ESTABLISHMENT LOCATION ECON 376 (by Prof. Kilkenny) 9/2/2003

The Classic Firm Location Problem

Consider a business that process an input from one location, and sells the processed output in a market in a different location. What is the best location for that business?

Prof. Kilkenny’s “5-Question Approach” to solving this problem:

Question 1: who =businesses owner

Question 2: their objective = maximize profit

Question 3: instruments = choose site "s" among (1) the input location, (2) the market, or (3) in-between

Question 4: constraints:

transport cost rates: t_o, t_i (subscripts: “o” for outputs, “i” for inputs)

distance: d_is + d_sm = d_im (subscripts: “i” for input location, “s” for chosen site, “m” output market location)

technology: q_i = amount of input i per unit output

L = amount of labor per unit output

K = amount of capital per unit output

output price: P_o

input price: P_i

wage rates at the site location: w_s

rent and fixed costs at the site: r_s

taxes less incentives at the site: T_s

Question 5: time horizon: period within which the constraints do not change

Given the five answers, the objective (profit, Π) is expressed as a function of the choices (distances to site) and the constraints (costs, prices, etc):

Π_s = P_o·Q - P_i·q_i·Q - w_s·L∙Q - r_s·K∙Q - T_s- t_i·q_i·d_is·Q - t_o·d_sm·Q

This formula can be simplified a lot:

(i) if labor costs, rents, taxes or subsidies do not vary across sites,

then w_s = w, r_s=r, and T_s = T:

Π _s = P_o·Q - P_i·q_i·Q - w·L∙Q - r·K∙Q - T- t_i·q_i·d_is·Q - t_o·d_sm·Q

(ii) also, prices paid for outputs and charged for inputs do not depend on the site of the plant.

When we are comparing sites, variables that are the same everywhere cancel out. So drop them from the site choice problem:

Π _s = P_o·Q - P_i·q_i·Q - w·L∙Q - r·K∙Q - T- t_i·q_i·d_is·Q - t_o·d_sm·Q

è Π _s = - t_i·q_i·d_is·Q - t_o·d_sm·Q

(iii) express the problem on a per unit output basis by dividing through by Q:

Π _s/Q = - t_i·q_i·d_is - t_o·d_sm

Now we can see above that to maximize profit, the business should choose a site to avoid high transport costs. Rewrite the problem as:

MIN t_i·q_i·d_is + t_o·d_sm

The solution to this “linear complimentarity” problem requires applying the “Kuhn-Tucker algorithm” (or use of a Varignon frame). It is a bit more complicated than the method of Lagrange for solving constrained optimization problems. Because ECON 301 is not a prerequisite for this class, we’ll just have to memorize the solution criteria:

SOLUTION Criteria:

if t_i·q_i > t_o, locate "on top of the inputs" (choose d_is = 0)

if t_i·q_i < t_o, locate "on top of the market" (choose d_sm = 0)

if t_i·q_i = t_o, locate anywhere in-between the input site and the market.

Mnemonic device: our “Varignon Frame” for this simple location problem in linear space is a teeter-totter. See below:

Illustrating the case of a weight-gaining or market oriented firm:

The horizontal axis represents a “road” between the market and where the input is. The distance between is d_i,m miles. The vertical axes measure transport costs.

The red line shows the input procurement transport costs. If a firm chooses its site “s” at the input location, d_i,s = 0, so that cost is zero. The farther away it is from the inputs, the higher are those costs. The slope of the red line is t_i∙q_I which is the transport cost rate per unit input per mile (t_i) times the amount of the input needed per unit output (q_i). The more input needed per unit output, or the higher is the transport rate, the “heavier” is the monetary weight of inputs in total transport costs and the more incentive a firm has to choose the site which minimizes those costs.

The blue line represents output distribution costs. If a firm chooses its site “s” at the market, d_s,m = 0, so that cost is zero. The farther away the site is from the market, the higher are those costs. The slope of the blue line is t_o, that is the transport cost rate per unit output. The higher is the output transport rate, the “heavier” is the monetary weight of outputs in total transport costs and the more incentive a firm has to choose the site which minimizes those costs.

The black line indicates the total transport costs per unit (T/Q) for each possible site along the “road.” It is the sum of procurement and distribution transport costs. The best site is where total per unit transport costs, the black line, is lowest. The above illustrates the case where the market is the least-cost site. This is because output transport is costlier than input transport, per unit output (t_o> t_iq_i). This higher cost can be avoided by locating at the market. This type of business is thus called “market-oriented.”

QUESTION 1: Under what circumstances might the minimum transport cost site NOT be the one that also maximizes profit? ANSWER: If input prices, output prices, or production costs vary across sites, those costs may reduce profit at the least-transport-cost site. For example, a downtown location may minimize transport costs if your customers are downtown, but land rents are higher there, so total costs may be higher downtown (and profits lower) than outside of town.

QUESTION 2: Under what circumstances would “somewhere between the input source and the market” be an optimal site for some firm? Only when per unit output and input transport cost rates are EXACTLY the same. This would be illustrated in the graph by the red and blue lines having the same slopes, and the black line would be flat—T/Q the same anywhere. This is unlikely! Also, if inputs are ubiquitous, then a “median location” may be optimal. (We need a much more complicated model to prove this case, and our one-site model does a pretty good job anyway .)

Example: Consider the meat-packing industry. One hundred years ago, it cost less to drive cattle to slaughter than to deliver cut beef to consumers, per pound. Where do you expect packing plants to have been located in 1899? When t_iq_i < t_o, the market is the optimal site, as we show above. (That’s why packing plants were in cities, like Chicago, the location of 1906 novel “The Jungle” by Sinclair Lewis.)

Nowadays, it costs more to move cattle than cut beef. So, nowadays, t_iq_i > t_o . Where do you expect to find packing plants now? Near feedlots, which are probably near feed and farms. Indeed, meat-packing is currently one of the few industries that optimally locates near farms and feed.

QUESTION 3. What does “the principle of median location” imply about the location of weight-losing industries, like food processing, that have multiple input source locations? Will such firms be more profitable in rural or urban locations? This question is more complex that it appears. The fact that food processing industries are “weight-losing” implies that rural/farm locations are optimal. The fact that food processors source inputs from many farms suggests that the most accessible, central location is optimal. Sites that are central to farm regions are, however, cities. Thus, food processing industry optimal locates in cities—that are central to farm regions. See the bar chart below: