ECON 361 Assignment 1 Solutions

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1 ECON 361 Assignment 1 Solutions Instructor: David Rosé Queen s University, Department of Economics Due: February 9th, [22] Poverty Measures: (a) [3] Consider a distribution of income over a sample of people: 5,000, 21,000, 45,000, 85,000,,000, 35,000, 115,000, 43,000, 37,000, 28,000. Let the poverty line be 24,000. Calculate the following: i. Headcount index. P R = Q N = 3 ii. Poverty gap index. P GI N y i d d y i d y i [( ) ( ) ( )], 000 5, , 000 = iii. Squared poverty gap index. SP GI y i d ( d yi N d = ) 2 ( ) 2 yi [ (24, ) 000, ( ) 5, ( ) ] 21, (b) [3] Consider a transfer from the person whose income is 21,000 of 5,000 to the person whose income is,000. Re-calculate the following given the new distribution of income: i. Headcount index. 1

2 P R = Q N = 3 ii. Poverty gap index. P GI N y i d d y i d y i [( ) ( ) ( )] 15, 000 5, , 000 = iii. Squared poverty gap index. SP GI y i d ( d yi N d = ) 2 ) 2 ( yi [ (24, ) , ( ) 5, ( ) ] 16, What, if any, poverty measurement axioms are violated in this example and by which indices? The axioms are focus, replication invariance, symmetry, monotonicity, transfer sensitivity, and continuity. In this example we have transferred income from the person with 21,000 to someone who has less income so we re looking at whether the above indices satisfy transfer sensitivity. In general, a poverty measure that satisfies the transfer sensitivity axiom should decrease when a transfer is made from one person in the distribution to another who is less well-off. In this example, the only measure that satisfies the transfer sensitivity axiom is the squared poverty gap index. Both the headcount rate and the poverty gap index remain unchanged. Also note, though, that the poverty gap index would have decreased had we transferred money from someone above the poverty line to someone below the poverty line (so long as the person transferring income remains above the poverty line). (c) [3] Now (returning to the original distribution from part (a)) imagine the person whose income is 5,000 receives a 1,000 increase in their income (perhaps they received a raise at work). Re-calculate the following given the new distribution of income: i. Headcount index. 2

3 P R = Q N = 3 ii. Poverty gap index. P GI N y i d d y i d y i [( ) ( ) ( )], 000 6, , 000 = iii. Squared poverty gap index. SP GI N y i d = ( d yi d ) 2 ) 2 ( yi [ (24, ) 000, ( ) 6, ( ) ] 21, What, if any, poverty measurement axioms are violated in this example and by which indices? This question deals with the monotonicity axiom. Here we see that the person with 5,000 has received an increase in income of 1,000 so that they now have 6,000. Any poverty index that satisfies monotonicity should decrease in this situation. We see that the both the poverty gap index and the squared poverty gap index satisfy monotonicity, although the headcount index does not. (d) [5] Based on parts (a) through (c) if you were a policy maker which of these three measures would you use to assess poverty in a given area? All poverty measures have upsides and downsides. In this example, the squared poverty gap index satisfies all the above axioms that we tested, but the downside of the squared poverty gap index is that it isn t intuitive and it is tedious to calculate. In practice, using a mix of all three measures is beneficial, as one measure can be appropriate where another is not. Or alternatively, we could use something like the SST index which combines various of these measures. (e) [8] Back to the original scenario in (a): 3

4 i. [6] Now imagine that the distribution of income changes over time, it is now the following: 8,000, 22,000, 45,000, 85,000, 11,000, 35,000, 115,000, 40,000, 37,000, 28,000. Hint: You can assume that ln(1 ĜP ) 0. A. Calculate the percentage change in the SST. Recall, Moreover, ln SST = ln P R ln P GI p ln(1 Ĝp ) % SST ln SST = ln SST t ln SST t 1 Using the hint and the equation for ln SST : % SST ln P R ln P GI p = (ln P R t ln P R t 1 ) (ln P GI p t ln P GIp t 1 ) We calculated P R t 1 in the previous question, to find and P GI p t 1, P R t and P GI p t : P GI p t 1 = 1 Q t 1 y i,t 1 d y i,t 1 24,000 d y i,t 1 d y i,t 1 3 [( ) (, 000 5, = ) ( )] 21, 000 P R t = Q N = 3 = 0.30 P GI p t Q y i d d y i d y i 3 [( 8, = ) Substituting in, we have ( ) % SST = ln(1/3) ln(1/3) = ( 22, 000 ) ( ) ln(0.431) ln(0.5) ( )] 11, 000 Thus, the SST decreased roughly 14.9 %. 4

5 B. What percentage of the change in the SST index is attributed to changes in the poverty rate and what percentage is attributed to changes in the poverty gap index applied to the poor? Since ln P R = 0 and assuming ln(1 ĜP ) 0, almost all of the change in the SST is driven by the ln P GI p. So roughly 0%. ii. [2] As a policy maker how would you target poverty reduction in this scenario? In this example the change in the SST index is driven entirely by the change in the poverty gap among the poor. Recall that the poverty gap measures the depth of poverty in a distribution. In this case it would be more beneficial to implement policies that target the people who are at the bottom of the distribution, or policies that aim to reduce poverty amongst the poor and not necessarily target people at or near the poverty line. 2. [15] Inequality Measures: (a) [5] Consider the Generalized Entropy Inequality Measures: GE(α) = [ 1 1 α(α 1) N N ( ) α yi 1] ȳ Show that lim α 0 yields the Theil-L index of inequality: L N N ( ) ȳ ln y i Using l Hopital s rule, as above, we have: f (0) N N ( ) yi ln ȳ And for the denominator: g (0) = 1 ( ) Dividing through by the denominator -1 and using the properties of logarithms (i.e. ln yi ȳ = ( ln ȳ y i )), we find that lim α 0 of GE(α) is: f (0) g (0) N N ( ) ȳ ln y i 5

6 (b) [] Briefly discuss the disadvantages of using the range, decile dispersion ratio, and Gini coefficients to measure inequality. What are the advantages of using a GE(α) measure of inequality? Range: disadvantage is that it relies on knowing the highest and lowest value in the income distribution which is not always realistic. Also, it ignores how income is distributed (other than what the two endpoints are). Decile Dispersion Ratio: ignores information about the middle of the distribution and also doesn t give us any information about how income is distributed within the top and bottom shares. Variance: if we double everyone s incomes the variance would quadruple even though the shape of the distribution would be unchanged. Gini coefficient: is not perfectly decomposable - we can t break it down into constituent parts that give us some idea of what s driving it. The main advantage of using a GE(α) measure of inequality is that they are perfectly decomposable so we can tell how much of total inequality is being driven by inequality between groups and how much is being driven by inequality within groups. 6

7 3. [30] Working with Aggregate Data: Access table from the CANSIM database and download market, after-tax and total income information for 2011 (both share of income and average income) for all income quintiles (use adjusted income to account for economies of scale) in Ontario. (a) [12] Using a computer or graph paper plot the Lorenz curves for market, after-tax and total income. Market Total After-tax Decile Inc Shr Cmltv Inc Shr Cmltv Inc Shr Cmltv 1 7, , , , , , , , , , , , , , , total 241, , ,800 i. In reference to these three distributions, what can be said about Lorenz dominance? The Lorenz curve for after-tax income lies completely above that for total income, which in turn lies completely above the Lorenz curve for market income. We can say that the after-tax Lorenz curve Lorenz dominates both the total and market Lorenz curves and furthermore the total income Lorenz curve Lorenz dominates the market income Lorenz curve. This implies that after-tax income is more equally distributed than both total income and market income, and that total income is more equally distributed than market income. 7

8 ii. Calculate the Gini coefficient for market, after-tax and total income. Recall that the Gini coefficient is calculated as: G = AB where A is the area between the Lorenz curve and the line of perfect equality and A B is the total area under the line of perfect equality. We can calculate this area using the Gini coefficient formula: G N (q i q i 1 )(p i p i 1 ) Here, recall that q i is the cumulative share of income and p i is the cumulative share of the population. Subbing in our cumulative values from the above table, the Gini coefficient for market income is: G M [( )(0.2 0) ( )( ) ( )( ) ( )( ) ( )(1 0.8)] = 0.40 By the same procedure, the total (G T ) and after-tax (G A ) Gini coefficient s are: G T = G A = A iii. Using the Gini coefficients you just obtained, calculate the Reynolds-Smolensky index of tax progressivety. RS = G T G A = = iv. Based on the set of calculations you just performed, comment on the redistributive properties of Canada s tax and transfer system. In the above calculations we see that G M > G T > G A. This means that once the government transfers income the level of inequality decreases. Once the effects of taxes are also taken into account, we see that inequality is reduced even further. This demonstrates the redistributive role that Canada s tax and transfer system plays. Both taxes and transfers help redistribute income to a more equitable outcome. Finally, the RS measure is positive implying that the tax system in place is progressive. As we discussed in class, the magnitude of the RS index tells us the redistribute potential of the tax. Here, we don t have any RS measure to compare to our calculated measure and therefore there is little that can be said about the particular tax s redistributive potential. 8

9 (b) [12] On a separate graph plot the Lorenz curve for pre-tax income (including transfers, i.e. total income). i. Plot the tax concentration curve in the same graph. First we need to calculate the cumulative share of taxes paid by each quintile: Decile Total After-tax Tax paid Share Cmltv 1 16,600 16, ,600 29,300 2, ,800 40,300 5, ,300 54,200, ,0 88,800 25, total 272, ,800 43,600 Then the tax concentration curve and the total income Lorenz curve are plotted as follows: ii. Now what can be said about Lorenz dominance? How does this apply to the discussion of tax progressivity/regressivity? In this plot the Lorenz curve for total income Lorenz dominates the tax concentration curve. The implication of this observation is that the tax incidence is more unequally distributed than the pre-tax income. Thus, the tax schedule is progressive, meaning that higher incomes are taxed at higher rates. iii. Is there anything you can add to the discussion in part 3, (a), (iv) based on the observations in part (b)? There isn t much that we can add to the overall discussion other than the fact that 9

10 the fact that total income Lorenz dominates the tax concentration curve, which also confirms that the Canadian tax system is progressive. (c) [6] How would your above discussion change if Canadian incomes were taxed proportionally for each income quintile? If we had a proportional tax on income, each bracket would be taxed at the same rate (e.g. everyone could be taxed at % of their income). The Reynolds-Smolensky measure would be 0, indicating the fact that the tax is proportional. In addition, if RS = 0 this tells us that the current-tax has redistributive potential. For the curious: Here I generate a very basic counter-factual with proportional tax, assuming no change in behaviour due to the new tax schedule.(it turns out that a proportional tax of 16% will generate almost the same amount of revenue as the actual tax schedule that we observe).

11 4. [] Working with Microdata: Download the Public Use Microdata File for the 2006 Canadian Census (follow instructions on the website). Note: This question should be done using Stata, an econometrics software available in the computer lab. The goal is to get you familiar with the tools that professional economists actually use for this type of analysis. Make sure to submit your stata do-file by . (a) Sample is properly trimmed. (b) Calculate the following measures of central tendency of pre-tax income for Canada, Ontario, and New Brunswick: i. Median: $38, (Canada), $41, (Ontario), and $31, (New Brunswick). ii. Mean: $29,000 (Canada), $30,000 (Ontario), and $24,000 (New Brunswick). (c) Plot the Lorenz Curves (for Ontario): i. Pre-tax total income ii. After-tax total income 11

12 (d) Calculate the Gini coefficient: i. Pre-tax total income G pre = ii. After-tax total income G after = (e) Briefly, comment on the redistributive impact Ontario s tax and transfer system. Since G pre > G after we can say that Ontario s pre-tax income is distributed more unequally than after-tax income. Although we can t say anything about the effect of transfers with the information at hand, this is evidence that in 2006 the tax system was progressive and resulted in a more equitable distibution of disposable income. 12