misc

Author: jstac

lectures/_toc.yml

@@ -8,6 +8,12 @@ parts:
   - file: long_run_growth
   - file: business_cycle
   - file: inequality
+- caption: Supply and Demand
+  numbered: true
+  chapters:
+  - file: intro_supply_demand
+  - file: supply_demand_multiple_goods
+  - file: supply_demand_heterogeneity
 - caption: Tools & Techniques
   numbered: true
   chapters:
@@ -29,7 +35,6 @@ parts:
   chapters:
   - file: schelling
   - file: solow
-  - file: sd_foundations
   - file: cobweb
   - file: olg
 - caption: Other

lectures/intro_supply_demand.md

@@ -0,0 +1,119 @@
+# Introduction to Supply and Demand
+
+This lecture is about some  linear models of equilibrium prices and
+quantities, one of the main topics of elementary microeconomics.
+
+Our approach is first to offer a  scalar version with one good and one price.
+
+## Outline
+
+We shall describe two classic welfare theorems:
+
+* **first welfare theorem:** for a  given a distribution of wealth among consumers,  a competitive  equilibrium  allocation of goods solves a  social planning problem.
+
+* **second welfare theorem:** An allocation of goods to consumers that solves a social planning problem can be supported by a competitive equilibrium with an appropriate initial distribution of  wealth.
+
+Key infrastructure concepts that we'll encounter in this lecture are
+
+* inverse demand curves
+* marginal utilities of wealth
+* inverse supply curves
+* consumer surplus
+* producer surplus
+* social welfare as a sum of consumer and producer surpluses
+* competitive equilibrium
+
+## Supply and Demand
+
+We study a market for a single good in which buyers and sellers exchange a  quantity $q$ for a price $p$.
+
+Quantity $q$ and price  $p$ are  both scalars.
+
+We assume that inverse demand and supply curves for the good are:
+
+$$
+p = d_0 - d_1 q, \quad d_0, d_1 > 0
+$$
+
+$$
+p = s_0 + s_1 q , \quad s_0, s_1 > 0
+$$
+
+We call them inverse demand and supply curves because price is on the left side of the equation rather than on the right side as it would be in a direct demand or supply function.
+
+
+
+We define **consumer surplus** as the  area under an inverse demand curve minus $p q$:
+
+$$
+\int_0^q (d_0 - d_1 x) dx - pq = d_0 q -.5 d_1 q^2 - pq
+$$
+
+We define **producer surplus** as $p q$ minus   the area under an inverse supply curve:
+
+$$
+p q - \int_0^q (s_0 + s_1 x) dx = pq - s_0 q - .5 s_1 q^2
+$$
+
+Sometimes economists measure social welfare by a **welfare criterion** that equals  consumer surplus plus producer surplus
+
+$$
+\int_0^q (d_0 - d_1 x) dx - \int_0^q (s_0 + s_1 x) dx  \equiv \textrm{Welf}
+$$
+
+or
+
+$$
+\textrm{Welf} = (d_0 - s_0) q - .5 (d_1 + s_1) q^2
+$$
+
+To compute a quantity that  maximizes  welfare criterion $\textrm{Welf}$, we differentiate $\textrm{Welf}$ with respect to   $q$ and then set the derivative to zero.
+
+We get
+
+$$
+\frac{d \textrm{Welf}}{d q} = d_0 - s_0 - (d_1 + s_1) q  = 0
+$$
+
+which implies
+
+$$
+q = \frac{ d_0 - s_0}{s_1 + d_1}
+$$ (eq:old1)
+
+Let's remember the quantity $q$ given by equation {eq}`eq:old1` that a social planner would choose to maximize consumer plus producer surplus.
+
+We'll compare it to the quantity that emerges in a competitive  equilibrium equilibrium that equates
+supply to demand.
+
+Instead of equating quantities supplied and demanded, we'll can accomplish the same thing by equating demand price to supply price:
+
+$$
+p =  d_0 - d_1 q = s_0 + s_1 q ,
+$$
+
+
+It we solve the equation defined by the second equality in the above line for $q$, we obtain the
+competitive equilibrium quantity; it equals the same $q$ given by equation  {eq}`eq:old1`.
+
+The outcome that the quantity determined by equation {eq}`eq:old1` equates
+supply to demand brings us a **key finding:**
+
+*  a competitive equilibrium quantity maximizes our  welfare criterion
+
+It also brings  a useful  **competitive equilibrium computation strategy:**
+
+* after solving the welfare problem for an optimal  quantity, we can read a competitive equilibrium price from  either supply price or demand price at the  competitive equilibrium quantity
+
+Soon we'll derive generalizations of the above demand and supply
+curves from other objects.
+
+Our generalizations will extend the preceding analysis of a market for a single good to the analysis
+of $n$ simultaneous markets in $n$ goods.
+
+In addition
+
+ * we'll derive  **demand curves** from a consumer problem that maximizes a **utility function** subject to a **budget constraint**.
+
+ * we'll derive  **supply curves** from the problem of a producer who is price taker and maximizes his profits minus total costs that are described by a  **cost function**.
+<!-- #endregion -->