
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.

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Green is quantity supplied, red is quantity demanded for $\displaystyle t = 0 $.
For (b) set $\displaystyle qd = qp $ or $\displaystyle 182p = 5p10, p = 4 $. So $\displaystyle t = 0, \ p = 4 $.
For (c) set $\displaystyle 182p = 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.).


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

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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.
