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Copy file name to clipboardExpand all lines: lectures/supply_demand_heterogeneity.md
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@@ -9,7 +9,7 @@ There are two consumers who differ in their endowment vectors $e_i$ and their bl
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The total endowment is $e_1 + e_2$.
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A competitive equilibrium requires that
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A competitive equilibrium requires that
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$$
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c_1 + c_2 = e_1 + e_2
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c_i = (\Pi^\top \Pi )^{-1}(\Pi^\top b_i - \mu_i p )
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$$
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Competitive equilibrium then requires that
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Competitive equilibrium then requires that
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$$
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e_1 + e_2 =
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for $\mu_i, i = 1,2$.
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**Exercise:** Show that, up to normalization by a positive scalar, the same competitive equilibrium price vector that you computed in the preceding two-consumer economy would prevail in a single-consumer economy in which a single **representative consumer** has utility function
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**Exercise:** Show that, up to normalization by a positive scalar, the same competitive equilibrium price vector that you computed in the preceding two-consumer economy would prevail in a single-consumer economy in which a single **representative consumer** has utility function
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$$
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-.5 (\Pi c -b) ^\top (\Pi c -b )
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* an $n \times 1$ vector $e$
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* a scalar "wealth" $W$ with default value $0$
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* **Production Costs** pinned down by
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* **Production Costs** pinned down by
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* an $n \times 1$ nonnegative vector $h$
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* an $n \times n$ positive definite matrix $J$
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The class will include a test to make sure that $b > > \Pi e $ and raise an exception if it is violated
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The class will include a test to make sure that $b > > \Pi e $ and raise an exception if it is violated
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(at some threshold level we'd have to specify).
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* **A Person** in the form of a pair that consists of
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* **Preferences** and **Endowments**
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* **A Pure Exchange Economy** will consist of
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* **A Pure Exchange Economy** will consist of
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* a collection of $m$ **persons**
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We can use the following steps to compute a competitive equilibrium:
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- First, we solve the single representative consumer economy by normalizing $\mu = 1$. Then, we renormalize the price vector by using the first consumption good as numeraire.
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- First we solve the single representative consumer economy by normalizing $\mu = 1$. Then, we renormalize the price vector by using the first consumption good as numeraire.
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- Next, we use the competitive equilibrium prices to compute each consumer's marginal utility of wealth:
Thus, we have verified that, up to choice of a numeraire in which to express absolute prices, the price vector in our representative consumer economy is the same as that in an underlying economy with multiple consumers.
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$$
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p=\tilde{\mu}^{-1}(\Pi^{\top}b-\Pi^{\top}\Pi e)
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$$
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Thus, we have verified that, up to choice of a numeraire in which to express absolute prices, the price vector in our representative consumer economy is the same as that in an underlying economy with multiple consumers.
To compute a competitive equilibrium for a production economy where demand curve is pinned down by the marginal utility of wealth $\mu$, we first compute an allocation by solving a planning problem.
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Then we compute the equilibrium price vector using the inverse demand or supply curve.
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Then we compute the equilibrium price vector using the inverse demand or supply curve.
Now we change the bliss point so that the consumer derives more utility from consumption.
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Now we change the bliss point so that the consumer derives more utility from consumption.
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```python
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PE.mu =1
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#### Example: Single Agent Two-Good Economy **with** Production
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* we'll do some experiments like those above
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* we can do experiments with a **diagonal** $\Pi$ and also with a **non-diagonal** $\Pi$ matrices to study how cross-slopes affect responses of $p$ and $c$ to various shifts in $b$
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* we can do experiments with a **diagonal** $\Pi$ and also with a **non-diagonal** $\Pi$ matrices to study how cross-slopes affect responses of $p$ and $c$ to various shifts in $b$
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```python
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```
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```python
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PE.b = np.array([12, 10])
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PE.b = np.array([12, 10])
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c, p =PE.competitive_equilibrium()
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```
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```python
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Pi = np.array([[1, 0.5],
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Pi = np.array([[1, 0.5],
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[0.5, 1]])
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b = np.array([10, 10])
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b = np.array([10, 10])
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h = np.array([0.5, 0.5])
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J = np.array([[1, 0.5],
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h = np.array([0.5, 0.5])
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J = np.array([[1, 0.5],
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[0.5, 1]])
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mu =1
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### A Monopolist
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Let's consider a monopolist supplier.
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Let's consider a monopolist supplier.
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We have included a method in our `productionEconomy` class to compute an equilibrium price and allocation when the supplier is a monopolist.
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We have included a method in our `productionEconomy` class to compute an equilibrium price and allocation when the supplier is a monopolist.
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Since the supplier now has the price-setting power
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- we first compute the optimal quantity that solves the monopolist's profit maximization problem.
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- Then we back out an equilibrium price from the consumer's inverse demand curve.
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- Then we back out an equilibrium price from the consumer's inverse demand curve.
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Next, we use a graph for the single good case to illustrate the difference between a competitive equilibrium and an equilibrium with a monopolist supplier.
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Recall that in a competitive equilibrium, a price-taking supplier equates marginal revenue $p$ to marginal cost $h + Hq$.
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Recall that in a competitive equilibrium, a price-taking supplier equates marginal revenue $p$ to marginal cost $h + Hq$.
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This yields a competitive producer's inverse supply curve.
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This yields a competitive producer's inverse supply curve.
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A monopolist's marginal revenue is not constant but instead is a non-trivial function of the quantity it sets.
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A monopolist's marginal revenue is not constant but instead is a non-trivial function of the quantity it sets.
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