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Cigarette taxes revenues and elasticity of demand

A few weeks ago the Nicholls Worth ran this article about Jindal’s plan to change the structure of taxation in Louisiana, moving away from income taxes and toward sales taxes.   More recently, more details of the governor’s tax proposals have been released, as we see in this article from the Baton Rouge Advocate.  In particular, notice that after fighting against keeping a 4-cent per pack temporary tax on cigarettes two years ago, Jindal has now proposed almost quadrupling the state’s cigarette tax to $1.41 per pack.  Of course, this time the increase in the cigarette tax is part of a plan to get rid of the income tax in the state.

A discussion of cigarette taxes and revenues seems to be especially in order, since we are now discussing something called the “price elasticity of demand” and the “income elasticity of demand” in two of my classes.  As it happens, I have done some research on estimating the effects of state cigarette taxes (for instance, see here and here).

Basic economic theory suggests that any unit tax (so many cents per unit, not based on selling price) on a specific good, as we see on cigarettes and beer, and we saw in our auction, the tax should either be passed on to buyers in higher prices, with sellers paying the rest of the tax in terms of the price received (after-tax price or price minus the tax).  Since the supply to any single state is a very small part of the total supply of cigarettes, state supply is completely flat and any state cigarette tax is completely passed on to buyers.

In Figure 1 we see the two types of demands noting that the X-axis measures the amount of cigarettes demanded in state j (where j could stand for any state) on a per capita basis, or total quantity demanded divided by the number of people in the state).  Demand 1 shows the demand for cigarettes in a state when prices in that state change, holding the prices of cigarettes in all other states constant, which is what happens when a state changes its tax rates while other states do not.  Demand 2 shows what happens when prices in all states increase the same, as is the case when the tax on cigarettes changes, affecting all states the same.   If the state tax on cigarettes increases, the supply curve shifts up by the amount of the tax, and so, with a flat supply curve, the supply shifts up in the price direction by the amount of the tax, resulting in the price changing (dP = change in price = tax) by the amount of the tax increase.

Figure 1.  Demand when prices rise together in all states and when prices rise in just one state

 fig 1

From Figure 1, using “d” to represent “change in,” b = inverse slope or dQ/dP of Demand 1 and a = inverse slope of Demand 2, or dQ/dP.  The coefficients “a” and “b” are estimated from data on cigarette, prices, cigarette taxes and cigarette sales state by state.  So, a state tax of the amount dP (dP stands here for change in price) will cause the quantity demanded to fall in the state by b*dP, where b is the change in quantity from a one-cent change in price along Demand 1 (what we can call the “state only” demand curve)—note that the change in Quantity Demanded, dQ = b*dP).    Based on Demand 2, where all states’ prices change together (the “all together” demand), we see that increases in cigarette prices because of the tax reduce consumption of cigarettes in the state by a*dP.  So the remaining reduction in cigarette sales in the state will be (b-a)*dP or s*dP.  This is the amount of sales reduced, not because people are cutting back on smoking, but because people are no longer buy cigarettes in this state, but buy them elsewhere.  Remember, the price increase of dP reduces the quantity demanded in the state only by a*dP.   So, we find the effect of a state tax hike in changing cross-state purchases by subtracting the slope coefficient a from the slope coefficient b.

Figure 2 shows the result of a federal or nation-wide cigarette tax increase from $.39 per pack to $1.01 per pack in 2009 as a movement along the blue demand curve, the all-together demand curve.  This all-together demand curve corresponds to Demand 2 in Figure 1.  If this increase in federal cigarette taxes of $.62 per pack is fully passed on to buyers in higher prices, the federal tax increase would boost the state average price for $4.08 to $4.70.  Here, the demand curve in blue, the “all-together” demand, corresponds to Demand 2 in Figure 1, the demand for cigarettes in the state when prices in all states go up together.  The two red demands show the demand when just this one state is raising its taxes, the state-only demand, corresponding to Demand 1 in Figure 1.  The lower state-only demand is the one for which the federal tax is only $.39 per pack, while the higher state-only demand is the one when the federal tax is $1.08  per pack.

Figure 2. Cigarette demand with elasticity estimates

fig 2

The state’s tax revenue gains from a state cigarette hike will be the amount of that tax hike times the number of units purchased after that tax hike, which is the rectangle labeled as the “Revenue rectangle.”  Notice that since the federal tax hike will reduce sales in the state, the same state tax hike generates less state revenue than before the federal tax hike.

The price elasticity of demand for the All-together demand is -0.296.  The price elasticity of demand for the state-only demand when state prices start at $4.08 is -1.085, but is -1.25 when the state price starts at $4.70.  The income elasticity of demand is estimated to be -0.58, meaning that across states, cigarettes are inferior goods.

Figure 3 shows the relationship between a given tax hike in Louisiana and the revenues gained by the state at different tax rate hikes.  Comparing the revenues to those in 2008, which we see as the red dotted line, the upper curve in blue shows the predicted gross revenues for cigarette taxes in Louisiana for various tax hikes (from along the horizontal axis).  Gross revenue is about total collections and has not taken the costs of collection and enforcement into effect.

Figure 3. Predicted Gross Revenues from Various State Cigarette Tax Hikes in Louisiana

fig 3

There are two points worth noting in Figure 3.  The first is that the revenues are predicted to grow with the size of the tax hikes, but that growth is at a decreasing rate.  We see that with blue and dashed green lines which have positive but falling slopes. The other point is that the state revenue gains from a given tax hike are lower after the federal tax hike than before it.

A last point to note is one that seems a contradiction from what we have seen with elasticities and firm revenues and elasticities and state tax revenues that we see here.  Notice from Figure 2, both state only demands are in the elastic range.  This means that an increase in price when demand is in the elastic range would result in lower revenues for sellers.  Yet, when taxes push prices up, they increase state tax revenues, as we see with the two Revenue rectangles.  Why is it that tax revenues go up even though the demand is elastic?  The first person to correctly answer this will get an extra half point for the answer and gets to earn a half point more from the blogs than other students.


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