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(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 + Gpe 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 pe=p+b- dp d²p dt² +c. dt and b> c> 0. If the current price is constant, so that = 0, then suppliers 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 price. How much higher depends on whether the current price is rising at an increasing rate, > 0; or at a decreasing rate, <0. The remainder of the market equilibrium model is a linear demand equation = qD = 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: dp = a(qº - q¹) 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. 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 + Gpe 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 pe=p+b- dp d²p dt² +c. dt and b> c> 0. If the current price is constant, so that = 0, then suppliers 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 price. How much higher depends on whether the current price is rising at an increasing rate, > 0; or at a decreasing rate, <0. The remainder of the market equilibrium model is a linear demand equation = qD = 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: dp = a(qº - q¹) 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. 3

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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(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 q' (t) = F + Gpe and p² 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 d²p dt² pe=p+b- and b> c> 0. If the current price is constant, so that = 0, then suppliers dt expect the prevailing price to equal the current price. If the current price is rising, so + c 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, d²p > 0; or at a decreasing rate, < 0. The remainder of the market equilibrium model is a linear demand equation qD = 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: dp dt = a (qº - qs) 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. 3 لا
Transcribed Image Text:(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 q' (t) = F + Gpe and p² 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 d²p dt² pe=p+b- and b> c> 0. If the current price is constant, so that = 0, then suppliers dt expect the prevailing price to equal the current price. If the current price is rising, so + c 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, d²p > 0; or at a decreasing rate, < 0. The remainder of the market equilibrium model is a linear demand equation qD = 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: dp dt = a (qº - qs) 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. 3 لا
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There are two markets in the Dornbusch model of exchange rate overshooting: an asset market in which the exchange rate, e, is determined and a goods market in which the domestic price level, p, is determined. In the asset market, the demand for money is given by: m² = −ar + by where r is the domestic interest rate, y is the domestic level of output, which is assumed to be constant and a and b are constant positive coefficients of the money demand function. The real supply of money is given by m - 0.5p, where m is the exogenous nominal supply of money. Since the economy is open to trade international capital movements will force the domestic interest rate to equal the world interest rate, r*, plus an interest rate differential equal to the expected rate of depreciation of the home currency, t dt In the goods market, domestic prices adjust sluggishly in response to excess demand. The rate of change of prices is given by: P = a(y-ys), a > 0, where yº and ys are aggregate quantity demanded and supplied respectively and a is the speed of adjustment coefficient. Under the assumption that supply is fixed at y and demand is given by yº = u + v(e − p) where e - p is the relative price of domestic goods and u and v are positive coefficients of the demand function. (i) 4 Derive a system of differential equation to represent the rate of change of the exchange rate and the price level.
Transcribed Image Text:There are two markets in the Dornbusch model of exchange rate overshooting: an asset market in which the exchange rate, e, is determined and a goods market in which the domestic price level, p, is determined. In the asset market, the demand for money is given by: m² = −ar + by where r is the domestic interest rate, y is the domestic level of output, which is assumed to be constant and a and b are constant positive coefficients of the money demand function. The real supply of money is given by m - 0.5p, where m is the exogenous nominal supply of money. Since the economy is open to trade international capital movements will force the domestic interest rate to equal the world interest rate, r*, plus an interest rate differential equal to the expected rate of depreciation of the home currency, t dt In the goods market, domestic prices adjust sluggishly in response to excess demand. The rate of change of prices is given by: P = a(y-ys), a > 0, where yº and ys are aggregate quantity demanded and supplied respectively and a is the speed of adjustment coefficient. Under the assumption that supply is fixed at y and demand is given by yº = u + v(e − p) where e - p is the relative price of domestic goods and u and v are positive coefficients of the demand function. (i) 4 Derive a system of differential equation to represent the rate of change of the exchange rate and the price level.
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