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Economics Dynamics Problems 198

# Economics Dynamics Problems 198 - Y with φ ± Y = c 1 c 2...

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182 Economic Dynamics Figure 4.30. solving. The Van der Pol equation takes the form: ¨ x µ (1 x 2 x + x = 0 (4.27) Let y = ˙ x , then ˙ y = ¨ x , so we have the two equations, ˙ x = y ˙ y = µ (1 x 2 ) y x (4.28) To illustrate the limit cycle, let µ = 1 . The phase portrait that results is shown in figure 4.30. Here we take two initial points: (a) point (0.5,0.5), which starts inside the limit cycle; and (b) point (1.5,4), which begins outside the limit cycle. Example 4.17 Walrasian price and quantity adjustment and limit cycles The presence of limit cycles is illustrated by a Walrasian model which includes both price and quantity adjustments (see Flaschel et al. 1997 and Mas-Colell 1986). Let Y denoteoutput of a one goodeconomy and L labour input. Y = f ( L ) is a production function which is twice differentiable and invertible with L = f 1 ( Y ) = φ ( Y ) and φ ( Y ) > 0. In equilibrium the price, p , is equal to marginal wage cost, where marginal wage cost is also given by φ ( Y ). Thus, p = φ ( Y ). For simplicity we
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Unformatted text preview: Y , with φ ± ( Y ) = c 1 + c 2 Y . Aggregate demand takes the form D ( p , L ) and in equilibrium is equal to supply, i.e., D [ p ∗ , φ ( Y ∗ )] = Y ∗ . ±inally, we have both a price and a quantity adjustment: ˙ p = α [ D ( p , φ ( Y )) − Y ] α > ˙ Y = β [ p − φ ± ( Y )] β > (4.29) These establish two differential equations in p and Y . Consider the following numerical example. Let φ ± ( Y ) = . 87 + . 5 Y D ( p ) = − . 02 p 3 + . 8 p 2 − 9 p + 50 then ( p ∗ , Y ∗ ) = (13 , 24 . 26) with isoclines: ˙ p = Y = − . 02 p 3 + . 8 p 2 − 9 p + 50 ˙ Y = p = . 87 + . 5 Y or Y = 1 . 74 + 2 p ±igure 4.31 reproduces the Fgures derived in ±laschel et al. 1997 using Mathe-matica , for α = 1 and different values of the parameter β . Not only do the Fgures...
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