L2 - consumption (Intertemporal Choice Model)

Lecture Information

• Instructor: Dawid Trzeciakiewicz

• Institution: Loughborough Business School

References

• Mankiw, N. G. (2022) - Macroeconomics, Worth Publishers, International edition, Chapter 20

• Mankiw, N. G. (2019) - Macroeconomics, Macmillan International, 10th ed., Chapter 19

• Mankiw, N. G., Taylor, M. P. (2014) - Macroeconomics European Edition, Worth Publishers, 2nd ed., Chapter 18

• Williamson, S.D. (2018) - Macroeconomics, Pearson Education Limited, 6th ed., Chapter 9

• Chamberlin, G., Yueh, L. Y. (2006) - Macroeconomics, Thomson Learning.

Major Theories of Consumption

This lecture surveys the most prominent work on consumption, touching upon various economic theories that explain how individuals decide what to consume and save:

• John Maynard Keynes: This theory, often called the Keynesian Consumption Function, focuses on how current consumption is directly and primarily determined by current income. It suggests that as income rises, consumption also rises, but not as much as the increase in income. This concept is captured by the Marginal Propensity to Consume (MPC), which is the fraction of an additional dollar of income that a household consumes rather than saves.

• Irving Fisher: Introduced the concept of intertemporal choice, which describes how rational individuals make decisions about consumption and saving over different periods of time (e.g., now vs. in the future). It considers income and expenditures at different points in time and how interest rates affect these choices.

• Franco Modigliani: Developed the Life-Cycle Hypothesis (LCH). This theory posits that individuals plan their consumption and savings over their entire lifetime to smooth out their consumption path. People tend to save during their working years (when income is high) to finance consumption during retirement (when income is low).

• Milton Friedman: Proposed the Permanent Income Hypothesis (PIH). This hypothesis asserts that people base their current consumption on their expected long-term average income ("permanent income") rather than on their current income alone, which might fluctuate temporarily. A temporary rise in income might be largely saved, while a permanent rise would lead to a significant increase in consumption.

• Robert Hall: Introduced the Random-Walk Hypothesis for consumption, which builds on the PIH and the concept of rational expectations. This theory suggests that if consumers have rational expectations about their future income, then changes in their consumption should be unpredictable (a "random walk"). This is because consumers have already incorporated all available information into their current consumption decisions, so only unexpected events would cause consumption to change.

• Borrowing constraints: These are limitations faced by consumers that restrict their ability to borrow money for current consumption, even if they expect higher future income. For example, a student might have high expected future income but limited access to loans today, thus constraining their current consumption. These constraints can cause consumption to track current income more closely than predicted by LCH or PIH.

• Precautionary savings: This refers to saving done to provide for future uncertainties or unexpected events, such as job loss, illness, or other financial shocks. People save even when they don't have a specific large purchase in mind, simply to create a buffer against unforeseen circumstances.

Intertemporal Budget Constraint

• The intertemporal budget constraint describes how consumers can allocate their total resources (current and future income) over multiple periods, allowing them to make choices between current consumption and future consumption. It essentially defines the total amount of consumption possible across different time periods.

• The fundamental equation that defines the intertemporal budget constraint, assuming two periods ($t$ for current and $t+1$ for future), is presented as the present value of consumption equals the present value of income:
  $c<em>t + \frac{c</em>{t+1}}{1 + r} = y<em>t + \frac{y</em>{t+1}}{1 + r}$ #### Explanation of symbols:

  • $c_t$: Current consumption (consumption in period $t$).

  • $c_{t+1}$: Future consumption (consumption in period $t+1$).

  • $y_t$: Current income (income in period $t$).

  • $y_{t+1}$: Future income (income in period $t+1$).

  • $r$: The real interest rate, which represents the return on savings or the cost of borrowing. It discounts future values to their present equivalent.
    #### Interpretation:
    The left side of the equation represents the present value of total consumption over both periods. The right side represents the present value of total lifetime income (also known as lifetime wealth). The constraint states that the present value of what you consume cannot exceed the present value of what you earn.

• Rearranging this equation to express future consumption in terms of current consumption and total wealth gives a more intuitive linear form:
  $c<em>{t+1} = -c</em>t \times (1 + r) + (y<em>t(1+r) + y</em>{t+1})$
  For simplicity, let $w = y<em>t(1+r) + y</em>{t+1}$ represent the future value of total wealth (current income saved at interest $r$ plus future income). Then, the equation becomes:
  $c<em>{t+1} = -c</em>t \times (1 + r) + w$
  This form shows a linear relationship between current and future consumption.

• The slope of the budget constraint, in this rearranged format, is given by:
  $\text{Slope} = -(1 + r)$
  This slope indicates the trade-off between current and future consumption. For every unit of current consumption given up, one can consume $(1+r)$ units in the future (the principal plus interest gained from saving).

Key Points for the Intertemporal Budget Constraint

1. Illustration: To plot the constraint, we can identify two extreme points, representing scenarios where all consumption occurs in one period:

   • If $c<em>t = 0$: This means all resources are saved for future consumption. In this case, $c</em>{t+1} = (1 + r) \times (y<em>t + \frac{y</em>{t+1}}{1 + r})$ which simplifies to $c<em>{t+1} = y</em>t(1+r) + y_{t+1}$. This represents the maximum possible future consumption, which is the future value of lifetime wealth ($w$).

   • If $c<em>{t+1} = 0$: This means all resources are consumed in the current period (or borrowed against future income to consume now). In this case, $c</em>t = y<em>t + \frac{y</em>{t+1}}{1 + r}$. This represents the maximum possible current consumption, which is the present value of lifetime wealth ($w/(1+r)$, in terms of our future wealth definition, or simply the right-hand side of the initial constraint).

2. Interpretation of the slope: The slope $-(1+r)$ represents the relative price of current consumption in terms of future consumption. It tells us how much future consumption one must give up to increase current consumption by one unit, or vice versa. A steeper slope (higher $r$) means current consumption is more "expensive" relative to future consumption because saving yields a higher return.

Factors Causing Shifts in the Intertemporal Budget Constraint

A shift means the entire budget line moves parallel, implying a change in total lifetime wealth, but the trade-off (slope) remains the same. Potential causes of a shift in the intertemporal budget constraint may include:

• Changes in current or future income: An increase in either current income ($y<em>t$) or expected future income ($y</em>{t+1}$) will increase the consumer's total lifetime wealth, shifting the entire budget constraint outward (to the right and up). Reduced income would cause an inward shift.

• Changes in autonomous consumption: If there's an exogenous change in the baseline level of consumption not related to income or interest rates, it can effectively alter the perceived starting wealth, causing a shift.

Dynamics of the Intertemporal Budget Constraint Rotating:

A rotation means the slope of the budget line changes, indicating a change in the relative price of current versus future consumption. The budget line typically pivots around the "endowment point" (the point where $c<em>t = y</em>t$ and $c<em>{t+1} = y</em>{t+1}$).

• A change in the real interest rate ($r$): This is the primary factor causing rotation. An increase in $r$ makes future consumption relatively cheaper (or current consumption relatively more expensive). The budget constraint rotates: it becomes steeper, as the opportunity cost of current consumption (foregone interest) increases. The point on the x-axis (maximum current consumption) moves inward, while the point on the y-axis (maximum future consumption) moves outward, presuming the consumer is a net saver. An decrease in $r$ would cause the budget constraint to become flatter.

• Changes in the slope of the indifference curve: While this impacts the optimal choice, it doesn't directly rotate the budget constraint itself. Consumer preferences (represented by indifference curves) determine where on the budget constraint the consumer chooses to consume, but they do not alter the constraint's position or slope.

• Changes in marginal utility of income: Similar to the above, this might affect optimal choice but not the budget constraint directly.

The Indifference Curves

• Indifference curves represent combinations of current and future consumption that yield the same level of total utility (satisfaction) for a consumer. Consumers are indifferent among all points on a single curve.

• Assumptions about these preferences ensure the standard shape and properties of indifference curves:

  1. More is better: Consumers prefer more consumption to less. Thus, indifference curves are downward sloping and higher indifference curves (further from the origin) represent higher levels of utility.

  2. Preference for diversity (Diminishing Marginal Rate of Substitution): Consumers prefer a mix of current and future consumption rather than consuming everything in one period. This causes indifference curves to be convex to the origin (bowed inward). As you consume more of one good (e.g., current consumption), you are willing to give up less of the other good (future consumption) to get an additional unit of the first.

  3. Both current and future consumption are considered normal goods: An increase in total lifetime wealth (holding prices constant) will lead to an increase in both current and future consumption.

• The Marginal Rate of Substitution (MRS) of current to future consumption is given by the formula:
  $\text{MRS}<em>{c</em>t, c<em>{t+1}} = -\frac{\partial U / \partial c</em>t}{\partial U / \partial c<em>{t+1}}$ This represents the rate at which a consumer is willing to trade current consumption ($c</em>t$) for future consumption ($c_{t+1}$) while maintaining the same level of utility. It is the absolute value of the slope of the indifference curve at any given point.

Optimal Choice

• The optimal choice for a consumer is the point where they achieve the highest possible level of utility, given their intertemporal budget constraint. Graphically, this occurs at the point where the highest attainable indifference curve is just tangent to the budget constraint.

• At the optimal point, the slope of the budget constraint equals the slope of the indifference curve: $(1 + r) = \text{MRS}<em>{c</em>t, c_{t+1}}$ #### Explanation:

  • The slope of the budget constraint ($-(1+r))$ represents the market trade-off between current and future consumption (what the market allows you to do through saving or borrowing).

  • The slope of the indifference curve ($\text{MRS}<em>{c</em>t, c_{t+1}}$ ) represents the consumer's personal trade-off (what the consumer is willing to do to maintain utility).

  • When these two slopes are equal, it means the consumer's willingness to substitute consumption across time periods matches the ability to do so offered by the market (the interest rate). This condition describes an equilibrium where consumers balance current and future consumption based on their preferences and available market rates, maximizing their utility.

Real-World Applications

1. U.S. Economic Stimulus: An example of consumption smoothing can be seen in the $1.9$ trillion economic stimulus signed into law by President Biden, which included $1,400$ checks to individuals. This aims to boost current consumption.

   • Analysis in a two-period model is required focusing on:

     • Instant transfers versus future transfers on consumption patterns: How would giving money now versus promising money in the future affect current consumption? An immediate transfer tends to have a larger impact on current consumption (especially for liquidity-constrained individuals), while a future transfer might encourage more saving today if consumers smooth consumption.

     • Comparison with the Keynesian consumption function: The Keynesian model would predict a significant increase in current consumption based on the MPC. However, models like PIH or LCH might predict less of an immediate boost if consumers view the transfer as temporary or if they are already smoothing consumption, leading them to save a portion.

2. Interest Rate Changes: On August $3$, $2023$, the Bank of England raised interest rates to $5.25\%$.

   • Analyze changes in lenders' consumption choices using a two-period model:

     • Look at where the optimal point is relative to the endowment point: The endowment point is where consumption equals income in each period ($c<em>t = y</em>t$, $c<em>{t+1} = y</em>{t+1}$). For a lender (someone who saves), their optimal point will be to the left of the endowment point, meaning ct < yt (they save in the current period) and c{t+1} > y{t+1} (they consume more in the future). A borrower would be to the right.

     • Assess the effect of the interest rate rise on the budget constraint: An increase in $r$ causes the budget constraint to become steeper. For a lender, this creates both an income effect and a substitution effect. The substitution effect encourages more saving (less current consumption) because saving is now more rewarding. The income effect, for a lender, means they are wealthier (their savings earn more), which could lead to increased consumption in both periods. The net effect on current consumption is ambiguous, though often lenders save more.

Mathematical Representation

• Lagrangian System: To solve for maximum utility under a constrained optimization scenario (maximizing utility subject to the intertemporal budget constraint), the following steps apply:

  1. Formulate the Lagrangian: This combines the utility function and the budget constraint into a single expression that can be optimized: \text{ℒ} = U(Ct) + \beta \times U(C{t+1}) + \lambda(Yt + \frac{Y{t+1}}{1+rt} - Ct - \frac{C{t+1}}{1+rt}) #### Explanation of symbols:

     • \text{ℒ}: The Lagrangian function.

     • $U(C<em>t)$ and $U(C</em>{t+1})$: Utility derived from current and future consumption, respectively. The utility function typically shows diminishing marginal utility (each additional unit of consumption provides less additional satisfaction).

     • $\beta$ (beta): The discount factor, which represents how much an individual values future utility relative to current utility. If 0 < \beta < 1, it implies a preference for current consumption over future consumption (time preference).

     • $\lambda$ (lambda): The Lagrange multiplier. It represents the marginal utility of wealth (how much utility would increase if the budget constraint were relaxed by one unit).

     • $Y<em>t$ and $Y</em>{t+1}$: Current and future income (notation consistent with budget constraint).

     • $r_t$: The real interest rate in period $t$. (Note: often simplified to $r$ for a constant interest rate).

  2. Calculate first-order conditions (FOCs) with respect to both current consumption ($C<em>t$), future consumption ($C</em>{t+1}$), and the Lagrange multiplier ($\lambda$): These conditions are found by taking the partial derivative of the Lagrangian with respect to each variable and setting it to zero. They represent the necessary conditions for optimality.

     • For current consumption ($C<em>t$): \frac{\partial \text{ℒ}}{\partial Ct} = U'(Ct) - \lambda = 0 \implies U'(Ct) = \lambda
       Explanation: The marginal utility of current consumption ($U'(C_t)$) must equal the marginal utility of wealth ($\lambda$). This means an extra unit of current consumption is worth exactly its cost in terms of utility from wealth.

     • For future consumption ($C<em>{t+1}$): \frac{\partial \text{ℒ}}{\partial C{t+1}} = \beta \times U'(C{t+1}) - \frac{\lambda}{1+rt} = 0 \implies \beta U'(C{t+1}) = \frac{\lambda}{1+rt}
       Explanation: The discounted marginal utility of future consumption ($\beta U'(C<em>{t+1})$) must equal the present value of the marginal utility of wealth ($\frac{\lambda}{1+r</em>t}$). This equates the benefit of future consumption (in today's utility terms) to its opportunity cost. Rearranging provides: $(1+r<em>t)\beta U'(C</em>{t+1}) = \lambda$

• These first-order conditions lead to the Euler equation, which states:
  $U'(C<em>t) = (1 + r</em>t) \times \beta \times U'(C<em>{t+1})$ #### Explanation of the Euler Equation: The Euler equation is a fundamental condition for intertemporal consumption optimization. It states that the marginal utility of current consumption ($U'(C</em>t)$) must equal the discounted (by $r<em>t$ and $\beta$) marginal utility of future consumption, adjusted for the interest rate. In simpler terms, the additional satisfaction from consuming an extra dollar today must be equal to the additional satisfaction you would get from saving that dollar, earning interest ($1+r</em>t$), and then consuming it in the future, adjusted for your time preference ($\beta$).

• From logarithmic utility functions (e.g., $U(C) = \ln(C)$), which often assume constant relative risk aversion, solving the Euler equation yields a specific relationship between current and future consumption:
  $C<em>{t+1} = (1 + r</em>t) \beta C<em>t$ This particular form shows that future consumption is proportional to current consumption, adjusted by the interest rate and the discount factor. A higher $r</em>t$ or $\beta$ (meaning higher value placed on future or lower discounting) leads to higher future consumption relative to current consumption.

Consumption Function Derivation

• Combining the Euler equation with the intertemporal budget constraint ultimately allows us to solve for the optimal consumption path and derive a consumption function that depends on lifetime wealth. For the specific logarithmic utility case, this yields:
  $C<em>t = \frac{1}{1 + \beta} (Y</em>t + \frac{Y<em>{t+1}}{1 + r</em>t})$
  #### Explanation:
  This derived consumption function shows that current consumption ($C_t$) is a constant fraction ($\frac{1}{1 + \beta}$) of the consumer's total lifetime wealth (the present value of current and future income). This highlights the concept of consumption smoothing – consumption is not just based on current income, but on the total resources available over one's lifetime, discounted to the present.

• Implications on sensitivity of consumption with respect to income and interest rates yield:
  [\frac{\partial Ct}{\partial Yt} = \frac{1}{1+\beta} > 0]
  Explanation: This derivative shows how current consumption ($C<em>t$) changes with a change in current income ($Y</em>t$). Since \beta > 0 (future utility is valued), the value $\frac{1}{1+\beta}$ will be between $0$ and $1$. This means an increase in current income leads to an increase in current consumption, but not by the full amount of the income increase. The remaining portion is saved (consistent with consumption smoothing, where a temporary increase in income is spread across both periods).

  [\frac{\partial Ct}{\partial Y{t+1}} = \frac{1}{(1 + \beta)(1 + rt)} > 0] Explanation: This derivative shows how current consumption ($C</em>t$) changes with a change in future income ($Y_{t+1}$). An increase in expected future income also leads to an increase in current consumption, but by a smaller amount than an increase in current income, because future income is discounted. This again demonstrates consumption smoothing: anticipating higher future income allows you to consume more today.

  [\frac{\partial Ct}{\partial rt} = -\frac{Y{t+1}}{(1 + \beta)(1 + rt)^2} < 0]
  Explanation: This derivative shows how current consumption ($C<em>t$) changes with a change in the real interest rate ($r</em>t$). An increase in the interest rate generally leads to a decrease in current consumption. This is primarily due to the substitution effect: saving becomes more attractive, so consumers substitute current consumption for future consumption. The income effect for borrowers would also cause a decrease in current consumption (they are poorer), but for savers, the income effect would increase current consumption (they are richer). The negative sign here implies that for many realistic scenarios (especially for net borrowers or when the substitution effect dominates for savers), a higher interest rate encourages less current consumption and more saving.

Conclusion

This lecture provides a comprehensive overview of key theories and concepts in consumption economics, emphasizing the fundamental importance of not just current income, but also future expectations and the interest rate in guiding consumption behaviors across various economic settings. Previous work by seminal economists continues to inform modern understanding of economic behavior concerning consumption and savings, and the intertemporal budget constraint framework provides a powerful tool for analyzing these decisions.