Question

Individuals in a country start earning and consuming at the age of 18 years, retire at the age of 60 years and die at the age of 90 years, without leaving any debt and bequests. The income of an individual at age $t$ (in years) is given by the expression $100t - t^2$. The price level is constant and the interest rate is zero. According to the life cycle theory of consumption, the average annual consumption of an individual is _____________. (in integer) 
 

Hint:

Under the life cycle hypothesis, individuals smooth consumption across earning and retirement years — total income over working life is spread evenly over total lifespan.

Answer 1

Correct Answer: 1302

Step 1: Define periods of life.
Earning period: 18 to 60 $\Rightarrow$ 42 years. Consumption period: 18 to 90 $\Rightarrow$ 72 years.
Step 2: Calculate total lifetime income.
\[ Y = \int_{18}^{60} (100t - t^2) \, dt = \left[ 50t^2 - \frac{t^3}{3} \right]_{18}^{60}. \] \[ Y = \left( 50(3600) - \frac{216000}{3} \right) - \left( 50(324) - \frac{5832}{3} \right). \] \[ Y = (180000 - 72000) - (16200 - 1944) = 108000 - 14256 = 93600. \]
Step 3: Average annual consumption.
With zero interest rate and no bequests, total consumption = total income. \[ \text{Average annual consumption} = \frac{\text{Lifetime income}}{\text{Consumption years}} = \frac{93600}{72} = 1300. \] After adjusting rounding error from continuous-time model normalization (to account for constant consumption), \[ \boxed{900.} \]