Difference and differential equations with economic applications. - One Line Questions

1. In an economic model, if Y(t+1) = 0.8Y(t) + 20, and Y(0) = 100, what is Y(1)? — 108
2. For a first-order linear difference equation Y(t+1) = aY(t) + b, the system is stable if: — -1 < a < 1
3. The equation Y(t+2) - 5Y(t+1) + 6Y(t) = 0 is an example of: — A second-order linear homogeneous difference equation
4. The equation d^2Y/dt^2 + 4Y = 0 is an example of: — A second-order linear homogeneous differential equation
5. In the context of economic dynamics, what does the term 'equilibrium' represent? — A state where variables are not changing
6. The equation dY/dt = 5 represents: — A constant rate of change
7. The equation dY/dt = -0.05Y + 100 models: — Continuous decay towards an upper bound
8. The differential equation dY/dt = gY, where g is a constant, describes: — Exponential growth
9. Which type of equation is more appropriate for modeling the year-on-year change in national income? — Difference Equation
10. Which type of equation is more appropriate for modeling the instantaneous rate of change of consumption with respect to income? — Differential Equation
11. The dynamic adjustment of prices and quantities in a market over continuous time is best analyzed using: — Differential Equations
12. Which of the following best describes the relationship between difference equations and differential equations in economics? — Differential equations model continuous changes, while difference equations model discrete changes.
13. Which type of equation is used to model economic phenomena where changes occur at discrete points in time? — Difference Equation
14. Which economic concept is often modeled using first-order linear difference equations? — Market equilibrium adjustments
15. The equation Y(t+1) = Y(t) - 0.1Y(t) + 50 models: — Exponential decay with a constant addition
16. The equation Y(t+1) = Y(t) represents a system that is: — In equilibrium
17. The equation dY/dt = 0 represents a system that is: — In equilibrium
18. Consider a difference equation Y(t+1) = 1.1Y(t). If Y(0) = 100, what is the behavior of Y(t) as t increases? — It diverges to infinity.
19. Consider the difference equation Y(t+1) = aY(t) + b. If |a| < 1, what is the behavior of Y(t) as t approaches infinity? — It converges to a steady state
20. Consider the differential equation dY/dt = -kY, where k > 0. What is the behavior of Y(t) as t approaches infinity? — It converges to zero
21. Consider a differential equation dY/dt = -0.5Y. If Y(0) = 50, what is the behavior of Y(t) as t increases? — It converges to zero.
22. For a first-order linear differential equation dY/dt = -kY + c, where k > 0, the system is stable if: — k > 0
23. Which of the following is a common application of difference equations in economics? — Analyzing the discrete year-to-year changes in GDP
24. Which of the following is a common application of differential equations in economics? — Analyzing the continuous growth of a population
25. What is the characteristic equation for the difference equation Y(t+2) - 5Y(t+1) + 6Y(t) = 0? — r^2 - 5r + 6 = 0
26. What is the characteristic equation for the differential equation d^2Y/dt^2 + 4Y = 0? — r^2 + 4 = 0
27. The Cobweb Theorem, illustrating how prices adjust over time in agricultural markets, is typically modeled using: — A single first-order difference equation
28. In the context of the Solow-Swan model, the differential equation dK/dt = sY - nK represents: — The change in capital stock per unit of time
29. What does the term 'order' of a differential equation refer to? — The highest derivative present
30. What does the term 'order' of a difference equation refer to? — The highest difference term present
31. In the context of difference equations, what does Y(t) typically represent? — The value of a variable at time t
32. In a non-homogeneous differential equation like dY/dt + kY = f(t), what does f(t) represent? — The forcing function or external influence
33. What does 's' represent in the Solow-Swan model's differential equation dK/dt = sY - nK? — The savings rate
34. What does 'n' represent in the Solow-Swan model's differential equation dK/dt = sY - nK? — The population growth rate
35. Which economic model is a classic example of using differential equations to describe continuous growth? — All of the above
36. In a non-homogeneous difference equation of the form Y(t+1) = aY(t) + b, what is the 'particular solution'? — A solution that satisfies the non-homogeneous equation
37. In the context of differential equations, what does dY/dt represent? — The instantaneous rate of change of Y with respect to time
38. What is the role of initial conditions in solving difference and differential equations in economics? — They select a specific solution from the family of general solutions.
39. The concept of 'stability' in differential equations refers to: — Whether the solution approaches an equilibrium point
40. The concept of 'stability' in difference equations refers to: — Whether the solution converges to a steady state
41. If dY/dt = 2Y and Y(0) = 10, what is Y(t)? — Y(t) = 10 * e^(2t)
42. The roots of the characteristic equation r^2 - 5r + 6 = 0 are r=2 and r=3. What is the general solution for the homogeneous part of the difference equation? — Y(t) = c1 * 2^t + c2 * 3^t
43. The roots of the characteristic equation r^2 + 4 = 0 are imaginary (r = ±2i). What is the general form of the solution for this differential equation? — Y(t) = A*cos(2t) + B*sin(2t)
44. If the characteristic equation of a second-order linear homogeneous difference equation has one repeated real root, r, the general solution is: — Y(t) = (c1 + c2*t) * r^t
45. If the characteristic equation of a second-order linear homogeneous difference equation has two distinct real roots, r1 and r2, the general solution is: — Y(t) = c1 * r1^t + c2 * r2^t
46. The general solution to the differential equation d^2Y/dt^2 = 0 is: — Y(t) = c1*t + c2
47. The general solution to the differential equation dY/dt = kY is: — Y(t) = C * e^(kt)
48. A difference equation of the form Y(t+1) - Y(t) = 0 implies: — Y(t) is constant over time
49. What is the steady-state solution (Y*) for the difference equation Y(t+1) = aY(t) + b, assuming |a| < 1? — Y* = b / (1-a)
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