The demand and supply functions have hte form qd=d0-d1p and qs=-s0+s1p, respecti

Question

The demand and supply functions have hte form qd=d0-d1p and qs=-s0+s1p, respectively, where p is the market price of the product, qdis the associated quatity demanded, qs is the associated quantity supplied and d0,d1,s0 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*=(d0+s0)/(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/dt=?(qd-qs) , where ? is greater than zero is a constant indicating the speed of adjustments. This follows the intuitive requirement 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---->?. - Find the price adjustment path: Subsitute the expressions for qd and qs into the price adjustment equation and show that the solution to the resulting differential equation is p(t)=[p(0)-p*]ect+p*, where c=-?(d1+s1). Additional questions to first question: Let d0=205, d1=7, s0=5, s1=7. a) If ?=0.1, use the graph of the p(t) you fround in part (a0 to estimate how long it will take for the price to reach the equilibrium value if it starts i) $5 higher than equilibrium ii) $5 lower than equilibrium. b) Graph you p(t) for multiple values of ? between 0 and 1. Explain how ? indicating the speed of adjustments is represented in the graph. - 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 qd(t)= d0- d1p(t)+d2p'(t) and qs(t)= -s0+s1p(t)- s2p'(t) , respectively, where p(t) is the market price of the product, qd(t) is the associated quantity demanded, qs(t) is the associated quantity supplied and d0,d1,d2,s0,s1, 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 qd(t)=qs(t), 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 bathers are possible. - Find a market clearing time path: Equate qd(t) and qs(t) and solve the resulting differential equation p(t) in terms of its initial value p0=p(0). - Is it true that for any market clearing time path we must have p(t) --->p* as t--->?? - If the price p(t) of a product is $5 at t=0 months and demand and supply funtions are modeled as qd(t)=30-2p(t)+4p'(t) and qs(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? Describe a situation where the expectations of suppliers and demanders would be evolving as they are in this situation.

Explanation / Answer

can you please post the question in a good format

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.