A two sector lagged-income model is defined byYt = Ct

A two sector lagged-income model is defined by

Y_t = C_t + I_t
C_t = 0.8Y_t-1 + 200
I_t  = 1000
where Y_t, C_t and I_t denote national income, consumption and planned investment in time period t. The initial value of national income is Y0 = 5000.
(a) Use these relations to work out Y₂ and C₂.
(b) Derive a first-order difference equation relating Y_t and Y_t-1.
(c) Solve the difference equation to express Y_t in terms of t.
(d) Verify that your general formula in part (c) agrees with the value of Y₂ calculated in part (a).
(e) Find an expression for C_t in terms of t and hence determine the time period in which the consumption first exceeds 4750.