1 Introduction
This set of lecture notes provides foundations of macro analysis, focusing particularly on the following issues:
- Fundamentals of Dynamics General Equilibrium
- Optimal Growth in Discrete Time
- Optimal Growth in Continuous Time
Basic references:
- Acemoglu (2009): chs. 5-7
- Aghion and Howitt (1998): chs. 1-2
- Barro and Sala-i-Martin: ch. 2, secs. 4.1-4.3
- Ljungqvist and Sargent (2000): chs. 3 and 11
- Stokey and Lucas with Prescott (1989): chs. 3-5
- Wang (2012), “Endogenous Growth Theory,” Lecture Notes, Washington University-St. Louis.
2 Fundamentals
Representative Household
To add to this section: Backus, Routledge, and Zin (2004)
Representative household is valid when the optimization of individual households can be represented as if there were a single household making the aggregate decisions using a representative preference subject to aggregate constraints.
Consider a particular preferences representations: \[ \max \sum_{t=0}^{\infty} \beta^{t} u(c(t)) \]
- Let the excess demand of the economy be \( \mathbf{x}(p) \).
- Key: whether this excess demand function \( \mathbf{x}(p) \) can be obtained as a solution to the single household optimization problem.
- Answer at the first glance: it cannot be so obtained in general as the weak axiom of revealed preferences for individuals need not hold for the aggregate.
- To yield a positive answer ensuring the excess demand function \( \mathbf{x}(p) \) to be obtained as a solution to the single household optimiation problem, we need to impose further restrictions, in particular, to remove strong income effects.
- A special bu useful case for such arepresnentation is to have linear value (indirect utility) function:
The class of preferences described in this theorem is referred to as “Gorman preferences” (1959 Econometrica)
In this class, the Engel curve of each household for each commodity is linear and its slop is identical to all individuals for the same commodity.
By Roy’s Identity, \[ x_j^h \left( p, w^h \right) = - \frac{1}{b(p)} \frac{\partial a^{h}(p)}{\partial p_j} - \frac{1}{b(p)} \frac{\partial b(p)}{\partial p_j} w^{h} \] Therefore, for each household, a linear relationship exists between demand and income and the slope \( - \frac{1}{b(p)} \frac{\partial b(p)}{\partial p_j} \),is independent of the household’s identity \( h \).
Even under this class of Gorman preferences, a representative household exists, typical macro models require further restrictions on:
- the abstract of distribution effects from the representative household’s concern (strong representation)
- the use of the representative household’s preferences as the welfare function of the aggregate economy (normative representation)
Normative representation requires convexity, interiority and price-invariant basic value (otherwise, once can transfer \( \varepsilon \) from low to high-valuation households for different \( p \)):
Consider an economy with a finite number \( N < \infty \) of commodities, a set \( \mathcal{H} \) of households, and a convex aggregate production possibilities est \( \mathcal{Y} \). Suppose that the preferences of each household \( h \in \mathcal{H} \) is represented by \( v^{h}(p, w^h) = a^h(p) + b(p) w^h \) with \( p = (p_1, \dots, P_N) \) and that each household \( h \in \mathcal{H} \) has a positive demand for each commodity.
- Then any feasible allocation that maximizes the utility of the representative household \( v(p, w)= \sum_{h \in \mathcal{H}} a^{h}(p) + b(p)w \), with \( w \equiv \sum_{h \in \mathcal{H}} w^h \) is Pareto optimal.
- If \( a^h (p) = a^h \) for all \( p \) and all \( h \in \mathcal{H} \) (price invariant basic value), then any Pareto optimal allocation maximizes the utilitiies of the representative household.