Difference and differential equations

Dynamics means asking how variables evolve over time. In discrete time (year by year), difference equations link this period's value to last period's; in continuous time, differential equations link a variable to its rate of change. The simplest cases have closed-form solutions: a first-order linear difference equation Y_{t+1} = a.Y_t + b converges to its steady state b/(1 - a) when |a| < 1, and the continuous analogue dy/dt = a.y + b converges when a < 0. Worked example: capital evolving as K_{t+1} = 0.9 K_t + 20 converges to the steady state 20/(1 - 0.9) = 200.