Economics Dynamics Problems 48

# Economics Dynamics Problems 48 - dp dt − ² α b − d 1...

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32 Economic Dynamics This is effectively a stock-adjustment model. Stocks (inventories) change accord- ing to the difference between supply and demand, and price adjusts according to the accumulation–decumulation of stocks. Thus, if i ( t ) denotes the inventory holding of stocks at time t , then di dt = q s q d i = i 0 + ± t 0 ( q s q d ) dt and prices adjust according to dp dt =− α di dt =− α ( q s q d ) = α ( q d q s ) α> 0 which is the third equation in the model. Substituting the demand and supply equations into the price-adjustment equation results in the following Frst-order linear nonhomogeneous differential equation
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Unformatted text preview: dp dt − ² α ( b − d ) 1 − α f ³ p = α ( a − c ) 1 − α f with solution p ( t ) = ´ c − a b − d µ + ² p − ´ c − a b − d µ³ e − α ( d − b ) t 1 − α f which satisFes the initial condition p (0) = p . ±or this model there are far more varieties of solution paths, depending on the values of the various parameters. Some typical solution paths are illustrated in Fgure 2.3. We shall discuss the stability of such systems later. Figure 2.3....
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