Profit Model from Ellison Glaeser 1997

Question

Can someone help me understand the following profit maximization problem (from Ellison Glaeser 1997)?

Manufacturing plants choose where to locate each of $N$ plants in one of $M$ locations. The profit from locating plant $k$ in location $i$ is given by

$log(\pi_{ki})=log(\bar{\pi}_{i})+\sum_{l\neq k}\delta_{kl}(1-\lambda_{li})(-\infty)+\epsilon_{ki}$

where $\bar{\pi}_{i}$ is the natural advantage from locating in $i$ to all plants (i.e. it doesn't depend on $k$), $\delta_{kl}$ is an Bernoulli random variable that is 1 with probability $\gamma^{s}$ (supposedly indicating the existence of a positive spillover possibility between plants $k$ and $l$), $\lambda_{li}$ is an indicator for whether plant $l$ is located in $i$, and $\epsilon_{ki}$ is an error term.

The model is meant to demonstrate the fact that plants choose to locate in certain locations both because of natural advantages of that location and because of agglomeration-type spillovers. My problem is with the second term, which doesn't seem to make much sense.

Thinking through the possible values of $\delta_{kl}(1-\lambda_{li})(-\infty)$ $\delta_{kl}(1-\lambda_{li})(-\infty)=\begin{cases} 0*1*-\infty=? & \text{ if }\delta_{kl}=0\text{ and }\lambda_{li}=0\\ 0*0*-\infty=? & \text{ if }\delta_{kl}=0\text{ and }\lambda_{li}=1\\ 1*1*-\infty=? & \text{ if }\delta_{kl}=1\text{ and }\lambda_{li}=0\\ 1*0*-\infty=? & \text{ if }\delta_{kl}=1\text{ and }\lambda_{li}=1 \end{cases}$

if we define $0*-\infty=0$ and $1*-\infty=-\infty$ we get $\delta_{kl}(1-\lambda_{li})(-\infty)=\begin{cases} 0*1*-\infty=0 & \text{ if }\delta_{kl}=0\text{ and }\lambda_{li}=0\\ 0*0*-\infty=0 & \text{ if }\delta_{kl}=0\text{ and }\lambda_{li}=1\\ 1*1*-\infty=-\infty & \text{ if }\delta_{kl}=1\text{ and }\lambda_{li}=0\\ 1*0*-\infty=0 & \text{ if }\delta_{kl}=1\text{ and }\lambda_{li}=1 \end{cases}$

which still makes no sense since the only possible value of $\sum_{l\neq k}\delta_{kl}(1-\lambda_{li})(-\infty)$ is zero or minus infinity. My next thought was that maybe I could transform this function to get profits alone:$\pi_{ki}=\bar{\pi}_{i}\prod_{l\neq k}e^{\delta_{kl}(1-\lambda_{li})(-\infty)}e^{\epsilon_{ki}}$

but now the term $\prod_{l\neq k}e^{\delta_{kl}(1-\lambda_{li})(-\infty)}$ can only possibly be zero or 1.

Answer 1

The convention: $0\times -\infty = 0$ and $1\times -\infty$ does make economic sense here. Yes, the term $\delta_{kl}(1-\lambda_{li})(-\infty)$ is either zero or minus infinity.

The story is the following: If $\delta_{kl}$ is one, it means that it is prohibitively costly to operate plant $k$ if plant $l$ is not present in the same location. If $\delta_{kl}$ is zero it means that plant $k$ can be operated independently of whether plant $l$ is or is not in the same location.

Caveats: I'm assuming the info. in the question is correct (I did not read the original). IMHO, the model assumes a very "inflexible" technology (think Leontieff). My guess is they are assuming this to help later if they will use empirical data to estimate stuff.