1. ## demand and supply

Demand and supply for a commodity are given by

qd = 18 - 2p and qs = -10 + 5p (1 - t)

respectively where t is a tax levied on each unit sold (to be paid by the producer).

a) Draw the supply and demand curves on the same diagram when t = 0

b) Find the equilibrium quantity sold when t = 0

c) Find the equilibrium quantity sold when t = 0.32 (i.e a tax rate of 32%).

d) Derive a functional relationship between p and t in equilibrium.

2. Green is quantity supplied, red is quantity demanded for $\displaystyle t = 0$.

For (b) set $\displaystyle qd = qp$ or $\displaystyle 18-2p = 5p-10, p = 4$. So $\displaystyle t = 0, \ p = 4$.

For (c) set $\displaystyle 18-2p = -10 + 3.4p$. So $\displaystyle t = 0.32, \ p = 5.\overline{185}$.

For (d) use (b) and (c) to try and find a functional relationship between $\displaystyle p$ and $\displaystyle t$ (i.e. is it a linear equation $\displaystyle y = ax+b$, a quadratic $\displaystyle y = ax^2 + bx +c$ etc.).

3. Linear?

4. So is that all I'll need to write? That the functional relationship is linear?

5. Actually I think its quadratic now that I have plotted a few more points. We have to be careful about overfitting/ extrapolating data.

$\displaystyle t = 0, \ p = 4$

$\displaystyle t = 1, \ p = 14$

$\displaystyle t = 2, p \approx - 9.4$

So the functional form is $\displaystyle p = at^2 + bt + c$ where $\displaystyle a,b,c$ are constants.

6. hmm... interesting.