Consider the two-period consumption-labor model. Suppose that lifetime utility of the representative household is given by:...

  

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Consider the two-period consumption-labor model. Suppose that lifetime utility of the representative household is given by: V(C₁,41, 02, l2)= u(c₁,41) + Bu(c₂, 12) where ß> 0 is the discount factor and %> 0 determines the relative preference for leisure. For both t= 1, 2, the period utility function given by: u(a, 4) Inc + In and the household faces the following real lifetime budget constraint: C₁ + w₂(1-1₂) 1+r w₁ (1-₁)+ 1+r (a) Using the lifetime Lagrangian, compute all relevant first-order conditions. (b) Derive the consumption-labor optimality conditions for periods 1 and 2, and the consumption-savings optimality condition across periods 1 and 2. (c) Solve for the optimal values (c₁*, 1₁, c2*, 12*) in terms of exogenous variables only. (d) Perform comparative static analysis to mathematically show an increase patience by the household will affect how each optimal choice variable. (HINT: An increase in patience is given by an increase in the discount factor, 3). Consider the two-period consumption-labor model. Suppose that lifetime utility of the representative household is given by: V(C₁,41, 02, l2)= u(c₁,41) + Bu(c₂, 12) where ß> 0 is the discount factor and %> 0 determines the relative preference for leisure. For both t= 1, 2, the period utility function given by: u(a, 4) Inc + In and the household faces the following real lifetime budget constraint: C₁ + w₂(1-1₂) 1+r w₁ (1-₁)+ 1+r (a) Using the lifetime Lagrangian, compute all relevant first-order conditions. (b) Derive the consumption-labor optimality conditions for periods 1 and 2, and the consumption-savings optimality condition across periods 1 and 2. (c) Solve for the optimal values (c₁*, 1₁, c2*, 12*) in terms of exogenous variables only. (d) Perform comparative static analysis to mathematically show an increase patience by the household will affect how each optimal choice variable. (HINT: An increase in patience is given by an increase in the discount factor, 3).

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