The present value of the continuous stream of income C(t) pounds per year, where t is the number of years from now, for T years at continuous interest rate r is given as Pv=0TC(t)ertdt Consider a small venture that generates income at the rate of 2t million pounds per year, where t is the number of years from now. Find the present value of this continuous stream for the next 5 years at the continuous interest rate of 10% Question Two The equation of motion for capital (K) is given as: tK=IoK Where Io=800=0.025K(0)=300 Where Io represent the constant level of investment and represents rate of depreciation of capital. a) Find the capital stock at time t[K(t)] b) Draw a graphical sketch of the capital time path c) Explain the dynamic stability of the time path. Question Three Consider the following data from a mortgage market. P0= borrowed amount by an individual at time t=0,= fixed repayment paid in each period, r= fixed interest rate per period, Pt is outstanding balance at time t. Further, at t=20,P20=0; that is, at t=20, all the loan is paid out. at t=0;Pt=P0 that is, at t=0, the outstanding balance is P0. This information is summarized by the following equation Pt+1=Pt+rPt a) Find the general solution and the unique solution to the difference equation b) Discuss the convergence properties of the system and draw a graphical sketch for the time path of Pt c) Find the formulae for computing P0 from the unique solution d) Find the formulae for computing from the unique solution. The present value of the continuous stream of income C(t) pounds per year, where t is the number of years from now, for T years at continuous interest rate r is given as Pv=0TC(t)ertdt Consider a small venture that generates income at the rate of 2t million pounds per year, where t is the number of years from now. Find the present value of this continuous stream for the next 5 years at the continuous interest rate of 10% Question Two The equation of motion for capital (K) is given as: tK=IoK Where Io=800=0.025K(0)=300 Where Io represent the constant level of investment and represents rate of depreciation of capital. a) Find the capital stock at time t[K(t)] b) Draw a graphical sketch of the capital time path c) Explain the dynamic stability of the time path. Question Three Consider the following data from a mortgage market. P0= borrowed amount by an individual at time t=0,= fixed repayment paid in each period, r= fixed interest rate per period, Pt is outstanding balance at time t. Further, at t=20,P20=0; that is, at t=20, all the loan is paid out. at t=0;Pt=P0 that is, at t=0, the outstanding balance is P0. This information is summarized by the following equation Pt+1=Pt+rPt a) Find the general solution and the unique solution to the difference equation b) Discuss the convergence properties of the system and draw a graphical sketch for the time path of Pt c) Find the formulae for computing P0 from the unique solution d) Find the formulae for computing from the unique solution