Derive the life-time budget constraint of this consumer. Set up the consumer’s problem. Solve for consumption (C and C′), leisure (l and l′), and saving (Sp). How does an increase in wage rate w affect C, Sp, and l?
Consider the problem of a consumer who chooses between consuming goods and enjoying leisure in the current and future periods. Denote the consumption and leisure in the current period as C and l, and the consumption and leisure in the future period as C′ and l′, respectively. The preference is summarized by the following utility function:
U(C,C′,l,l′)=lnC+ψlnl+β(lnC′ +ψlnl′).
This individual is endowed with h units of time in each period. Wage rate per unit of labour time is w and w′ in the current and future period. In addition, the consumer receives profit transfer π and π′ and pays lump-sum taxes T and T′ in the current and future periods. Denote the saving in the current period as Sp. Answer the following questions.
- Derive the life-time budget constraint of this consumer.
- Set up the consumer’s problem.
- Solve for consumption (C and C′), leisure (l and l′), and saving (Sp).
- How does an increase in wage rate w affect C, Sp, and l?
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