G G Market Equilibrium: Stability and Time Paths Courtesy of James E. Foster, George Washington University A perfectly competitive market is made up of many buyers and sellers of an economic product, each of whom has no control over the market price. In this model, the overall quantity demanded by the buyers of the product is taken to be a function of the price of the product (among other things) called the demand function. Similarly, the overall quantity supplied by the sellers of the product is a function of the price of the product (among other things) called the supply function. A market is in equilibrium at a price where the quantity demanded is just equal to the quantity supplied. The linear model assumes that the demand and supply functions have the form qa = do – dp and q, = –so + sip, respectively, where p is the market price of the product, qą is the associated quantity demanded, q, is the associated quantity supplied, and do, d1, so, and s1 are all positive constants. The functional forms ensure that the “laws" of downward sloping demand and upward sloping supply are being satisfied. It is easy to show that the equilibrium price is p* = (do + so)/(d1 + s1). Economists typically assume that markets are in equilibrium and justify this assumption with the help of stability arguments. For example, consider the simple price adjustment equation dp A(qa – q,) , dt where A > 0 is a constant indicating the speed of adjustments. This follows the intuitive require- ment that price rises when demand exceeds supply and falls when supply exceeds demand. The market equilibrium is said to be globally stable if, for every initial price level p(0), the price adjustment path p(t) satisfies p(t) → p° as t → ∞. (a) Find the price adjustment path: Substitute the expressions for qa and q, into the price adjustment equation and show that the solution to the resulting differential equation is p(t) = [p(0) – p* ]ed + p*, where c = -X(d, + s1). (b) Is the market equilibrium globally stable? Now consider a model that takes into account the expectations of agents. Let the market demand and supply functions over time t > 0 be given by qalt) = do – dip(1) + d¿p'(t) and q,(t) = –so + sip(t) – s2p'(t) , respectively, where p(t) is the market price of the product, qa(t) is the associated quantity demanded, q,(t) is the associated quantity supplied, and do, d1, d2, so, 81, and s2 are all positive constants. The functional forms ensure that, when faced with an increasing price, demanders will tend to purchase more (before prices rise further) while suppliers will tend to offer less (to take advantage of the higher prices in the future). Now given the above stability argument, we restrict consideration to market clearing time paths p(t) satisfying qa(t) = q,(1), for all t> 0, and explore the evolution of price over time. We say that the market is in dynamic equilibrium if p'(t) = 0 for all t. It is easy to show that the dynamic equilibrium in this model is given by p(t) = p* for all t, where p* is the market equilibrium price defined above. However, many other market clearing time paths are possible. (c) Find a market clearing time path: Equate q.(t) and q,(t) and solve the resulting differen- tial equation p(t) in terms of its initial value po = p(0). (d) Is it true that for any market clearing time path we must have p(t) → p* as t → ∞? (e) If the price p(t) of a product is $5 at t = 0 months and demand and supply functions are modeled as q.(t) = 30 – 2p(t) + 4p'(t) and q,(t) = -20 + p(t) – 6p'(t), what will be the price after 10 months? As t becomes very large? What is happening to p'(t) and how are the expectations of demanders and suppliers evolving?