Basic Types of Goods

  • From last lecture, there exist luxury and necessary goods
  • Inferior goods are goods that have demands which fall with income
  • Normal goods are goods that have demands which rise with income
  • In other words, normal goods have positively sloped Engel curves while inferior goods have negatively sloped Engel curves

Equilibrium Chapter

  • A market is in equilibrium when the total quantity demanded by buyers is equal to the total quantity supplied by sellers
    • Equilibrium must be maintained in order to avoid shortages or surpluses such that there is no waste
  • Mathematically, given a price p, the quantity demanded is D(p) and the quantity supplied is S(p)
    • Equilibrium is defined as the point (p*, q*)

Equilibrium with Linear Supply + Demand Curves

  • In this situation, D(p) = a - bp and S(p) = c + dp
    • To find p*, set the equations equal to each other
At the equilibrium, D(p∗)=S(p∗)a−bp=c+dp→p∗=a−cb+dq∗=a−b(a−c)b+d=a(b+d)b+d−b(a−c)b+d=ab+adb+d−ab−bcb+d=ad+bcb+d\text{At the equilibrium, } D(p^*) = S(p^*) \\ a - bp = c + dp \rightarrow p^* = \frac{a - c}{b + d} \\ q^* = a - \frac{b(a - c)}{b + d} = \frac{a(b + d)}{b + d} - \frac{b(a - c)}{b + d} \\ = \frac{ab + ad}{b + d} - \frac{ab - bc}{b + d} = \frac{ad + bc}{b + d}

We can do the same thing with the inverse demand and supply functions.

q=D(p)=a−bp  ⟺  p=a−qb=D−1(q)q=S(p)=c+dp  ⟺  p=−c+qd=S−1(q)At the equilibrium quantity, D−1(q∗)=S−1(q∗)a−q∗b=−c+q∗d→d(a−q∗)=b(−c+q∗)da−dq∗=−bc+bq∗→q∗=ad+bcb+dp∗=a−ad+bcb+db→p∗=a−cb+dq = D(p) = a - bp \iff p = \frac{a - q}{b} = D^{-1} (q) \\ q = S(p) = c + dp \iff p = \frac{-c + q}{d} = S^{-1} (q) \\ \text{At the equilibrium quantity, } D^{-1} (q^*) = S^{-1} (q^*) \\ \frac{a - q^*}{b} = \frac{-c + q^*}{d} \rightarrow d(a - q^*) = b(-c + q^*) \\ da - dq^* = -bc + bq^* \rightarrow q^* = \frac{ad + bc}{b + d} \\ p^* = \frac{a - \frac{ad + bc}{b + d}}{b} \rightarrow p^* = \frac{a - c}{b + d}

Special Cases

  • Quantity supplied is fixed, independent of the market price; that is, the supply curve is now a vertical line, and q* = c for some fixed quantity c
    • In our original equations, S(p) = c + dp, so in this case, d = 0
    • p^* = D-1(q*) = (a - c)/b
  • Quantity supplied is extremely sensitive to the market price; that is, the supply curve is now a horizontal line, and S-1(q) = p*
    • Because p* is fixed, we need to find q*
    • q* = a - bp*
    • There is no “easy” way to find the optimal price because d would be infinity in this case