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So I have this question and answer to it which one part i'm stuck understanding, here's the question.

Given demand & supply functions as = 86 − 0.8 and = −10 + 0.2 −1 Find:

a) the market price in any time.

Here's the answer which I understand:

Equating demand and supply, we will have:

86 − 0.8 = −10 + 0.2 −1 ⟹ −0.8 = 0.2 −1 − 96

Or

= −. t-1 +

Using the iterative method, we have:

= 1 ⟹ 1 = (−. ) o +

= 2 ⟹ 2 = (−0.25)1 + 120 = −0.25(−0.25 o + 120) + 120

= 2 ⟹ 2 = (−. )^2.o + ( − . )

= 3 ⟹ 3 = (−. )^3.o + ( − . + (. )^2)

= ⟹ = (−. )^t.o + ( − . + ⋯ + (−)^(t-1) .^(t-1))

and then here comes the part which I'm stuck understanding:

We can re-write this as: = (o −(/( + . ))) (−. )^t +/(1+0.25)

How do we derive this?

1 Answers1

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I'll clear up the notation a little bit. We have a recursive equation $$ P_t = a+qP_{t-1} $$ where $a$ and $q$ are constants. In your case, $a>0$ and $q<0$. If we assume that we know $P_0$ (the initial value of $P$), then we can calculate $$ P_1 = a+qP_{0} $$ and of course $$ P_2 = a+qP_{1} = a+q(a+qP_{0}) = a + qa + q^2 P_0 = a(1+q) + q^2 P_0 $$ Going further, we have $$ P_3 = a(1+q+q^2) + q^3 P_0 $$ $$ P_3 = a(1+q+q^2+q^3) + q^4 P_0 $$ We can see a pattern forming. The expression in the parenthesis is a geometric sum. Let's denote $$ S(n) = 1 +q + q^2 + \ldots + q^n \tag{1} $$ then, multiplying both sides by $q$, we get $$ qS(n) = q +q^2 + q^3 + \ldots + q^n + q^{n+1} \tag{2} $$ Let's subtract equation ($2$) from equation ($1$). The result is $$ S(n)-qS(n) = 1-q^{n+1} $$ from which we can solve $S(n) = \frac{1-q^{n+1}}{1-q}$.

Returning to the original question, we can easily see that the answer is $$ P_n = a\frac{1-q^{n}}{1-q} + q^n P_0 $$

Matti P.
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