A Primer on Tax Incidence Analysis (Part 2) — My wages are lower. But how much lower?

In part 1, I wanted to show that the burden of a tax is shared between the producer and the consumer no-matter who sends the check to the IRS. Once we know that there is a burden and that the burden is shared, it may be of interest as to how it is shared between the consumer and the producer.

Let’s return to our widget market. This time however, we will have slightly more general supply and demand curves. S = Es P where S is how may widgets producers are ready to produce at a price P and Es is the elasticity of supply. D = Cd + Ed P where D is the number of widgets people are ready to buy at a price P and Ed is the elasticity of demand and C is some constant.

I’m sure by now you’re wondering what I mean by elasticity. It’s actually a simple concept. The elasticity is just the change in quantity demanded or quantity supplied given a change in price. So for instance above, if the price goes up by $1, the producers will be willing to produce an extra Es widgets but the consumers will be willing to buy (negative) Ed fewer widgets. The elasticity of demand is negative because the higher the price is, the fewer widgets customers are willing to buy.

Let’s put some numbers to understand elasticity. Let’s say Es = 0.2, Ed =  – 2 and the price of widgets goes up by $10.

S’ = 0.2 (P + 10)

S’ = 0.2(10) + 0.2P

S’ = 2 + 0.2P

Producers will be willing to produce 2 more widgets.

D’ = C – 2 (P + 10)

D’ = C – 2P – 20

Consumers will be willing to purchase 20 less widgets.

So let’s find the equilibrium for the general case. We set S = D.

Es P = C + Ed P

C = Es P – Ed P

C = (Es – Ed) P

P = C / (Es – Ed)

S = Es P

S = Es C / (Es – Ed)

In other words, the equilibrium price is equal to some constant divided by the elasticity of supply minus the elasticity of demand. It makes intuitive sense. The elasticity of supply is how responsive producers are to prices. The more responsive producers are to prices, the less consumers will be able to demand a lower price without producers scaling back production. The elasticity of demand is how responsive consumers are to prices. The more responsive consumers are to prices, the less producers will be able to raise prices without loosing customers.

So now let’s put some numbers on this and calculate the consumer and supplier surplus in two different cases. In the first case, Es = 0.5 and Ed = – 2 and C = 100. We plug this in to find the price

P = 100 / (0.5 + 2) = 40

S = 0.5 * 40 = 20

In case one 20 widgets are produced and purchased at $40 each. Now let’s look at a different case. In the second case, Es = 1, Ed = – 0.5 and C = 100.

P = 100 / (1 + 0.5) = 67

S = 1 * 67 = 67

In case two, 67 widgets are sold at $67 each.

Now let’s add a $10 tax per widget to the first case above. Suppliers are still paid P but consumers now pay P+10. Es = 0.5 and Ed = – 2 and C = 100. So let’s reflect that in the demand curve and compute the new equilibrium and surplus.

100 – 2 (P + 10) = 0.5P

P = 32 ($8 less for the producer and $2 more for the consumer)

Let’s now add the same $10 per widget tax to case two where demand is much less elastic. Es = 1 and Ed = – 0.5 and C = 100.

100 – 0.5(10 + P) = P

P = 63 ($4 less for producer and $6 more for consumers)

Comparing the two cases, you can see the effect of the relative effects of elasticity of demand and supply. In the first case where demand is much more elastic than supply, producers are only able to pass on 20% of the tax as a price increase to consumers. On the other hand, when you lower the relative elasticity of demand in the second case, producers are able to pass on 60% of the tax as price increase.

This is actually a fairly intuitive result. If a small change in price scares off many potential customers without inducing much more product, producers are reluctant to raise the price even when a tax is imposed. On the other hand, if customers don’t react much to price changes compared to producers, producers will be able to pass on the tax to the customers without fear of losing much business.

So how might this apply in the labor market? It is of course difficult to say as there are many different labor markets and estimating elasticities is no easy feat. But as a first-order approximation, you can look at the relative negotiating power of your employer and yourself. If you and your colleagues feel confident that you can get a raise when you ask for it, it is likely that your employer is bearing more of the burden than you are. On the other hand, if you feel that your employer can dictate the terms of your employment, it is likely that your employer is passing more of the burden on to you.

[Note] I accidentally published this post before I intended to. I have decided that I would rather leave it up and update it later rather than confuse readers by having people wonder where the post went. Mostly, expect some graphs to be added tonight or tomorrow.

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2 Responses to A Primer on Tax Incidence Analysis (Part 2) — My wages are lower. But how much lower?

  1. Aceofwhat? says:

    Yeahhhh…so this is already one of my favorite blogs. It’s the dispassion, really. Anyone can do snark. But disagreeing without being disagreeable? A rare quality, one which eludes even the occasional Nobel laureate. (Found you through Mankiw’s kind props on his blog, of which I am an avid reader.)

    Keep up the great work. I promise to (usually) give you credit when I use your work as a platform to more eloquently express these topics to friends and acquaintances.

  2. Captcook says:

    I concur. Keep up the good work and thanks for sharing!

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