Melnikov (2001)

Demand for Differentiated Durable Products

These notes are based on the following article:

Melnikov, Oleg (2001). Demand for Differentiated Durable Products: The Case of the U.S. Computer Printer Market. Unpublished manuscript, Yale University.




Consumer’s Problem

Consumer’s Problem: Reformulation

(1)J(v t,I t)=max{v t,c+βE tJ(v t+1,I t+1)}.



Diffusion Process

r t+1=μ(r t)+σ(r t)ν t+1

μ(r) and σ(r) must satisfy the following properties:

  1. μ(r) and σ(r) are continuous and differentiable a.e.
  2. 0σ(r) for all r.
  3. r t is a weak submartingale: μ(r t)r t.
  4. lim nβ nμ n(r) where 0β1, μ 0(r)=μ(r), and μ n(r)=μ(μ n1(r)).

Solving the Consumer’s Problem

Under the previous assumptions, we can write (1) as J(v,r)=max{v,W(r)} where v has a Type 1 Extreme Value distribution with mode r and W(r)=c+betaE[J(v,r)|r] is the reservation utility. This is an optimal stopping problem with stopping set 𝒮={v|vW(r)}.


Hazard Rate

Hazard Rate: Numerical solutions


Econometric specification

The econometrician observes:

Define the following:

Estimation: Static Parameters

Estimation: Transition Kernel

Estimation: Dynamic Parameters

(2)Q^ t(θ v,q,N t;r^ t,θ^ r)=N th(r^ t;θ^ r,θ v).
(3)N t+1=N tπ 0t(r^ t;θ^ r,θ v)+q(M tN t)+(M t+1M t).

Monte Carlo Results

U.S. Printer Market

Data Sources

Descriptive Statistics


Empirical Results