Question

Differential equations problem

8. Solve the following macroeconomic model. Let C(t) be the consumption, I(C) the investment of capital and Y(t) the national rent. C(1)+I() = Y(), 10=kC'(), CO)=aY() + b Y(0) = Y 0<a<1,5, k > 0 Solve for Y(t) & I(t). Calculate the long run value of y(t)/I(t)

Answer 1

Solution- Given that c(t) +16+) = y(t) i(t) = Kc'lt) ------ CCH) = a y(+6 ----- Given Assumption - Y(0) = Yo Olact, b, K>0 steps- CCE )+1() = y(t) Substituting t82 aylt) +b+kc'(t) = Y(t) --- on Solving equation 3 for ylt) bt ke'le) =YLE)-YLt) bukelt) = 4C4) [1-a] YCE )= b+ ke'l+)/(-0) ---(0) on Solving equation 3 Por Ilt) akc'lt ) Kert) =Y14). a YCE)-b Kc'ct) = 4(1)(1-a7 -6 -6 1(t)= 444)[1-a] -b ---© Solving for YLE) TILL)

Sub Stilelting from equation 4 and 6 YCU)/I/+) = [b+ke'ct)/(1-0)] [4() 1-a] -b Substituting from . equation 5 4Ct)/11t)=[b+84C4) [1-a) -b} /(1-0)37 YC4) [1-a7-6 Y(+)/114) = 4(2)(1-9)/(1-2)/4C4) [i-a)-b Y(+)/1(t) = 44+) 74()[7-a)-b 414)1724) =1/4- a] - Y(+)/6 Therefore 460), ... 41034110)/-) -Y60916 y (0)/160) = 7/11 -a] -1/b..