Difference and Differential Equations

Question 1

What is a linear first-order autonomous difference equation?

Answer 1

x_t = bx_t-1 + a is linear because x_t is a linear function of x_t-1, first-order because the largest difference between indices is 1, and autonomous because a and b are independent of t

Question 2

What is the solution to x_t = bx_t-1 + a?

Answer 2

x_t {b^tx₀ + (1 - b^t)a/(1 - b) if b ≠ 1, x₀ + ta if b = 1

Question 3

What is a steady state?

Answer 3

The quantity x_n when x_n+1 = x_n

Question 4

When does x_t converge to x = a/(1 - b)?

Answer 4

Iff |b| < 1

Question 5

What is the cobweb model?

Answer 5

A model of a market where supply follows prices with a lag while demand responds immediately so S_t = G(p_{t - 1}) and D_t = F(p_t)
The market clears in each period so D_t = S_t so F(p_t) = G(p_t-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

Question 6

What is a differential equation?

Answer 6

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

Question 7

What is the relative rate of change of x?

Answer 7

x’(t)/x(t) = d/dt ln(x(t)) as long as x(t) assumed to be positive

Question 8

What form must a function with constant relative rate of change take?

Answer 8

Integrating d/dt ln(x(t)) = r wrt t yields ln(x(t)) = rt + C so x(t) = Ae^rt
This method can be used to find that constant PED can be expressed D(p) = Ap^ε

Question 9

How do you solve x’(t) = bx(t) + a?

Answer 9

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 + Ae^bt

Question 10

What is a stationary solution?

Answer 10

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 –> ∞

Question 11

How can you solve a non-autonomous differential eqn in the form x’(t) = bx(t) + a(t)?

Answer 11

Multiply by e^-bt and rearrange to get the derivative of x(t)e^-bt on one side so that the solution is e^bt times the integral of the other side

Question 12

How do you solve x’(t) = b(t)x(t) + a(t)?

Answer 12

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