Dynamic Macroeconomic Modeling with Matlab
Dynamic Macroeconomic Modeling with Matlab
Dynamic Macroeconomic Modeling with Matlab
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4 Numerical Simulation of <strong>Macroeconomic</strong> Models<br />
C<br />
1.01<br />
1.005<br />
1<br />
0.995<br />
0.99<br />
0 5 10 15 20 25<br />
t<br />
Figure 7: impulse response function for anticipated and unanticipated tax cut<br />
4.2.4 Numerical Calculation of Utility<br />
Consider a household that maximizes<br />
max U <strong>with</strong> U =<br />
c(t)<br />
∞<br />
0<br />
u(c)e −ρt dt (43)<br />
<strong>with</strong> per-capita consumption c, discount rate ρ, and standard utility function u(·). I define<br />
Ũ(t) =<br />
t<br />
0<br />
u(c)e −ρτ dτ (44)<br />
and assume u(·) to equal u(c) = c1−σ.<br />
Moreover, I define the balanced growth rate of c to equal<br />
1−σ<br />
γ and I define the scale-adjusted variable ˜c as ˜c := ce−γt . Then I get<br />
Note that<br />
Ũ(t) =<br />
Differentiating <strong>with</strong> respect to time yields<br />
=<br />
t<br />
0<br />
t<br />
0<br />
c1−σ 1 − σ e−ρτdτ (45)<br />
˜c 1−σ<br />
1 − σ e(γ(1−σ)−ρ)τdτ (46)<br />
Ũ(0) = 0. (47)<br />
˙Ũ = ˜c1−σ<br />
1 − σ e(γ(1−σ)−ρ)t<br />
(48)
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