Walking in Memphis

Memphians don't like to walk. What does this mean for Beale Street Landing?

Memphis is not a city that likes to move. Combine oppressive summer heat with world famous barbecue and a population of which only half report exercising frequently, and the result is not surprising: Memphis is a plump place. In fact, it was the fattest city in the nation in 2012.

Memphis is a city that likes to invest large sums of public money into pie-in-the-sky projects. After years of delays and budget overruns, Beale Street Landing is finally nearing completion. But a crucial piece of the Beale Street Landing puzzle is still missing. The Riverfront Development Corporation planned to anchor the project with a restaurant, but to the frustration of all, a restaurant tenant has yet to be found.

So what does Memphis' lethargy have to do with Beale Street Landing?

Consider that parking issues ensure that most downtown patrons will be on foot. Consider also that Memphis' hot summers and large stomachs mean that there is a real cost to walking, a cost paid with sore feet and a sweaty brow. Now add the fact that Beale Street Landing is several blocks away from any other restaurants or bars, and the result is a lack of incentive for a restaurant to move in.

We can illustrate the situation with a simple game theory location model. But before we can apply the model, we need to make some simplifications. Imagine Beale Street as a one dimensional strip with one end, 0, at Riverside Drive and the other end, 1, at Danny Thomas Boulevard. Assume customers are uniformly distributed along Beale Street (this is equivalent to saying that the point where customers enter Beale Street is random with each point just as likely as any other point). Assume also that each restaurant and bar on Beale Street sells identical meals and drinks and charges an identical price (we’ll relax this assumption later) and that consumers buy exactly 1 or 0 meals. Finally, assume that customers will go to the cheapest option, including the cost of walking, and that every customer can afford a meal.

No restaurant has committed to moving into Beale Street Landing yet, so let’s first consider the simplest case of a restaurant that can pick any location along Beale. If prices are identical, consumers will choose the closest place. Divide the bars and restaurants of Beale Street into two groups: the prospective Beale Street Landing restaurant (“B.S.L.”) and everything else (“Bars”). Beale Street Landing is at the very bottom of Beale, while the other bars and restaurants are about halfway between Riverside (0) and Danny Thomas (1). If the restaurant moves into Beale Street Landing, it will get one 1/4th of the customers on Beale, while the bars farther up will get 3/4ths.

A rational restaurateur will move in closer to the other restaurants. If the restaurant moves to the halfway point between Riverside (0) and the bars, it will get 1/4 + 1/8 = 3/8 of the customers, while the bars will get 1/2 + 1/8 = 5/8 of the customers. If the restaurant chooses an even closer location, it will further increase its share; to get the maximum share, it would choose a location immediately next to the bars at the halfway point between Riverside and Danny Thomas. Both will then get 1/2 of the customers. (Side note: this analysis treats all the “other” restaurants and bars as a single entity. If this single entity is also allowed to move, the equilibrium will still be at the halfway point. If we treat each bar as a separate entity, though, a stable equilibrium does not exist!)

Now let’s assume the restaurant is already committed to moving in to Beale Street Landing, but is free to set its own price. At this point, we need to formalize the full cost of each option, including the cost of walking. Let t be the physical toll of walking one unit of distance. Let d be the distance a consumer walks. Let’s make the cost of walking quadratic: each step is more painful than the last. The total transportation cost then is td^2. Suppose a customer enters Beale Street at random distance x from Riverside Drive. The cost of walking to Beale Street Landing and buying a meal is the price of a Beale Street Landing meal, P_BSL, plus the transportation cost, tx^2: P_BSL + tx^2. The cost of walking to the bars and buying a meal is P_Bars + t(1/2 – x)^2.

 

At what point will a customer face the same cost for both options? That would be where the cost of the meal plus walking is the same for each, the point x_i. If the prices are the same, then we get our initial result: the indifferent customer is at location 1/4, or halfway between the bars and Beale Street Landing. If Beale Street Landing lowers their price below the bars, they will attract customers from farther away as the lower price offsets some of the misery of walking. The indifferent customer would then be farther to the right on our [0, 1] diagram.

What does the Beale Street Landing restaurant need to do to maximize its profits given its distant location? To answer this, we need demand functions that tell us what share each would get given the two prices and the cost of walking. Distances on the [0,1] line can be thought of as shares of customers, so we generate demand functions by finding the point where the prices are equal; the consumers to the left of the equilibrium point go to B.S.L., while the consumers to the right go to the Bars. As expected, the demand functions show that if the prices are the same, Beale Street Landing will get 1/4th and the bars will get 3/4th. To optimize, we use the usual profit function given by the price times the quantity sold (in this case, demand) minus the cost of producing a meal (c) times the number of meals produced (also demand).

Maximising profit for each and solving the system of equations gives the optimal prices, P*_BSL and P*_Bars. Plug these prices into the demand functions to get shares: if Beale Street Landing sets its price optimally, the most it can get is 5/12 of the customers, while the bars will get 7/12. Comparing profits, Beale Street Landing will only make about half what the bars make. Of course, these numbers are a result of our stylized model and shouldn't be taken as exact predictions, but they illustrate the point.

So it’s not looking good for Beale Street Landing. Less mathematical versions of this same argument have members of the press calling the project a fiasco. Of course, our simple model doesn’t capture all the complexities of real life: (1) meals and drinks are not identical up and down Beale; (2) a solution may come along that improves access, like bikes or more parking; (3) people may be willing to pay a premium, both in terms of price and walking costs, for the novelty of eating at a restaurant on the riverfront. And with the new Greenline and Greenway expansions, maybe Memphians are finally warming up to the idea of walking.

But for now, this simple model predicts that the RDC will have a hard time finding a restaurant to move in to Beale Street Landing. So far, the predictions match reality.