Stuggling with this one: In a simple economic model, time t is set up as a discrete parameter (possibly referring to days or months), while demand $D_{t}$ and supply $S_t$ are described in terms of the price $p_{t-1}$ and $p_{t}$ as follows: since prices influence demand "instantaneously", one puts $D_{t}=\gamma-\delta p_{t}$, but one sets $S_{t}=(p_{t-1}-\alpha)/\beta$ because the production of new commodities consumes time. Assume that all parameters $\alpha, \beta, \gamma, \delta$ are positive, and set $p_{0}=1.$
1) Dervive and solve the difference equation describing economic equilibrium, i.e the situation when $D_{t}=S_{t}$.
$\Rightarrow$ $\gamma-\delta p_{t}= \frac {p_{t-1}-\alpha}{\beta}$ $\Rightarrow$ $\gamma(\beta)-\delta(\beta)\ p_{t}=p_{t-1}-\alpha$ $\Rightarrow$ $p_{t}=\frac {-p_{t-1}+\alpha+\gamma\beta}{-\delta\beta}$
Not sure if this is on the correct lines?
2)Compute the steady states $p^{*}$ and determine a condition on the parameters $\alpha, \beta, \gamma, \delta$ that ensures stabilty of $p^{*}$.
Steady states $p^{*}$:
$p^{*}=\frac {-p^{*}=\alpha+\gamma\beta}{-\delta\beta}$ $\Rightarrow$ $p^{*}-p^{*}(\delta\beta)=\alpha+\gamma\beta$ $\Rightarrow$ $p^{*}=\frac{\alpha+\gamma\beta}{1-\delta\beta}$
Then we want $\alpha, \beta, \gamma, \delta$ that ensures stabilty:
So we call $f(p)= \frac {-p+\alpha+\gamma\beta}{-\delta\beta}$ $\Rightarrow$ $f^\prime= \frac{1}{\delta\beta}$ as being stable $\frac{1}{\delta\beta}<1$ $\Rightarrow$ $1<\delta\beta$.
3) Assume t is measured in days and $p_{t}$ in Pounds sterling. Setting $\alpha=1, \beta=\frac{3}{2}, \gamma=2, \delta=\frac {4}{5}$, how long does it take until $p_{t}$ stays within a penny of $p^{*}$?
Not sure if I'm going down the right lines here, any help will be much appreciated, many thanks in advance.