One way to generate disequilibrium prices in a simple model of supply and demand is to incorporate a lag into producer’s supply response. To examine this possibility, assume that quantity demanded in period t depends on price in that period (QDt= a − b Pt) but that quantity supplied depends on the previous period’s price – perhaps because farmers refer to that price in planting a crop (QSt= c+dPt−1).
a. What is the equilibrium price in this model (P*=Pt= Pt– 1) for all periods, t?
b. If P0 represents an initial price for this good to which suppliers respond, what will the value of P1 be?
c. By repeated substitution, develop a formula for any arbitrary Pt as a function of P0 and t.
d. Use your results from part (a) to restate the value of Pt as a function of P0, P* and t.
e. Under what conditions will Pt converge to P* as t →∞?
f. Graph your results for the case a= 4, b= 2, c=1, d= 1 and P0= 0. Use your graph to discuss the origin of the term cobweb model.