Answer to Question #212957 in Differential Equations for hunka

Question

The supply S and demand D for a particular commodity satisfy the equations

S(p)=100+p+p^t and D(p)=200-p-p^t

1. Find the equilibrium price p(t) at any time t
2. Find the long range equilibrium price
3. graph the particular solution p(0)=75 and p(0)=25 hence discuss the long term behaviour of the price of this commodity

Answer 1

1.- The equilibrium price can be found when S(p)=100+p+p^t and D(p)=200-p-p^t are both equal to the same value and then:

$S(p)= D(p) \\ 100+p+p^t =200-p-p^t \\ \implies p+p^t = 50$

Then, if we consider the last we find that:

$\implies p+p^t = 50 \\ 100+(p+p^t) =200-(p+p^t) \\ 100+50= 200-50 \\ S(p)= D(p)=150$

2.- We find first the average of the supply to then differentiate it and find the price:

$\overline{S}= S(p)/p=\frac{100}{p}+1+p^{t-1}=1+\frac{100+p^t}{p}$

$\dfrac{d\overline{S}}{dp}=\dfrac{p^t ( t-1)-100 }{p^2}=0$

$\implies p^t=\dfrac{ 100 }{( t-1)} \implies p=[\dfrac{ 100 }{( t-1)}]^{1/t}$

The only condition that has to be satisfied as well is that $t>1$.

3.- If we make the graphs by using the value of p(0) we find

• With p(0)=75 the supply is higher than the demand after several times.

• On the other hand, if we analyze p(0)=25 the supply converges with the demand in the first year and then its behavior is like in the prior example.