Deadweight loss of a tax Consider the standard problem of an excise tax, that is

Question

Deadweight loss of a tax

Consider the standard problem of an excise tax, that is, a tax t imposed on each unit of a good. To unify the notation, the tax is paid by the consumers, so p(t) denotes the price received by the producers and p(t) + t is the price paid by consumers. The inverse demand is P(q) and the industry marginal cost is C?(q).

a)Derive the formula for the deadweight loss of the tax as a function of the tax rate t.

b)Derive the change in DWL due to an increase in the tax rate t.

c)For policy analysis, we are frequently interested in the marginal welfare costs of taxes, that is, theincrease in DWL due to an increase in the tax revenue by 1 euro. Thus, derive the formula for themarginal welfare cost of the tax. (You need to derive the marginal change in revenue due an increaseintas a by-product.)

Explanation / Answer

Part-a)

The price recieved by the producers after the imposition of the excise tax is p(t) and the price paid by consumers is p(t)+t.Now let us assume that the quantity produced/supplied by the producers was Q0 before the imposition of the tax and it becomes Q1 after the imposition of the tax.Due to tax imposition by the government the supply will expectedly decrease as notice that the producer recieves a lower price now at p(t) and the tax rate t is collected by the government.

Hence,Deadweight Loss of the tax can be represented as:DWL=0.5*({p(t)-(p(t)+t)}*(Q0-Q1)

=0.5*(t)*(Q0-Q1)

=0.5(tQ0-tQ1)

=0.5tQ0-0.5tQ1

The above equation represents the deadweight loss due to excise tax in terms of the tax rate t.

Part-b)

We already derived:DWL=0.5tQ0-0.5tQ1

dDWL/dt=0.5Q0-0.5Q1

dDWL/dt=0.5(Q0-Q1)

Marginal change in DWL due to change in tax rate t is 0.5(Q0-Q1).Hence,any marginal increase in tax rate t would cause the DWL to change by 0.5(Q0-Q1) ammount.

Part-c)

Before the tax increase the tax rate was t and the price recieved by the producer was p(t).Now we assume that per unit of output produced/supplied by the producer is q and the total output produced by the producer is Q1.

Hence,Total Tax Revenue(TR0) obtained by the producer before tax increase:TR0=tQ1 or {p(t)-(p(t)+t}Q1

Now,following an increase in tax rate the tax revenue obtained by the government increases by 1 Euro.

Now,as the tax revenue increases by 1 Euro,the new Tax Revenue becomes=(TR0+1)

={p(t)-(p(t)+t}Q1+1

=Q0p(t)-Q0P(T)+tQ1+1

=tQ1+1

We already know that TR0=tQ1

dTR0/dt=Q1 or the marginal change in total tax revenue due to an increase in t is Q1.

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