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Copy file name to clipboardExpand all lines: lectures/gorman_heterogeneous_households.md
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# Gorman Aggregation
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{cite:t}`gorman1953community` described a class of preferences with the useful property that there exists a "representative household"
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in the sense that competitive equilibrium allocations can be computed by following a recursive procedure:
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## Overview
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{cite:t}`gorman1953community` described a class of models with preferences having the useful property that there exists a "representative household" in the sense that competitive equilibrium allocations can be computed in the following way:
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* take the heterogeneous preferences of a diverse collection of households and from them synthesize the preferences of a single hypothetical "representative household"
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* collect the endowments of all households and give them to the representative household
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```
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Chapter 12 of {cite:t}`HansenSargent2013` described how to adapt the preference specifications of {cite:t}`gorman1953community`
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to a linear-quadratic class of environments.
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This lecture uses the `quantecon.DLE` class to study economies that satisfy necessary conditions for Gorman
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aggregation of preferences.
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Chapter 12 of {cite:t}`HansenSargent2013` described how to adapt the preference specifications of {cite:t}`gorman1953community`
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to the linear-quadratic class of environments assumed in their book.
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The first step in implementing the above recursive algorithm will be to form a representative agent economy and then apply our DLE tools to compute its competitive equilibrium.
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Thus, this lecture builds on tools and Python code described in {doc}`hs_recursive_models`, {doc}`growth_in_dles`, and {doc}`irfs_in_hall_model`.
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To compute a competitive equilibrium, our first step will be to form a representative agent.
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In addition to what's in Anaconda, this lecture uses the `quantecon` library
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```{code-cell} ipython3
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:tags: [hide-output]
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After that, we can use some of our DLE tools to compute competitive equilibrium
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!pip install --upgrade quantecon
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```
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* prices without knowing the allocation of consumption to individual households
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* households' individual wealth levels
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* households' consumption levels
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We make the following imports
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Thus, this lecture builds on tools and Python code described in {doc}`hs_recursive_models`, {doc}`growth_in_dles`, and {doc}`irfs_in_hall_model`.
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```{code-cell} ipython3
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import numpy as np
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from scipy.linalg import solve_discrete_are
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from quantecon import DLE
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from quantecon import LQ
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import matplotlib.pyplot as plt
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```
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## Overview
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When conditions for Gorman aggregation of preferences are satisfied, we can compute a competitive equilibrium of heterogeneous-household economy in two steps: solve a representative-agent linear-quadratic planning problem for aggregates, then recover household allocations via a sharing-formula that makes each household's consumption a household-specific constant share of aggregate consumption.
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In a little more detail, when conditions for Gorman aggregation of preferences are satisfied, we can compute a competitive equilibrium of a heterogeneous-household economy in two steps: solve a representative-agent linear-quadratic planning problem for aggregates, then recover household allocations via a sharing-formula that makes each household's consumption a household-specific constant share of aggregate consumption.
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* a household's share parameter will depend on the implicit Pareto weight implied by the initial distribution of endowment.
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2. Compute household-specific policies and the Gorman sharing rule.
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For the special case in Section 12.6 of {cite:t}`HansenSargent2013`, where preference shocks are inactive, we can also
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For a special case described in Section 12.6 of {cite:t}`HansenSargent2013`, where preference shocks are inactive, we can also
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3. Implement the Arrow-Debreu allocation using only a mutual fund (aggregate stock) and a one-period bond.
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3. Implement the Arrow-Debreu allocation by allowing households to trade a risk-free one-period bond and a single mutual fund that owns all of the economy's physical capital.
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We shall provide examples of these steps in economies with two and many households.
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We shall provide examples of these steps in economies with two and then with many households.
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Before studying these things in the context of the DLE class, we'll first introduce Gorman aggregation in a static economy.
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In addition to what's in Anaconda, this lecture uses the `quantecon` library
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```{code-cell} ipython3
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:tags: [hide-output]
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!pip install --upgrade quantecon
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```
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We make the following imports
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```{code-cell} ipython3
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import numpy as np
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from scipy.linalg import solve_discrete_are
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from quantecon import DLE
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from quantecon import LQ
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import matplotlib.pyplot as plt
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```
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Before studying these things in the context of the DLE class, we first introduce Gorman aggregation in a static economy.
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(static_gorman)=
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### Gorman aggregation in a static economy
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To see where the sharing rule comes from, start with a static economy with $n$ goods, price vector $p$, and households $j = 1, \ldots, J$.
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To indicate where our competitive equilibrium sharing rule originates, we start with a static economy with $n$ goods, price vector $p$, and households $j = 1, \ldots, J$.
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Let $c^a$ denote the aggregate amount of consumption to be allocated among households.
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These restrictions enable Gorman aggregation by ensuring that household demands are affine in wealth.
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This will allow us to solve for aggregate allocations and prices without knowing the distribution of wealth across households as we shall see in {ref}`sharing_rules`.
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This will allow us to solve for aggregate allocations and prices without knowing the distribution of wealth across households, as we shall see in {ref}`sharing_rules`.
Aggregates are economy-wide totals: $c_t := \sum_j c_{jt}$, $b_t := \sum_j b_{jt}$, $d_t := \sum_j d_{jt}$, and similarly for $(i_t, k_t, h_t, s_t, g_t)$.
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Under the Gorman/LQ restrictions, we can compute equilibrium prices and aggregate quantities by synthesizing a fictitious *representative agent* whose first-order conditions reproduce the competitive equilibrium conditions for aggregates.
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Under the Gorman/LQ restrictions, we can compute equilibrium prices and aggregate quantities by synthesizing a fictitious *representative agent* whose first-order conditions generate equations that determine correct equilibrium prices and a correct *aggregate* consumption allocation and aggregate capital process.
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* these equations will not be sufficient to determine the allocation of aggregate consumption among individual households.
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This representative problem is an aggregation device: it is chosen to deliver the correct *aggregate* allocation and prices.
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Posing a social planning problem for a representative agent is thus a device for computing the correct *aggregate* allocation along with correct competitive equilibrium prices.
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The representative agent maximizes
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The social planner in our representative agent economy maximizes
This section studies the special case from Section 12.6 of {cite:t}`HansenSargent2013` in which the Arrow-Debreu allocation can be implemented by opening competitive markets only in a mutual fund and a one-period bond.
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This section studies a special case from Section 12.6 of {cite:t}`HansenSargent2013` in which the Arrow-Debreu allocation can be implemented by opening competitive markets only in a mutual fund and a one-period bond.
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* So in our setting, we don't literally require that markets in a complete set of contingent claims be present.
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Household $j$'s time-$t$ resources are its mutual fund dividend $\mu_j d_t$ plus the return on its bond position $R(\mu_j k_{t-1} + \hat{k}_{j,t-1})$.
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Its uses of funds are consumption $c_{jt}$ plus the new bond position $(\mu_j k_t + \hat{k}_{jt})$:
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Its uses of funds are consumption $c_{jt}$ plus the new bond position $(\mu_j k_t + \hat{k}_{jt})$, so
The DLE framework represents the economy as a linear state-space system.
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After solving the optimal control problem and substituting the policy rule, we obtain a closed-loop system:
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We can use our DLE tools first to solve the representative-household social planning problem and represent equilibrium quantities as a linear closed-loop state-space system that takes the usual Chapter 5 {cite:t}`HansenSargent2013` form
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$$
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x_{t+1} = A_0 x_t + C w_{t+1},
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\end{bmatrix},
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$$
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with $h_{t-1}$ the household service stock, $k_{t-1}$ the capital stock, and $z_t$ the exogenous state (constant, aggregate endowment states, and idiosyncratic shock states).
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with $h_{t-1}$ the aggregate household service stock, $k_{t-1}$ the aggregate capital stock, and $z_t$ the exogenous state (constant, aggregate endowment states, and idiosyncratic shock states).
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Any equilibrium quantity is a linear function of the state.
label=f'Mean across {d_households.shape[0]} households', alpha=0.8)
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axes[1].set_xlabel('Time (after burn-in)')
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axes[1].set_ylabel('Endowment')
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axes[1].set_title(f'Average of Individual Household Endowments')
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axes[1].set_xlabel('time (after burn-in)')
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axes[1].set_ylabel('endowment')
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axes[1].legend(loc='upper right', ncol=2)
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plt.tight_layout()
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## Redistributing by adjusting Pareto weights
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This section analyzes Pareto-efficient tax-and-transfer schemes by simply taking competitive equilibrium allocations and then using
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a set of nonnegative Pareto weights that sum to one.
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This section analyzes Pareto-efficient tax-and-transfer schemes by starting with competitive equilibrium allocations and using a specific set of nonnegative Pareto weights that sum to one.
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```{note}
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There are various schemes that would deliver such efficient redistributions, but in terms of what interests us in this example,
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they are all equivalent.
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There are various tax-and-transfer schemes that would deliver such efficient redistributions, but in terms of what interests us in this example, they are all equivalent.
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