# Supply and Demand Functions with Tax

I've been given the below supply and demand functions:

$$q^s(p)=50p~~~~~~q^d(p)=100(\frac{12}{p}-1)$$

I've answered the first few questions, which include finding the equilibrium etc, and inverting the above:

$$p^s(q)=\frac{q}{50}~~~~~~p^d(q)=\frac{1200}{q+100}-1$$

Now it says the price is 1/unit and the government introduces a tax on the production of t per unit. It also says "after these changes, the demand function remains the same, but the new inverse supply function is $$p^s(q)= 1+t$$

The part I'm stuck on is where it asks for a function of the total tax collected $$T$$, in terms of $$t$$. I assume that $$~T=q^d*t$$, so do I just sub $$p=1+t$$ into my original demand equation for $$q^d$$ then multiply that by $$t$$ for the answer?

$$q^d(1+t)=100(\frac{12}{1+t}-1)=\frac{100(11-t)}{1+t}$$

$$T=q^d*t=\frac{100t(11-t)}{1+t}$$

• $p^d(q)$ should be $\frac{1200}{q+100}$ Mar 22, 2019 at 0:04
• Thanks, not sure how that made its way in there Mar 25, 2019 at 19:37