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HT6 Inter-temporal Macroeconomics Notes
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  • Course
  • Macroeconomics
  • Institution
  • University Of Oxford

These notes were prepared based on the lectures and supplemented by information from textbooks and tutorials where parts of the lecture were unclear. Graphs, equations, and bullet-point explanations included. Prepared by a first class Economics and Management student for the FHS Macroeconomics pape...

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  • June 27, 2024
  • 31
  • 2022/2023
  • Class notes
  • -
  • Ht6 inter-temporal macroeconomics

Subjects

  • consumption
  • euler equation
  • hall random walk
  • precautionary saving
  • intertemporal model of consumption

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  • University of Oxford
  • Unknown
  • Macroeconomics
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HT6 Macroecons (Inter-temporal Macro)
Week's outline
 Introduction to intertemporal macroeconomics modelling
 Go beyond what intro UG courses/textbooks typically cover, get closer to what advanced/state
of the art modelling looks like
 Discuss further consumption because it is a key determinant of aggregate demand
 Present the RBC model an alternative to the 3-equation model to explain macroeconomics
fluctuations but also and perhaps more importantly a possible foundation for more general
models (DSGE models, although we will not cover these)

How do these lectures fit with the rest of the course?
 In some way complementary, we discuss consumption and consumption matters for the IS
 Intertemporal model of consumption can be thought of as an alternative derivation of the IS
 The RBC model brings together the Solow model (LR growth) and short run fluctuations
 Each topic can be seen also as interesting in itself
 Some of the material is inevitably rather technical, but it gives a sense of how more advanced
models really look like

Lecture 15 & 16: Consumption
Refresher: two period consumption model
Two period consumption model
 Households solve:

o subject to
o Maximise total utility: dependent on consumption levels in both periods c 1 and c2, with
utility in the next period discounted by β
 If β = 1 current and future treated equally
 If β < 1 future is discounted and today’s consumption is valued more than future
consumption
o Subject to income in both periods y1 and y2, with consumption transferable across
periods by investing/ borrowing at rate r
 First order condition:
o
 There is a choice to consume today or save and consume (1+r) units in future
 When you save, there are 2 opposing forces: your impatience (when β < 1) and
savings interest earned (1+R)
 Equate cost of waiting (LHS) with benefit of waiting (RHS)
 Utility is maximised when we are indifferent between moving consumption from
today to the future (or vice versa)

o Can also be expressed as MRS = MRT:
 Slope of indifference curve = slope of intertemporal budget constraint
 Graphically:

, o
o Intertemporal budget constraint
 From the initial income (y1, y2) point, move along the line with slope (1 + r) by
borrowing/ saving
 X-intercept equals to discounted lifetime resources
 Y- intercept as all resources compounded to the next period
o Agents are consumption smoothing: optimise utility by moving along the constraint

Towards a more general model
 Much of the intuition from the more general model we will now look at is very much like in the
two-period model setting
 There are more than two periods in reality
 We might want to be more explicit in the treatment of uncertainty

Standard intertemporal model of consumption
Standard intertemporal model of consumption
 Consumption is modelled as the decision of an optimizing rational agent
 The key elements are:
o Objectives of the decision maker (the household)
o Constraints faced
 Objectives are summarized by a utility function


o
o u(.): instantaneous utility function
o ct: consumption at time t
o βt: discount for period t's utility
 Intertemporal budget constraint


o
o Derivation below

Breaking down the objective utility function
 To analyze the choice of the household we must describe its willingness to substitute
consumption across periods. This is captured by 2 elements in the expression.
o Element 1: curvature of u(.)

,  Measure curvature with the elasticity of intertemporal substitution (EIS)


 This is the inverse of the CRRA (in micro course)
 Smaller EIS <> larger CRRA <> more risk averse
 It measures how the MRS (the indifference curve gradient) varies with the ratio
ct+1/ct (the axes). How MRS change as you substitute consumption from one
period to another.
 In other words, EIS measures how convex the indifference curves are
 More concave utility function, u''(.) more negative, positive EIS is
smaller, more convex indifference curve
 The more convex the indifference curve (smaller EIS), the higher the desire for
consumption smoothing (and the more risk averse)
 Recall: more concave utility function, more risk averse, prefer many
small deviations vs less but more extreme deviation, prefer smooth
consumption vs high variance consumption
 Risk attitudes are thus intrinsically linked to willingness to substitute
temporally
 The EIS measures the elasticity of the ratio of consumption w.r.t. a change in the
ratio of marginal utilities (the indifference curve slope). It is therefore a key
determinant of the sensitivity of consumption decision/ growth w.r.t. the
interest rate.
 Convexity of indifference curve captured in EIS determines sensitivity of
consumption decisions (changes in the tangency point) as the interest
rate (intertemporal budget slope) changes
 Visualise: higher EIS, less convex indifference curve, the more the
consumption decision (tangency point) would change as interest rate
(intertemporal budget slope) changes





o Element 2: the discount factor 0 < β < 1
 Captures the idea that households value future consumption less than present
consumption
 Due to psychological factors, such as impatience;
 Also due to a chance that future consumption will not be enjoyed,
because of death of the decision maker
 Assuming that future utility is discounted by β t (exponential discounting) implies
intertemporally consistent (time consistent) choices.

,  A planned course of action remains optimal unless new information
arrives
 Alternative assumptions exist in the literature (eg. hyperbolic
discounting), where choices may not be time consistent. More on that
at the end of the lecture.
 Σ: why we consider a sum of terms up to infinity
o Nobody argues that real life individuals live literally forever
o Justification 1: infinity is to be taken as a convenient approximation for a finite but very
long horizon
o Justification 2: intergenerational altruism
 The individual taking the consumption decision cares for their offspring, and
they intend to leave bequests to them; they also know that their offspring will
care about their own offspring, and so on.
 This argument has been made precise by Barro (1974), and we will return to it
when discussing fiscal policy
 Σ: why we consider sums
o An important “hidden” assumption is that instantaneous utility is independent of
consumption in other periods
 We could instead have assumed more generally: u(..., c t−1, ct, ct+1, ...)
o This is called intertemporal separability or additivity over time
 It implies the MRS between any two periods is independent of consumption in
any other period.
o This might be seen as more restrictive than it first appears as it rules out, for example:
 Habit formation (satisfaction may be dependent on prior periods' levels of
consumption, like a person who is used to a rich lifestyle)
 Goods like treats or holidays where the benefits last after the act of
consumption (benefits of consumption lasts over multiple periods)
o Example:
 Gives the standard additive function if γ = 0
 But for γ ≠ 0, the MRS between consumption at times 1 and 2 depends on
consumption at times 0 to 3:


δU δU
 MRS above is: ÷
δ c1 δ c2
 Substitutability of consumption (MRS) between 2 periods (1 and 2) is no
longer independent of consumption in other periods (0 and 3)
 This example utility function, if γ ≠ 0, could show the case of habits (less benefit
now from a given consumption level if prior consumption level is very high, as
the agent is used to high consumption level)
 E [.]: taking expectations
o The future is uncertain, so agents must forecast the value of relevant future variables

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