Difference and Differential Equations Flashcards
What is a linear first-order autonomous difference equation?
xt = bxt-1 + a is linear because xt is a linear function of xt-1, first-order because the largest difference between indices is 1, and autonomous because a and b are independent of t
What is the solution to xt = bxt-1 + a?
xt {btx0 + (1 - bt)a/(1 - b) if b ≠ 1, x0 + ta if b = 1
What is a steady state?
The quantity xn when xn+1 = xn
When does xt converge to x = a/(1 - b)?
Iff |b| < 1
What is the cobweb model?
A model of a market where supply follows prices with a lag while demand responds immediately so St = G(pt - 1) and Dt = F(pt)
The market clears in each period so Dt = St so F(pt) = G(pt-1) which is a difference equation for price
If demand and supply are linear, the market clearing condition makes it possible to solve the difference equation and to determine convergence behaviour
If demand and supply have constant price elasticity, it can be shown that the system converges iff demand is more price elastic than supply
What is a differential equation?
A ‘functional equation’ (holds for all t in the domain of f) which relates a function to its derivatives
Solving a differential equation means identifying all f which satisfy the equation
What is the relative rate of change of x?
x’(t)/x(t) = d/dt ln(x(t)) as long as x(t) assumed to be positive
What form must a function with constant relative rate of change take?
Integrating d/dt ln(x(t)) = r wrt t yields ln(x(t)) = rt + C so x(t) = Aert
This method can be used to find that constant PED can be expressed D(p) = Apε
How do you solve x’(t) = bx(t) + a?
Note that -a/b is a particular solution and the difference y(t) between two solutions will satisfy the homogenous equation y’(t) = by(t) therefore x(t) = -a/b + Aebt
What is a stationary solution?
A constant solution to a differential equation
If b < 0 for x’(t) = bx(t) + a then the solution converges to the stationary one as t –> ∞
If b > 0 and x(t) ≠ -a/b, |x(t)| –> ∞ exponentially fast as t –> ∞
How can you solve a non-autonomous differential eqn in the form x’(t) = bx(t) + a(t)?
Multiply by e-bt and rearrange to get the derivative of x(t)e-bt on one side so that the solution is ebt times the integral of the other side
How do you solve x’(t) = b(t)x(t) + a(t)?
Multiply the equation by some integrating factor h(t) which has h’(t) = -h(t)b(t) so that integrating both sides yields x(t) = 1/h(t) ∫h(t)a(t)dt