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(1) Derive the linear second-order differential equation implied by this model. (ii) Given a = 0.1, G = 25, B = -20,b= 0.5 and c = 0.1, solve the differential

(1) Derive the linear second-order differential equation implied by this model. (ii) Given a = 0.1, G = 25, B = -20,b= 0.5 and c = 0.1, solve the differential

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 3 (a) Consider a model of market equilibrium in which the current supply of firms is a function of the price that is expected to prevail when the product is sold. Assume that the market supply equation is q³(t) = F + Gp and pe is the expected price and F and G are constant parameters of the supply equation. Assume further that suppliers use information about the actual current price and its first and second derivatives with respect to time to form their prediction of the price that will prevail when their product reaches the market. In particular, assume that dp pe=p+b+c- dt d²p dt² tdpd²p=0, then suppliers and b> c> 0. If the current price is constant, so that dt expect the prevailing price to equal the current price. If the current price is rising, so that > 0, then suppliers expect the prevailing price to be higher than the current dt price. How much higher depends on whether the current price is rising at an increasing rate, > 0; or at a decreasing rate, p < 0. dt² dt² The remainder of the market equilibrium model is a linear demand equation q² = A + Bp and a linear price-adjustment equation that says that price rises when there is excess demand and falls when there is excess supply: (ii) dp dt = a(qº - q³) and a > 0 is a constant which determines how rapidly price adjusts when the market is out of equilibrium and A and B are constant parameters of the demand equation. (1) Derive the linear second-order differential equation implied by this model. Given a = 0.1, G = 25, B = -20, b = 0.5 and c = 0.1, solve the differential 3
Transcribed Image Text:Question 3 (a) Consider a model of market equilibrium in which the current supply of firms is a function of the price that is expected to prevail when the product is sold. Assume that the market supply equation is q³(t) = F + Gp and pe is the expected price and F and G are constant parameters of the supply equation. Assume further that suppliers use information about the actual current price and its first and second derivatives with respect to time to form their prediction of the price that will prevail when their product reaches the market. In particular, assume that dp pe=p+b+c- dt d²p dt² tdpd²p=0, then suppliers and b> c> 0. If the current price is constant, so that dt expect the prevailing price to equal the current price. If the current price is rising, so that > 0, then suppliers expect the prevailing price to be higher than the current dt price. How much higher depends on whether the current price is rising at an increasing rate, > 0; or at a decreasing rate, p < 0. dt² dt² The remainder of the market equilibrium model is a linear demand equation q² = A + Bp and a linear price-adjustment equation that says that price rises when there is excess demand and falls when there is excess supply: (ii) dp dt = a(qº - q³) and a > 0 is a constant which determines how rapidly price adjusts when the market is out of equilibrium and A and B are constant parameters of the demand equation. (1) Derive the linear second-order differential equation implied by this model. Given a = 0.1, G = 25, B = -20, b = 0.5 and c = 0.1, solve the differential 3
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