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We set up and solve a rich life-cycle model of household decisions involving consumption of both perishable goods and housing services, stochastic and unspanned labor income, stochastic house prices, home renting and owning, stock investments, and portfolio constraints. The model features habit formation for housing consumption, which leads to optimal decisions closer in line with empirical observations. Our model can explain (i) that stock investments are low or zero for many young agents and then gradually increasing over life, (ii) that the housing expenditure share is age- and wealth-dependent, (iii) that perishable consumption is more sensitive to wealth and income shocks than housing consumption, and (iv) that non-housing consumption is hump-shaped over life.
We show that the optimal consumption of an individual over the life cycle can have the hump shape (inverted U-shape) observed empirically if the preferences of the individual exhibit internal habit formation. In the absence of habit formation, an impatient individual would prefer a decreasing consumption path over life. However, because of habit formation, a high initial consumption would lead to high required consumption in the future. To cover the future required consumption, wealth is set aside, but the necessary amount decreases with age which allows consumption to increase in the early part of life. At some age, the impatience outweighs the habit concerns so that consumption starts to decrease. We derive the optimal consumption strategy in closed form, deduce sufficient conditions for the presence of a consumption hump, and characterize the age at which the hump occurs. Numerical examples illustrate our findings. We show that our model calibrates well to U.S. consumption data from the Consumer Expenditure Survey.