Question

Substituting the expressions for C_t and ( I_t ) into Y_t = C_t + I_t gives:

Y_t = (0.8Y_{t-1} + 100) + 200 = 0.8Y_{t-1} + 300
The complementary function is given by ( CF = A(0.8)^t ) and for a particular solution, try ( Y_t = D ) for some constant ( D ). Substituting this into the difference equation gives ( D = 0.8D + 300 ), which has a solution ( D = 1500 ). The general solution is therefore ( Y_t = A(0.8)^t + 1500 ). If the initial condition ( Y_0 = 1700 ) is given, what is the value of ( A )?
a. 100
b. 200
c. 300
d. 400

Answer 1

By substituting the initial condition into the given equation and solving for A, we find that the value of A is 200.

Step-by-step explanation:

The student is working with a differential equation to determine the value of a constant A in the equation Y_t = A(0.8)^t + 1500, given the initial condition Y_0 = 1700. To find the value of A, we substitute t = 0 and Y_0 = 1700 into the equation:

1700 = A(0.8)^0 + 1500

Since (0.8)^0 is 1, the equation simplifies to:

1700 = A + 1500

Subtracting 1500 from both sides of the equation results in:

A = 1700 - 1500

A = 200

Therefore, the value of A is 200, which corresponds to option b.