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Suppose the government imposes a tax on each unit of output produced of $t. What is the firm’s new profit maximizing level of output?

The question suggests the use of comparative to determine how a change in tax affects a change in output. The firm would operate under perfect competition conditions.

I started to answer it this way.

Before Tax:
π = revenue - costs
π = pq - C(q)
dπ/dq = p - C'(q)

Set dπ/dq for profit-maximisation
0 = p - C'(q)
p = C'(q)
p = MC

After Tax:
π = revenue - costs - tax
π = pq - C(q) -tq
dπ/dq = p - t - C'(q)

Set dπ/dq for profit-maximisation
0 = p - t - C'(q)
p = t + C'(q)
p = t + MC

Next I would try to find out how quantity changes with tax t by taking derivates? Any ideas?

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you can do this using comparative statics and envelope theorem. The profit function specification was correct given the assumptions, let me just rewrite it as:

$$\pi (q,t) = p q - C(q) - tq$$

However, given the above we also know that the optimal quantity will depend also on a tax rate, so we will have $q^* = q(t)$.

Now by envelope theorem:

$$\frac{\pi^*(t)}{dt} = \pi'_2(q^*(t),t) = - Q^*(t) $$

So the optimal level of output given the tax $t$ will depend inversely on change in profit when the tax rate changes or $-\pi'_2(q^*(t),t) = Q^*(t)$.

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Intuitive/graphical answer:

A tax per unit of production basically increases marginal cost of production at each level of production. Effectively, it shifts MC curve up by $t$ units. Now we know that in a competitive market, firms are price takers (so horizontal demand curve). So they'll produce till the point MC touches $P$. Given that MC is increasing$^\dagger$, an upward shift in in MC curve means decrease in output. More the shift more the decrease in output.

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$^\dagger:$ In a perfect competition, decreasing MC is not possible because there is infinite demand for each firm - so firm will always operate in increasing MC part

Mathematically: You have already found the FOC, which states that at tax rate $t$, output supplied $q^*(t)$ will satisfy the following condition:

\begin{align} C'(q^*(t))+t-P&=0 \\ \end{align}

Differentiating this equation w.r.t $t$: \begin{align} \frac{dC'(q^*(t))}{dt}+1&=0 \\ \frac{\partial C'(q^*)}{\partial q^*} \cdot \frac{dq^*}{dt}&=-1 \\ \frac{dq^*}{dt}&=-1/C''(q^*)<0 \end{align}

The last step uses the SOC:

$$\frac{d^2\pi}{dq^2}=-C''(q)<0$$

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