Economics Dynamics Problems 53

# Economics Dynamics Problems 53 - Continuous dynamic systems...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Continuous dynamic systems 37 Figure 2.5. which satisﬁes the initial condition 1 v0 = k0−α This allows us to solve for k(t) as follows k1−α = i.e. as as 1 + k0−α − e−(1−α)(n+δ)t n+δ n+δ k(t) = as as 1 + e−(1−α)(n+δ)t k0−α − n+δ n+δ 1 1−α The solution path for different initial values of k is illustrated in ﬁgure 2.5. 2.2 Solutions to ﬁrst-order linear differential equations Solutions to ﬁrst-order linear differential equations are well discussed in the mathematical texts on differential equations (see Boyce and DiPrima 1997; Giordano and Weir 1991). Here our intention is simply to provide the steps in obtaining a solution. In doing this we shall suppose y is a function of t. This is useful since most economic examples are of this type. The general form for a ﬁrst-order linear differential equation is then dy + p(t)y = g(t) dt Notice that in this formulation both p and g are functions of time. This also allows for the case where p(t) and g(t) are constants, in which case we have dy + by = a dt The four-step procedure is as follows. Step 1 Write the linear ﬁrst-order equation in the standard form dy + p(t)y = g(t) dt ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online