Dynamic Macroeconomic Modeling with Matlab

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4 Numerical Simulation of Macroeconomic Models Blueprints are sold to an intermediate firm with price PA. Profits of an intermediate firm Firm i faces an intermediate demand function px = αxα−1L1−α−βH β Y . Maximization of π = pxx − rx implies an optimal monopolistic price px = r α Hence, monopolistic firms add the markup 1/α > 1 to marginal costs r. Equilibrium profits can be written as π = pxx − αpx = (1 − α)αx α L 1−α−β H β Y = (1 − α)αKα A −α L 1−α−β = (1 − α)α Y A Capital compensation From intermediate firms we get r α = αxα−1L1−α−βH β Y r = α 2 x α−1 L 1−α−β H β Y = α2 and, hence, α−1 K L A 1−α−β H β Y Note that capital is underpaid relative to a competitive equilibrium. Equilibrium HY allocation Profit maximization in the final output sector implies w H = β Y = α2 Y K The R&D sector is perfectly competitive. Production in this sector exhibits a positive external effect because of the ‘standing on shoulders’ effect. We introduce government subsidies for researchers working in the R&D sector. The government pays the fraction sA of the R&D firm’s wage bill: Equating both equations yields HY (73) (74) (75) (76) (1 − sA)w H = pAηA (77) pAηA (1 − sA) Equilibrium in the capital market = β Y Equilibrium in the capital market requires the no-arbitrage condition to hold HY ˙ PA + π = rPA (78) (79)
4 Numerical Simulation of Macroeconomic Models In other words, the price of a blueprint PA at time t equals the present value of discounted profits of the time interval [t, ∞]. Household optimization Household optimization yields the Keynes Ramsey rule ˙c = Households’ budget constraint is given by c(r − ρ) σ (80) ˙a = ra + w H HY + w L L + w H HA − c − S (81) with total subsidies S = sAw H HA. Differentiating a = K + PAA with respect to time one can rewrite the equation as Summary of Notation Dynamic system ˙K = Y − c (82) Y final output K capital L labor H human capital A number of blueprints x(i) amount of intermediate i c consumption price of blueprint PA a household wealth π profit per intermediate firm r interest rate wH wage rate of human capital wL wage rate of labor ρ time preference rate σ inverse of intertemporal elasticity of consumption α, β, η technology parameters The dynamic system can be summarized as ˙K = Y − c (83)
Page 1 and 2: Dynamic Macroeconomic Modeling with
Page 3 and 4: 1 Introduction 1 Introduction The a
Page 5 and 6: 2 An Outline of the Theory x(t) 1 0
Page 7 and 8: 2 An Outline of the Theory South-Ea
Page 9 and 10: 2 An Outline of the Theory whereby
Page 11 and 12: 2 An Outline of the Theory c 0.68 0
Page 13 and 14: 2 An Outline of the Theory Figure 5
Page 15 and 16: 3 A Short Introduction to Matlab 3
Page 17 and 18: 3 A Short Introduction to Matlab 4.
Page 19 and 20: 4 Numerical Simulation of Macroecon
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Page 37 and 38: References References Ascher, U. M.
Page 39: References Solow, R. M. (1956): “

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