Question

1 Life Cycle Model (4 points) 1. Suppose that a person expects to live five periods and receives a wage income of 200,000 each of the first 3 periods, and the interest rate is 60 percent. After period 3 the household retires and receives no wage income. Assuming this household follows the life cycle consumption function, calculate consumption in each period of his life. (2 points) 1 2. Now suppose the interest was 50 percent instead of 60 percent, recom- pute the agent's consumption for each period of his life. (2 points)

Answer 1

1. The first 3 period income has pa present value of $251,953.13 Detailed calculation as follows,

ДА в 1 Present Value of Annuity Payments at the end of each period 2 3 Present value of annuity is calculated using the formula PVA=P*{(1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r= Rate of interest per period in decimals All amounts in $ Given that 9 Periodical payments (P)= 200000 No. of periods= 10 Rate of interest (% p.a)=R= 60 No. of payments a period (t)= 11 12 Calculations: Formula Cell reference Value 14 Interest rate per period in decimals (r)= R/(100*t) D10/(100*G10) 0.6 15 No. of periods (n)= N*t G9*G10 (1+r)= 1+G14 1.6 (1+r)^-n= G164-G15 0.244140625 [1-(1+r)^-n]= 1-G17 0.755859375 (1-(1+r)^-n]/r= G18/614 1.259765625 20 Present value of annuity PV= P*{[1-(1+r)^-n]/r]} D9*G19 251953.125 Rounded to $251,953.13 19 21 22

As per Life cycle Model of consumption, discounted present value of lifetime income must equal the discounted present value of lifetime consumption.

Therefore, total present value of the consumption in all the five years should be equal to this amount. The amount of yearly consumption is arrived at as an annuity for 5 years at $167,168.60 as follows:

A 1 Annuity Payment B C D E Payments at the end of each period 3 Amount of periodical payments is calculated using the formula PMT= (PV*r)/(1-(1+r)^-n] 4 Where PMT= Periodical payments required, PV= Present value in lumpsum 6 n= Number of payments. 7 r=Rate of interest per period in decimals All amounts in $ Given that 10 Present value (PV) 251953.13 No. of periods (N)= 11 Rate of interest (% p.a)=R= 60 No. of payments a period (t)= 12 13 Calculations: Formula Cell reference 15 Interest rate per period in decimals (r )= R/(100*t) 011/(100*G11) 0.6 16 No. of periods (n)= N*t G10*G11 (1+r)= 1+G15 1.6 18 (1+r)^-n= G174-G16 0.095367432 (1-(1+r)^-n] = 1-G18 0.904632568 PV*r= D10*G15 151171.878 21 Periodical payment (PMT)= (PV*r)/(1-(1+r)^-n] G20/G19 167108.5956 22 Rounded to $167,108.60 22 17 19

2. If the interest rate is 50% instead of 60%, present value of the income during the first 3 periods is $281,481.48 as shown below:

с р 1 Present Value of Annuity Payments at the end of each period 3 Present value of annuity is calculated using the formula PVA=P*{[1-(1+r)^-n]/r]} 4 Where P= Periodical payments, 5 n= Number of payments. 6 r=Rate of interest per period in decimals All amounts in $ Given that 9 Periodical payments (P)= 200000 No. of periods 10 Rate of interest % p.a)=R= 50 No. of payments a period (t)= 3 12 Calculations: 14 Interest rate per period in decimals (r ) = 15 No. of periods (n)= Formula R/(100*t) N*t (1+r)= (1+r)^-n= [1-(1+r)^-n]= [1-(1+r)^-n]/r= P*{[1-(1+r)^-n]/r]} Cell reference D10/(100*G10) G9*G10 1+G14 G16A-G15 1-G17 G18/614 D9*G19 Rounded to Value 0.5 3 1.5 0.296296296 0.703703704 1.407407407 281481.4815 $281,481.48 co 20 Present value of annuity PV=

The resultant consumption during each of the 5 periods shall be $162,085.31 as given below:

B A 1 Annuity Payment C D Payments at the end of each period 9 3 Amount of periodical payments is calculated using the formula PMT=(PV*r)/(1-(1+r)^-n] 4 Where PMT= Periodical payments required, 5 PV= Present value in lumpsum 6 n= Number of payments. 7 r= Rate of interest per period in decimals All amounts in $ Given that 10 Present value (PV) 281481.48 No. of periods (N)= 11 Rate of interest (% p.a)= R= 50 No. of payments a period (t)= 12 13 Calculations: 14 Formula Cell reference Value 15 Interest rate per period in decimals (r )= R/(100*t) D11/(100*G11) 0.5 16 No. of periods (n)= N*t G10*G11 5 17 (1+r)= 1+G15 1.5 18 (1+r)^-n= G174-G16 0.131687243 [1-(1+r)^-n]= 1-G18 0.868312757 PV*r= D10*G15 140740.74 21 Periodical payment (PMT)= (PV*r)/(1-(1+r)^-n] G20/619 162085.3072 22 Rounded to $162,085.31 20 23