20.8 Summary

Things you should know

Intertemporal Budget Constraint

• Good 1 ($c_1$) is consumption now (or “period 1,” or “year 1,” or “this year,” or “today”); good 2 ($c_2$) is consumption in the future (or “period 2”, or “year 2,” or “next year,” or “tomorrow”). For the purposes of this introductory analysis, we will measure both of these in dollars (i.e. nominal terms).
• Endowment $(m_1,m_2)$ an exogenously given income stream – that is, the agent has $m_1$ now and expects to receive $m_2$ in the future.
• We assume that agents can save money and earn interest, or borrow money and pay it back with interest. The nominal interest rate is $r$.
• Saving means consuming less than $m_1$ in period 1 and more than $m_2$ in the future. If the agent saves $s$, then that money grows at interest rate $r$ to become $(1+r)s$; hence the amount consumed in period 2 is \(c_2 = m_2 + (1+r)s\) Since $s = m_1 - c_1$, this can be rearranged to form the intertemporal budget line \((1+r)c_1 + c_2 = (1+r)m_1 + m_2\) This is the same as an endowment budget line with $p_1 = 1+r$ and $p_2 = 1$.
• Borrowing means consuming more than $m_1$ in period 1 and less than $m_2$ in the future. If the agent borrows $b$, then the money they owe grows at interest rate $r$ to become $(1+r)b$; hence the amount consumed in period 2 is \(c_2 = m_2 - (1+r)b\) Since $b = c_1 - m_1$, this can be rearranged to form the same intertemporal budget constraint as before.
• Present value The budget line above is given in “future value” (there is no coefficient before $c_2$ and $m_2$. If we divide through by $(1 + r)$ we get \(c_1 + \frac{c_2}{1+r} = m_1 + \frac{m_2}{1+r}\) This is the same as an endowment budget line with $p_1 = 1$ and $p_2 = 1/(1+r)$. Either the present value or future value budget constraint has a price ratio of $p_1/p_2 = 1 + r$.

Intertemporal Preferences

• Utility is of the form $u(c_1,c_2) = v(c_1) + \beta v(c_2)$ – e.g., $u(c_1,c_2) = \ln c_1 + \beta \ln c_2$
• The function $v(c_t)$ is the utility in period $t$ from consuming $c_t$
• The parameter $\beta$ measures the “discount rate” at which the agent discounts future utility
• If $\beta < 1$ then the agent is impatient - they prefer current consumption to future consumption
• Marginal rate of substitution: \(MRS(c_1,c_2) = \frac{MU_1}{MU_2} = \frac{v^\prime(c_1)}{\beta v^\prime(c_2)}\)Note that when $c_1 = c_2$, $MRS = 1/\beta$; and the lower $\beta$ is, the more the consumer is willing to give up future income for an additional dollar today.

Optimization

• Borrow or Saving Decision: The MRS at the endowment is $v^\prime(m_1)/\beta v^\prime(m_2)$. The price ratio is $1 + r$. Therefore the agent will: \(\text{Borrow if }MRS(m_1,m_2) > 1 + r \iff r < \frac{v^\prime(m_1)}{\beta v^\prime(m_2)} - 1 \iff \beta < \frac{v^\prime(m_1)}{(1 + r)v^\prime(m_2)}\) \(\text{Save if }MRS(m_1,m_2) < 1 + r \iff r > \frac{v^\prime(m_1)}{\beta v^\prime(m_2)} - 1 \iff \beta > \frac{v^\prime(m_1)}{(1 + r)v^\prime(m_2)}\) More likely to borrow if $r$ and/or $\beta$ is low (i.e., cheap to borrow and/or are very impatient).
• The rest of optimization/supply/demand is the same as before: for example, if $c_1^\star(r)$ is the consumer’s optimal consumption in period 1, then their net demand is $c_1^\star(r) - m_1$; if this is positive, it’s their demand for borrowing, and if it’s negative, its absolute value is their supply of saving.

Refinements to the Intertemporal Choice Model (covered in lecture, not in problem sets or on exam)

• Nominal vs. Real Interest Rates: If $c_1$ and $c_2$ are “composite goods” rather than dollars, then the slope of the budget line becomes the real interest rate $\rho$, where \(1 + \rho = \frac{1 +r}{1 + \pi}\) The inflation rate $\pi$ measures the fact that goods might get more expensive between the two periods, reducing the purchasing power of savings in period 2.
• Net present value over multiple periods: The present-value budget line may be extended indefinitely to many periods: since savings $s$ grows to $(1+r)s$ in one year and $(1+r)^2s$ in two, we can have the more general budget constraint \(c_1 + \frac{c_2}{1+r} + \frac{c_3}{(1+r)^2} + \frac{c_4}{(1+r)^3} + \cdots = m_1 + \frac{m_2}{1+r}+ \frac{m_3}{(1+r)^2} + \frac{m_4}{(1+r)^3} + \cdots\)
• Different interest rates for borrowing/lending: If an agent faces a different interest rate for borrowing and lending, their budget constraint will have a kink at the endowment. If this is the case, they will borrow if their MRS at the endowment point is greater than $1 + r^\text{borrow}$, and save if it is less than $1 + r^\text{save}$. This may lead to a broad range of cases where it is optimal to consume at the endowment.
• Borrowing constraints: If an agent cannot borrow at all, their budget line simply ends at their endowment; so they cannot consume $c_1 > m_1$. If their MRS is greater than the interest rate at their endowment, this means they would be better off if they could borrow money.

Things you should be able to do

• Given an endowment $(m_1,m_2)$ and interest rate $r$, and inflation rate $\pi$ draw the endowment budget line
• Illustrate how a change in $m_1$, $m_2$, or $r$ affects a budget line
• Given a utility function and budget constraint, determine whether an agent will choose to borrow, save, or neither
• Given a utility function and budget constraint, determine the agent’s optimal choice