# 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

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 Y_{2} and C_{2}.

(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 Y_{2} 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.

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