Question

Consider a two-period consumption model in which a representative household lives for two periods only. In period 1, he earns income \(y_1\) and consumes \(c_1\). In period 2, he earns \(y_2\) and consumes \(c_2\). He can borrow and lend at the same rate of interest (\(r\)). His lifetime utility function is as follows: \[ U(c_1, c_2) = \ln(c_1) + \beta \ln(c_2) \] where \(\beta>0\) measures the sensitivity of future period’s consumption. What will be the marginal propensity to consume of current consumption, i.e. \(\frac{\partial c_1}{\partial y_1}\)?

• A. \(\frac{1}{1 + \beta}\)
• B. \(\frac{1}{1 + \beta^2}\)
• C. \(\frac{y_1}{1 + \beta}\)
• D. \(\frac{y_1}{1 + \beta^2}\)

Hint:

The marginal propensity to consume current consumption depends on the relative importance of future and current consumption, as indicated by \(\beta\).

Answer 1

The Correct Option is A

Step 1: Utility maximization

The representative household maximizes its utility with respect to the consumption choices \(c_1\) and \(c_2\). The utility function is given by:

\[ U(c_1, c_2) = \ln(c_1) + \beta \ln(c_2) \]

The household’s budget constraint for the two periods is:

\[ c_1 + \frac{c_2}{1 + r} = \frac{y_1 + y_2}{1 + r} \]

Step 2: Solve for \(c_1\) and \(c_2\)

To find the marginal propensity to consume, we differentiate the utility function with respect to \(y_1\) while considering the budget constraint. Solving this optimization problem gives the marginal propensity to consume current consumption \(c_1\) as:

\[ \frac{\partial c_1}{\partial y_1} = \frac{1}{1 + \beta} \]

Step 3: Conclusion

Thus, the correct answer is \( \frac{1}{1 + \beta} \).