Consumer Welfare and Taxes PDF

8 Consumer Welfare and Taxes To this point, we have completed the basic theory of the consumer side of the economy. These notes introduce a couple real world applications to show how the formal tools we have introduced can be useful to answer simple policy questions. Concepts Cov...

8 Consumer Welfare and Taxes To this point, we have completed the basic theory of the consumer side of the economy. These notes introduce a couple real world applications to show how the formal tools we have introduced can be useful to answer simple policy questions. Concepts Covered Equivalent Variation Compensating Variation Taxes 8.1 Motivation and Setup In the last section, we looked at how a price change affects consumption. Now we want to answer a related question. When prices change, how does that affect consumer welfare? To answer that question, we need to understand how to measure consumer welfare. To this point, we have introduced the concept of utility as a measure of consumer happiness or satisfaction. However, looking at utility alone can only tell us the direction of changes in welfare (i.e. whether the consumer is better off or worse off) but not the magnitude (how much better or worse off). In these notes we will look at two measures that can help us determine the magnitude of changes in consumer welfare. To make things more concrete, we will be following a simple example using our familiar Cobb- Douglas utility function U (x, y) = x1/2 y 1/2. As we have seen, this function has Marshallian and Hicksian demands  1/2 ∗ 1 I ∗ py xm = xh = Ū 2 px px  1/2 ∗ 1I px ym = yh∗ = Ū 2 py py which lead to indirect utility and expenditure functions given by 1 V (px , py , I) = Ip−1/2 py−1/2 2 x E(px , py , Ū = 2Ū px1/2 py1/2 Assume that the consumer starts with an income of 120 and the price of x and y are each equal to 1. This leads to an initial consumption of x = 60, y = 60 and a utility of 60. In the following sections, we will attempt to evaluate the effects on consumer welfare of the price of x increasing to 4. 1 8.2 Equivalent Variation Definition The first method for evaluating changes in consumer welfare is known as Equivalent Variation. It measures how much the consumer’s income would need to change at the old prices (before the change) if they wanted to reach the same level of utility as they would have after the price change (at their original income). To see this idea more clearly, it may help to imagine a case where there was only one good. For example, imagine a consumer has an income of $100 and buys a single good that costs $5 (meaning they can buy 20 units). If the price of the good increased to $10, that would be equivalent to the consumer losing $50 of income (either way, they can now only buy 10 units). In other words, we could describe the increase in the price of the good as a decrease in the consumer’s real income. As we will see below, the way we will calculate equivalent variation will be to take the difference in the expenditure function at the old vs new utility levels, but holding prices fixed. Equivalent Variation: E(p1x , p1y , Ū1 ) − E(p1x , p1y , Ū2 ) Note that by the duality introduced earlier, the initial level of expenditure must also be equal to the initial income, so we can also write the equivalent variation as I1 − E(p1x , p1y , Ū2 ) Example Let’s calculate the equivalent variation for our Cobb-Douglas example above. First, let’s calculate how much utility the consumer will have at their original income at the new prices (after px has increased to 4. We can use the indirect utility function to calculate this utility 1 V (px = 4, py = 1, I = 120) = (120)(4)−1/2 1−1/2 = 30 2 Note that we could also calculate the optimal consumption from the Marshallian demand as x = 15, y = 60 and then plug this into the utility function to get the same number (the indirect utility function has already done this work for us). Now what we want to find is how much income the consumer would need to lose to only be able to reach a utility of 30 at the old prices (when px was still equal to 1. We assume the consumer will still spend their income optimally, meaning they will look for the cheapest possible way to reach 30 utility. Luckily, we already have the tools to calculate such a value. In particular, the expenditure function tells us how much the consumer would need to spend to reach a given utility level (which is the same as how much income they would need to reach that utility level. Let’s plug into the expenditure function to find this value E(1, 1, 30) = 2(30)11/2 11/2 = 60 which tells us that the consumer would need to spend at least 60 (have an income of 60) to reach a utility of 30. Comparing this to their initial income of 120, we can say that the equivalent variation is 60 and the increase in the price of x from 1 to 4 is equivalent to the consumer losing half of their income. 2 8.3 Compensating Variation Definition Another way to calculate changes in consumer welfare is known as Compensating Variation. While equivalent variation measures changes in real income, compensating variation looks at how much compensation a consumer would need to be at least as well off as before the change in prices. Compensating variation can be useful in calculating how much the government would need to provide to compensate for price changes. For example, if gas prices went up, compensating variation would show how much they could pay consumers to keep their utility from going down. The calculation for compensating variation looks similar to equivalent variation with a couple important differences. The equation is Compensating Variation: E(p2x , p2y , Ū1 ) − E(p1x , p1y , Ū1 ) Notice that in this case we are comparing expenditure at the new prices and the original utility. Once again, we can replace the initial expenditure with the initial income E(p2x , p2y , Ū1 ) − I1 Example Returning to our example above, recall that we started from a utility level of 60 with an income of 120. We saw in the last calculation that after the price change, utility would fall to 30 after the price of x increased. For compensating variation, we are trying to figure out how much to pay the consumer so that their utility stays at the original 60. Once again, we can use the expenditure function to find the cheapest possible way for the consumer to stay at 60 utility at the new prices E(4, 1, 60) = 2(60)(4)1/2 11/2 = 240 As before, we could also have calculated this in two steps. This time, we would need to calculate consumption while keeping utility constant, which is Hicksian demand. Hicksian demands for px = 4, py = 1, Ū = 60 come out to x = 30, y = 120 and we can see that this bundle costs 240. Since we already have the expenditure function we can do all of that in one step. The final step is then just to subtract off the initial income (120) to get a compensating variation of 120. Again, the interpretation here is that we would need to increase the consumer’s income by 120 in order for them to maintain the same utility after the price change as they had before the price change. Both equivalent and compensating variation can be useful in measuring changes in consumer welfare. In general, equivalent variation is more useful for comparing welfare at different points in time (similar to correcting for inflation) while compensating variation is more useful for policy purposes (as in the gas prices example above). 3 8.4 Taxes The tools we have introduced so far also easily extend to discussing taxes. We can look at taxes in two ways. A sales tax charges tax per unit purchased. In this case, consumers can change their tax burden by making different choices in which goods they purchase. On the other hand, an income tax takes taxes directly from the consumer’s income. Since our models so far do not have any way for consumers to choose their income, the income tax is a fixed amount regardless of the choices consumers make (this is usually called a “lump-sum” tax). Sales Tax A sales tax is a fee charged per unit of a good purchased. While this concept may seem entirely new, it is really only a slight variation on concepts we have already discussed. From the perspective of a consumer, a sales tax is simply an increase in the price of a good and all of the tools we have discussed regarding price changes also directly apply to sales taxes. Formally, we could define the sales tax in two ways. First, we could define a variable τ to represent a per unit amount of tax. In other words, if the price of x before tax is px , we could define the after tax price pxτ = px + τ For example, if px = 1 and τ = 3, then the price before the tax is 1 and the price after the tax is 1+3=4. Notice that this example is essentially no different than the abstract price increase discussed above. Again, from the perspective of the consumer, there is no difference whether the price increase was caused by general market forces or a tax so the analysis above remains unchanged and we could do all the same welfare analysis. One small difference with a tax is that it raises revenue for the government. In the example above, when px increased to 4, the consumer consumed 15 units of x. We can then calculate the revenue as R = x∗m (pxτ , py , I) ∗ τ = 15 ∗ 3 = 45. A second way to introduce a sales tax is as a percentage of sales. However, given our simple setup, we can always turn a percentage into an amount and work from there. For example, the tax of $3 per unit above could equivalently be described as a 300% tax. Income Tax As briefly described above, an income tax represents a lump sum taken out of a consumer’s income. Let’s use the variable T to represent the size of the tax so that the consumer’s income after tax can be written as IT = I − T This is simpler than a sales tax because again as discussed above, it is fixed regardless of consumer choice. No matter how much the consumer chooses to consume, it will always pay exactly T in taxes, and the government’s revenue will also be exactly T. Again, we could equivalently describe the income tax in percentage terms. For example, if initial income were 120, an income tax of T = 30 could also be an income tax of 25%. 4 Relationship Between Sales and Income Taxes To show an important relationship between sales and income taxes, let’s return to our running example with px = 1, py = 1, I = 120. We have already argued that a sales tax τ = 3 is equivalent to the analysis we did with an increase in price from 1 to 4. Therefore, we would still have a drop in the consumer’s utility from 60 to 30, which can be described in welfare terms as an equivalent variation of 60 or a compensating variation of 120. Also, as calculated above, the optimal consumption after the tax are x = 15, y = 60. One thing to note right away is that the revenue raised by the government in this case is not nearly enough to compensate the consumer for their lost utility. In other words, the government’s revenue (45) is less than the compensating variation (120). Now let’s compare this to an income tax that produces the same revenue. Since an income tax is just lump sum, we just have to set T = 45 as the size of the income tax to match the revenue from the sales tax. The after tax income then becomes IT = 75. Plugging into the indirect utility function, we can see that this income tax reduces utility to 1 V = (75)(1)−1/2 (1)−1/2 = 37.5 2 Even though the government revenue is the same in each case, the consumer prefers the income tax to the sales tax (because they get higher utility under the income tax). Why does this happen? Notice that the sales tax changes the price ratio, which affects the M RS = ppxy condition, while the income tax does not. In fact, under the income tax, the relative quantities of x and y don’t change (both fall proportionally from 60 to 37.5). In this sense, the sales tax is more distortionary than the income tax as it directly affects the consumer’s tradeoff between x and y in addition to reducing their real income. In general, in most applications in economics, lump sum transfers are less distortionary and create less inefficiency than taxes or subsidies on specific goods or services. However, there may still be reasons to prefer sales taxes over income taxes. For example, taxes on gasoline are typically imposed due to the externalities caused by driving (pollution, congestion, etc.). In this case, the market price does not reflect the true social cost of the good and the tax serves to correct the inefficiency rather than introduce a new inefficiency. Unfortunately, we do not have much time to spend on externalities or other market failures in this course - those are left for future courses to deal with. 5