The optimum tax under competitive market and diseconomies of scale

This note deals with optimum taxes under competitive markets and diseconomies of scale. The demand curve come from consumer theory and they can be Marshallian or Hicksian demand, they are in some cases equal (under quasilinear preferences).  We going to ignore the type of demand, the main issue concerned is the consumer welfare can be measure through consumer surplus. The market demand  of good j comes from summation of individual demands of each i which are equal for every consumer but not in income distribution, it means:

$Q^{d}(\bf{p},y)=\sum^{N}_{i=1}q_{i}(\bf{p},\alpha_{i}y)$

This market demand $Q^{d}(\bf{p}, y)$ does not change if income distribution changes see Jahle and Reny (2011) exercise 4.1 pag 188.  Now the point is to work with inverse demand, as this market demand function is onto and one to one for $p_{j}$, then its inverse exits. Ma-Colell, Whinston and Green (1995) deals this issue in detail pag 116 and Jahle and Reny (2011) pag 505.

Firms are in their long run production that means no fixed costs in their cost structure, then we can calculate the total cost (C) through marginal cost (MC) integration:

$C=\int \limits_0^{q_{j}}MC[s]ds$

As theory of firm points, the supply curve of firm h comes from $CM_{h}(q)=p_{j}$, then the market supply can be written:

$Q^{o}_{j}[\bf{p}]= \sum_{h=1}^{H}q_{h, j}(\bf{p})$

The above lines is an attempt to aggregate consumers consumption of good j and firms production of good j. Now with this market demand and market supply the tax t ($ per unit) has to be charged. Consumer and firms are interested in getting the maximum welfare, then they will maximize:

$SB^{*}=\underset{Q}{Max}[\int_0^{Q}[p^{d}[s]-p^{o}[s]]ds-tQ]$

The Firts Order Conditions (F.O.C) are: 

$\frac{dSB}{dQ}=0$, therefore,

$p^{d}[Q^{E}]- p^{o}[Q^{E}]=t$,       E.1

It means the society gets the maximum social welfare where the difference between consumers price and firms price is equal to tax "t".

To be sure that $Q^{E}$ is a maximum one takes the Second Order Conditions (S.O.C):

$\frac{d^{2}SB}{d^{2}Q}$,

and one gets

$[\frac{dp^{d}[Q]}{dQ} - \frac{dp^{o}[Q]}{ds}]<0$   $\forall {Q}$   as it is required for a maximum.

There is an interesting question about what is the "t" that maximizes the consumer and firms welfare?, in other words what is the tax that maximize the change in society welfare. To answer this question we follow few steps:

The solution above E.1 can be written as $Q^{E}=Q[t]$,

The initial state is with tax equal to zero, then we start in $Q^{E}=Q[0]$

The final state is $Q^{E}=Q[t^{*}]$ where we need to find the $t^{*}$ that makes the maximum positive change between these two states, then the optimisation problem is:

$SBT^{*}=\underset{t}{Max}[\int_{Q[0]}^{Q[t]}[p^{d}[s]-p^{o}[s]-t]ds$

It is fair to work on the last expression a bit. It is the maximum difference between social welfare under optimal tax $t^*$ and social welfare under tax equal to zero, in this case a negative difference, positive difference or null difference make sense. As the problem is solved under optimisation tools see Chiang, A (1992) pag 29-32, one gets:

$\frac{dSBT}{dt}=0$, therefore

$\frac{dQ}{dt}[p^{d}[Q[t^{*}]]-p^{o}[Q[t^{*}]]-t^{*}]-\int_{Q[0]}^{Q[t^{*}]}ds=0$

$\frac{dQ}{dt}\underline{[p^{d}[Q[t^{*}]]-p^{o}[Q[t^{*}]]-t^{*}]}-({Q[t^{*}]}-{Q[0]})=0$

The underline text is always zero no matter the tax value, see above problem E.1 and Post of "Maximum social surplus in competitive market - November 17 of 2012 wrote by me". Therefore,

${Q[t^{*}]}={Q[0]}$,

We got the best tax for society is $t^{*}=0$ due to loss of welfare under positive tax. Figure 1 shows the result.

Figure 1. Maximun social welfare under any tax and optimum tax

With this result the government is not agree due to she does not get any revenue, then she optimises her revenue $R[t]=tQ[t]$ no matter the social welfare value. Moreover, government knows that $\frac{dQ[t]}{dt}<0$, then

$\frac{dR[t]}{dt}=0$

$Q[t]+\frac{dQ[t]}{dt}t=0$

$[\frac{dQ[t]}{dt}][\frac{t}{Q[t]}]=-1$

Therefore the optimum tax for government is where tax elasticity of the volume is equal to -1. The existence of this maximum depends on volume function form.

Special case "The Laffer Curve"

The Laffer Curve is an interesting economic thought about the optimum tax see Laffer Curve. The next lines I try to set up this curve under above results.

Demand:

$P^{d}[Q]=a-bQ$   where $a>0$ and $b>0$.

Supply:

$P^{o}=\alpha Q$  where $\alpha >0$.

$P^{d}-P^{o}=t$

$a-bQ- \alpha Q =t$

The solution is

$Q[t]= \frac{a-t}{b+ \alpha}$   as one can see a must be greater than t, it is normal due to demand curve intercepts on P axis.

This optimum volume $Q[t]$ follows above results. Now the government revenue is:

$R[t]=t[\frac{a-t}{b+ \alpha}]$

But government optimizes for "t", then

$ \frac{dR[t]}{dt}=0$,

the solution is

$t^{*}= \frac{a}{2}$

This is where the tax elasticity of volume is equal to -1.

Now it is a maximum due to

$ \frac{d^{2}R[t]}{d^{2}t}<0$   $ \forall t$,   Figure 2 shows the result

Figure 2. Laffer curve a special case of optimum tax

The conclusion is a tax greater than 0 lets a lower optimum volume than tax equal to zero under diseconomies of scale, it means a positive tax lets getting Allocative Inefficiency. However, government has to get revenues from market transactions, therefore she looks for optimum tax that lets getting maximum revenue and she found an optimum tax where tax elasticity of volume equal to -1 but in some cases there is not solution, it depends on volume function characteristics. I showed a special case where the result is Laffer curve.

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