Consider a household choosing a plan for current consumption C, and future consumption C according to...

  

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Consider a household choosing a plan for current consumption C, and future consumption C₂ according to the Fisher model with two time periods. Both cur- rent and future consumption are assumed to be normal goods. The household receives current income Y, and expects to receive future income Y₂. Taxes and transfers may be ignored in this question. The household's net saving in the cur- rent period is S=Y₁-C₁, and the real interest rate on any saving or borrowing between the current and future time periods is r. Assume the household already has incurred debts D (including interest) that are due for repayment in the current time period. Given saving S, the net debt of the household at the end of the current period is D-S. The budget constraint on consumption in the future time period is C₂=Y₂-(1+r)(D-S). (a) [2 marks] Using the budget constraint on C₂ and the definition of S, derive the present-value budget constraint of the household in terms of C, and C₂. (b) [3 marks] Illustrate the budget constraint in a diagram with Cy and C₂ on the axes. What is its gradient? What point does it pass through for every possible interest rate r? (c) [2 marks] Illustrate the household's indifference curves and the optimal choice of consumption plan in a diagram for the case where the choice of saving S is positive, but less than existing debt D. Suppose there is an increase in the interest rate r. In what follows, assume that the household is initially choosing saving S with 0 <S<D as in part (c). (d) [4 marks] What is the effect of higher r on the present-value budget con- straint? Analyse the effect of higher r on current consumption C₁ by break- ing down the response into income and substitution effects. Now suppose the household starts with a long-term debt contract that specifies (1 + r)D is due for repayment in the future period, but no payments are required in the current period. The fixed interest rate written into this debt contract is F. The budget constraint on future consumption is now C₂=Y₂-(1+r)D+(1+r)S. (e) [3 marks] Derive the new present-value budget constraint faced by the house- hold. What is its gradient, and which point does it pass through for all r? (f) [2 mark] If the interest rate on new saving or borrowing is the same as the fixed interest rate from the long-term debt contract, that is, r = r, how does the optimal choice of consumption C, compare to that in part (c)? Explain. (g) [4 marks] Starting from r = r, suppose there is an increase in r. What is the effect on the present-value budget constraint? Compare the household's response of C₁ to that with short-term debt in part (d), referring to income and substitution effects. Consider a household choosing a plan for current consumption C, and future consumption C₂ according to the Fisher model with two time periods. Both cur- rent and future consumption are assumed to be normal goods. The household receives current income Y, and expects to receive future income Y₂. Taxes and transfers may be ignored in this question. The household's net saving in the cur- rent period is S=Y₁-C₁, and the real interest rate on any saving or borrowing between the current and future time periods is r. Assume the household already has incurred debts D (including interest) that are due for repayment in the current time period. Given saving S, the net debt of the household at the end of the current period is D-S. The budget constraint on consumption in the future time period is C₂=Y₂-(1+r)(D-S). (a) [2 marks] Using the budget constraint on C₂ and the definition of S, derive the present-value budget constraint of the household in terms of C, and C₂. (b) [3 marks] Illustrate the budget constraint in a diagram with Cy and C₂ on the axes. What is its gradient? What point does it pass through for every possible interest rate r? (c) [2 marks] Illustrate the household's indifference curves and the optimal choice of consumption plan in a diagram for the case where the choice of saving S is positive, but less than existing debt D. Suppose there is an increase in the interest rate r. In what follows, assume that the household is initially choosing saving S with 0 <S<D as in part (c). (d) [4 marks] What is the effect of higher r on the present-value budget con- straint? Analyse the effect of higher r on current consumption C₁ by break- ing down the response into income and substitution effects. Now suppose the household starts with a long-term debt contract that specifies (1 + r)D is due for repayment in the future period, but no payments are required in the current period. The fixed interest rate written into this debt contract is F. The budget constraint on future consumption is now C₂=Y₂-(1+r)D+(1+r)S. (e) [3 marks] Derive the new present-value budget constraint faced by the house- hold. What is its gradient, and which point does it pass through for all r? (f) [2 mark] If the interest rate on new saving or borrowing is the same as the fixed interest rate from the long-term debt contract, that is, r = r, how does the optimal choice of consumption C, compare to that in part (c)? Explain. (g) [4 marks] Starting from r = r, suppose there is an increase in r. What is the effect on the present-value budget constraint? Compare the household's response of C₁ to that with short-term debt in part (d), referring to income and substitution effects.

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