Demand for discrete goods

Summary

  • Setup:
    • Good 1 is discrete (integer)
    • \(p_2=1\) , \(p_1x_1+x_2=m\)
  • Reservation price: \(r_n\) is the price where you are indifferent between \(x_1=n-1\) and \(x_1=n\)
  • With strictly convex preferences: \(r_1>r_2>…\)
  • Optimal choice \(x_1\) :
    • If \(p_1=r_n\) : \(x_1=n-1\) or \(x_1=n\) (indifferent)
    • If \(r_{n+1}<p_1<r_n\) : \(x_1=n\)
    • If \(p_1=r_{n+1}\) : \(x_1=n\) or \(x_1=n+1\) (indifferent)
  • Optimal choice \(x_1\) :
    • \(x_1=n\) if and only if \(r_{n+1}≤p_1≤r_n\)

  • Implication for utility function:
    • \(p_1=r_n\) Indifferent between \(x_1=n-1\) and \(x_1=n\)
    • If \(x_1=n\) then \(x_2=m-r_n⋅n\)
    • If \(x_1=n-1\) then \(x_2=m-r_n⋅\left( n-1 \right)\)
    • \(u\left( n-1,m-r_n⋅\left( n-1 \right) \right)=u\left( n,m-r_n⋅n \right)\)
  • Quasilinear preferences:
    • \(u\left( x_1,x_2 \right)=v\left( x_1 \right)+x_2\) where \(v\left( x_1 \right)\) strictly concave
    • \(u\left( n-1,m-r_n⋅\left( n-1 \right) \right)=u\left( n,m-r_n⋅n \right)\) im plies
    • \(v\left( n-1 \right)+m-r_n⋅\left( n-1 \right)=v\left( n \right)+m-r_n⋅n\) which implies
    • \(r_n=v\left( n \right)-v\left( n-1 \right)\)
    • Can normalize \(v\) such that \(v\left( 0 \right)=0\)