6	5	6	6	6	6	5	5	5	5	4	5
T&F	4	3	5	4	4	5	3	3	2	3	3	2	3	3	2	3	2	2
Serv&Oth	6	6	4	6	6	4	5	6	5	5	5	5	6	6	3	6	6	3
Merit MRCs ($e=1$)
Food	4	5	3	4	4	5	3	5	1	2	4	1	5	5	1	3	4	1
Alcohol	1	1	1	1	1	2	1	2	2	1	2	2	2	2	2	1	3	3
Tobacco	5	4	6	5	5	1	5	4	6	5	5	6	6	4	6	5	5	6
Clothing	3	2	4	3	3	6	2	1	4	3	1	4	3	1	5	4	1	5
T&F	2	3	2	2	2	3	4	3	3	4	3	3	1	3	3	2	2	2
Serv&Oth	6	6	5	6	6	4	6	6	5	6	6	5	4	6	4	6	6	4

1. A value of 1 indicates that the good had the lowest ranking MRC, and a value of 6 indicates that the good had the highest ranking MRC
2. NA Nonadjusted demand system, PS demographically adjusted demand system, ZA zero-expenditure adjusted demand system