MVDT: Simultaneous change in supply and demand.

[Bipolarity]

Consider S for supply, and D for demand, both of which are functions of price P.
By the law of supply, we know that
$$\frac{dS}{dP} > 0$$
By the law of demand, we know that
$$\frac{dD}{dP} < 0$$Suppose that

$$S(P_{1}) = D(P_{1})$$

This is the equilibrium point of supply and demand.

Now suppose that supply decreases by some positive amount $ε$.
Suppose that demand increases by the same amount $ε$.

Consider the new equilibrium, given the transformations above:

$$S(P_{2})-ε = D(P_{2})+ε$$

Prove that

$$S(P_{1}) = D(P_{1}) = S(P_{2})-ε = D(P_{2})+ε$$

In other words, prove that if supply decreases by some amount, and demand increases by the same amount, then the equilibrium quantity traded will be unaffected.

You should use the MVDT to prove this.

If it is not true, show a counterexample defining any function for S and D which satisfies the constraints of the problem.

BiP

 

[HallsofIvy]

I moved this since such things as "You should use MVDT to prove this" indicate it is homework.