[-/1 Points] DETAILS MY NOTES
In this problem, is the price per unit in dollars and is the number of units.
If the demand and supply functions of a product are and , respectively, find the tax per unit that will maximize the tax revenue .
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To find the tax per unit that maximizes tax revenue , we first need to determine the equilibrium quantity and price without tax by setting the demand equal to the supply:

Given:

Combining terms, we get:

Rearranging gives us:

Using the quadratic formula :

(since we can’t have a negative quantity).

Now substituting back into either the demand or supply function to find :

With the equilibrium values, we can analyze the impact of tax on quantity . The new demand equation becomes:

The new supply equation (considering tax) becomes:

Setting these equal to find adjusted equilibrium, however, requires a deeper dive into differential revenue maximization which can get technical but essentially boils down to optimizing:

To maximize , further steps involving calculus would be needed, involving deriving the equation of total revenue with respect to . Therefore, numerical or computational methods might be optimal for precise determination of maximizing .

In general practice, the determination of taxes for maximizing revenue is an application seen in both public finance and economics, making this segment quite relevant in policymaking.

But for understanding real-world applications, think of how businesses adjust prices based not on only supply and demand, but to accommodate added costs like taxes—this ultimately can influence everything from consumer trends to corporate strategy. Taxes can be toolkits for moderation or incentives in various market dynamics.

If you’re curious about deeper learning, textbooks on microeconomics or public finance could dive into more real-world implications of tax policies on markets. Online platforms like Khan Academy or Coursera even offer courses specifically addressing economic principles, which can be both fun and enlightening!