A household must decide between consuming today, C1, and consuming tomorrow, C2. The intertemporal budget constraint for this household is: C1 + C2/R = Y1 + Y2/R. The life-time utility function of this household is: U(C1,C2) = log C1 + beta*log C2. If the parameter beta

Question

A household must decide between consuming today, C1, and consuming tomorrow, C2. The intertemporal budget constraint for this household is: C1 + C2/R = Y1 + Y2/R. The life-time utility function of this household is: U(C1,C2) = log C1 + beta*log C2. If the parameter beta<1, then the household is _____ and _____.

BlackTom AI · Accepted Answer

We begin by restating the scenario and candidate outcomes to set the stage for evaluation.
Question: A household chooses between consuming today (C1) and tomorrow (C2) under the intertemporal budget constraint C1 + C2/R = Y1 + Y2/R, with lifetime utility U(C1,C2) = log C1 + beta*log C2. Given beta < 1, which description best fits the household?
Options:
1) Patient; saves all of their resources for use in period 2.
2) Impatient; only consumes in period 1.
3) Impatient; prefers to consume in period 1 relative to consuming in period 2.
4) Patient; prefers to consume in period 2 relative to consuming in period 1.

Now, assess each option one by one.
Option 1: Patient; saves all of their resources for use in period 2.
- This asserts long-run patience and a strict saving behavior. However, beta < 1 denotes a lower weight on future consumption in the utility, i.e., the household discounts the future more heavily. This is the opposite of patient saving in the sense of valuing future consumption equally or more than present consumption. Therefore this description is inconsistent with beta < 1.

Option 2: Impatient; only consumes in period 1.
- Claiming that the household consumes only in period 1 ignores the intertemporal choice structure and the possibility of trading off some C1 for C2 within the budget. While beta < 1 signals a preference for present consumption, it does not imply that C2 is zero; there could still be positive consumption in period 2 depending on prices and returns. So this is an overstatement and not necessarily accurate.

Option 3: Impatient; prefers to consume in period 1 relative to consuming in period 2.
- This aligns with the standard interpretation of beta < 1 in a two-period utility U = log C1 + beta log C2: the discount factor on future utility is less than one, making current consumption more attractive relative to future consumption. The Euler equation 1/C1 = (beta/R)*(1/C2) indicates that the relative marginal utilities are scaled by beta; since beta < 1, present consumption tends to be preferred over future, all else equal. Thus, the household is impatient and has a relative preference for C1 over C2.

Option 4: Patient; prefers to consume in period 2 relative to consuming in period 1.
- This would require a high valuation of future consumption or a beta > 1 or some reversal in relative impatience, which contradicts beta < 1. Therefore this option incorrectly describes the agent’s time preference.

In summary, the only option that coherently reflects beta < 1 as a discounting of future utility, yielding a preference for current consumption over future consumption (i.e., impatience and a relative preference for C1), is option 3.

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