Haggett, Hawtrey, Heckscher, Hitchcock, Holyrod, Hoover, Hotelling, Hyman, Isard,. Koopmans, Lam ... Harold Hotelling (1921). John Gordon Skellam (1951) ...
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Tönu Puu, CERUM, Umeå University, Sweden
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1826 - ???? Geometry of Shape and Size in Economic Space (including some Topological Aspects)
There was life before Krugman Alonso, Angel, Beckmann, Berry, Bos, Bunge, von Böwenter, Christaller, Dafermos, Enke, Haggett, Hawtrey, Heckscher, Hitchcock, Holyrod, Hoover, Hotelling, Hyman, Isard, Koopmans, Lam, Launhardt, Leontief, Lerner, Lösch, Marschak, Miksch, Miller, Mills, Moses, Mosler, Newell, Ohlin, Palander, Paelinck, Ponsard, Predöhl, Sax, Schneider, Sombart, Smeed, von Stackelberg, Tanner, von Thünen, Tinbergen, Tobler, Vaughan, Wardrop, Warndts, Weber, Weigmann, Wilson, Wingo, Zipf, a. o.
I SPATIAL TRADE AND PRICING Martin Beckmann (1952)
flow: vector field 𝜙=
𝜙1 𝑥1 , 𝑥2 , where 𝑥1 , 𝑥2 ∈ 𝑅 ⊂ ℝ2 are space coordinates 𝜙2 𝑥1 , 𝑥2
𝑑𝑥1 𝜙 cos𝜃 it has direction = = 𝑑𝑠 and volume |𝜙| = 𝑑𝑥2 |𝜙| sin𝜃 𝑑𝑠
𝜙12 + 𝜙22
1) the direction is the actual direction of trade flow on a map of the region studied 2) the volume is the actual volume of commodities transported in the trade flow due to Gauss′s (Green′s) theorem the change of flow is 𝛻 · 𝜙 =
𝜕𝜙1 𝜕𝜙2 + 𝜕𝑥1 𝜕𝑥2
this divergence 𝛻 · 𝜙 is related to local excess supply/demand 𝑧 𝑥1 , 𝑥2 and guarantees interregional market equilibrium with trade z can be taken as a given datum on the region; a scalar field other scalar fields are transportation cost 𝑘 𝑥1 , 𝑥2 , and commodity price 𝜆 𝑥1 , 𝑥2 ; the latter turns up as a Lagrange function, related to the interregional equilibrium constraint
Minimize
𝑅
𝑘|𝜙| 𝑑𝑥1 𝑑𝑥2 where
|𝜙|= 𝜙12 + 𝜙22 subject to
𝜕 2 𝜙1 𝜕 2 𝜙2 𝛻 · 𝜙 + 𝑧 = 0 where 𝛻 · 𝜙 = + 𝜕𝑥12 𝜕𝑥22
𝑘
𝜙 = 𝛻𝜆 where 𝛻𝜆 = |𝜙|
PDE
𝜕𝜆 𝜕𝑥1
2
+
𝜕𝜆 𝜕𝑥2
2
𝜕𝜆 𝜕𝑥1 𝜕𝜆 𝜕𝑥2
1) goods are transported in the price gradient direction 2) in this direction prices increase with transportation cost
𝜕𝜆 𝑑𝑥1 𝜙 1 𝜕𝑥1 = 𝑘 𝑥1 , 𝑥2 2 ODE 𝑑𝑠 = = 𝑑𝑥2 |𝜙| 𝑘 𝜕𝜆 𝑑𝑠 𝜕𝑥2
The one commodity model can be extended to a production economy with multiple commodities, where profits are imputed to land as rent. The picture shows a regular land rent landscape, which generalizes von Thünen’s theory. It can be proved that due to transversality in each location there is just one commodity which is both produced and traded interregionally and whose profitability determines rent. Beckmann and Puu (1985)
II MIGRATION AND POPULATION GROWTH Harold Hotelling (1921) John Gordon Skellam (1951)
Original Hotelling 1921 model with Malthusian growth and spatial diffusion, rediscovered in ecology by Skellam in 1951. Sustainable population s is given 𝜕𝑝 𝜕2𝑝 𝜕2𝑝 2 2 = 𝛾𝑝 𝑠 − 𝑝 + 𝛿𝛻 𝑝 where 𝛻 𝑝 = 2 + 2 𝜕𝑡 𝜕𝑥1 𝜕𝑥2 A possible variant where humanpower produces its means of subsistance by a production function with increasing/decreasing returns. 𝜕𝑝 𝑞 rephrase = 𝛾𝑝 − 𝜔 + 𝛿𝛻 2 𝑝 where 𝑞 = 2𝛼𝛽𝑝2 − 𝛼𝑝3 𝜕𝑡 𝑝 substitute, factorize, and define 𝑎 = 𝛽 − 𝛽2 −
𝜔 𝜔 , 𝑏 = 𝛽 + 𝛽 2 − 𝛼 𝛼
𝜕𝑝 then, rescaling to 𝛾 = 𝛿 = 1, = 𝑝 𝑏 − 𝑝 𝑝 − 𝑎 + 𝛻 2 𝑝, 𝜕𝑡 𝑏
which has closed form solution 𝑝 = 1+𝑒
−
𝑏 𝑏 𝑥1 +𝑥2 + 2 2 −𝑎 𝑡 2
Case of standing wave (a = b/2), with regions of high and low population density; where low local population growth can be compensated by immigration, high growth by emigration. When the wave moves, population density can go to maximum or extinction.
III DIFFUSION OF ECONOMIC GROWTH IN SPACE
Consider Harrod’s (1948) growth model, where savings balance investment, only generalized to an open economy. The constant import/export propensity is denoted m;it is applied to spatial income differences. Income in a location triggers imports, incomes in the surrounding points trigger exports. The difference is measured by the Laplacian of income, and export surplus adds to investment as an expansive element. 𝜕𝑌 𝜕2𝑌 𝜕2𝑌 2 2 𝑎 − 1 − 𝑐 𝑌 + 𝑚𝛻 𝑌 = 0 where 𝛻 𝑌 = 2 + 2 𝜕𝑡 𝜕𝑥1 𝜕𝑥2 𝑌 = 𝑇 𝑡 𝑆 𝑥1 , 𝑥2 , substitute, and divide by 𝑇𝑆:
Separate time and space:
1 𝑑𝑇 1 2 𝑎 − 1−𝑐 +𝑚 𝛻 𝑆 =0 𝑇 𝑑𝑡 𝑆
1 𝑑𝑇 define 𝜆=− 𝛻 2 𝑆, then 𝑎 − 1 − 𝑐 + 𝑚𝜆 𝑇 = 0 and 𝛻 2 𝑆 + 𝜆𝑆 = 0 𝑆 𝑑𝑡 Suppose we have a unit square region with eternal rest on the boundary. Separating space coordinates: 𝑆 𝑥1 , 𝑥2 = 𝑋1 𝑥1 𝑋2 𝑥2 , and dividing by 𝑋1 𝑋2 : 1 𝑑 2 𝑋1 1 𝑑 2 𝑋2 + = −𝜆 𝑋1 𝑑𝑥12 𝑋2 𝑑𝑥22
Then 𝑆 𝑥1 , 𝑥2 = sin 𝜋𝑖𝑥1 sin 𝜋𝑗𝑥2 , where 𝜆 = 𝑖 2 + 𝑗 2 𝑖, 𝑗 ∈ ℑ Solving also the linear temporal equation: 𝑌 = 𝐴𝑒
1 𝑣
𝑠+ 𝑖 2 +𝑗 2 𝑚 𝑡
sin 𝜋𝑖𝑥1 sin 𝜋𝑗𝑥2
∞
∞ 1 𝑠+ 𝑖 2 +𝑗 2 𝑚 𝑡 𝑣 𝐴𝑖𝑗 𝑒 sin
can be combined 𝑌 = 𝑖=1
𝜋𝑖𝑥1 sin 𝜋𝑗𝑥2
𝑗=1
Growth and stagnation regions on a square with boundary conditions of constancy.
IV BUSINESS CYCLES IN SPACE
Business cycles, unlike growth, were usually modelled in discrete time. The reason was probably their backgound in Keyenesian macroeconomics, and its relation to the emergent periodized national accounting. As Phillips (1954) showed, the same phenomena as focused in Samuelson (1939) could also be put in continuous time. In the Phillips model, instead of equating investments to saving, let the rate of increase of income be proportional to the difference between investments and saving: 𝑑𝑌 𝑑𝐼 𝑑𝑌 = 𝐼 − 1 − 𝑐 𝑌, also let = 𝑎 − 𝐼 , investments change in proportion to 𝑑𝑡 𝑑𝑡 𝑑𝑡 the difference of accelerator triggered investments and actual invetsments. There are adaptation coefficients, but we have the units of time and income as degrees of freedom and can put these coeficients equal to unity. One can differentiate the first equation once more, substitute from the second for dI/dt, and use the first as it is to eliminate I, so getting the reduced form: 𝑑2𝑌 𝑑𝑌 − 𝑎+𝑐−2 + 1 − 𝑐 𝑌 = 0 with solution 𝑌 = 𝐴𝑒 𝛼𝑡 cos 𝜔𝑡 + 𝜑 , where 2 𝑑𝑡 𝑑𝑡 1
1
𝛼= 2 (𝑎 + 𝑐), 𝜔= 2 4𝑎 − (𝑎 + 𝑐)2 , 𝑎𝑛𝑑 𝐴, 𝜑 are arbitrary, determined by initial conditions.
As in the growth model consider an open economy where export surplus enters as an expansive element along with investments. Also as in the growth case, export surplus can be obtained though an import coefficient m, multiplied with the Laplacian of income 𝛻 2 𝑌 as a measure of spatial income difference in a point and its surroundings. Then the Phillips model becomes: 𝜕2𝑌 𝜕𝑌 𝜕2𝑌 𝜕2𝑌 2 2 − 𝑎+𝑐−2 − 𝑚𝛻 𝑌 + 1 − 𝑐 𝑌 = 0 where 𝛻 𝑌 = 2 + 2 𝜕𝑡 2 𝜕𝑡 𝜕𝑥1 𝜕𝑥2 It is linear, like the Harrod growth model for an open economy, and can be handled through separating time and space coordinates. Put 𝑌 = 𝑇 𝑡 𝑆 𝑥1 , 𝑥2 , substitute, and 1 divide by 𝑇𝑆. Again, defining 𝜆=− 𝑆 𝛻 2 𝑆, we get two equations: 𝑑2𝑇 𝑑𝑇 − (𝑎 + 𝑐 − 2) − 1 − 𝑐 + 𝑚𝜆 𝑇 = 0 and 𝛻 2 𝑆 + 𝜆𝑆 = 0 2 𝑑𝑡 𝑑𝑡 The temporal equation is now slightly more complicated, second order, and thus produces oscillations rather than growth. The spatial eigenvalue/eigenfunction problem is the same. Its solution depends on the shape of the region and the boundary conditions. Again, a square or rectangle can be dealt with by trigonometric functions, a disc by Bessel functions, and a sphere (the whole globe) by Legendre polynomials. Each solution results in an Eigenvalue 𝜆 that feeds back in the temporal equation. Combining different solutions, more or less complex nets of node lines, as in the picture, can be obtained where the cycles move in opposite phase on either side.
Actually, the growth picture could illustrate cycles at a frozen moment of time equally well. A difference is that growth is at a uniform speed over all space, whereas in the case of cycles the smaller subdivisions move faster than the larger. Both models suffer from the default common to all linear models – they either go to rest or explode. In the case of growth, in some regions income grows (or decreases) beyond any limit, in the case of cycles, the amplitude either goes to zero or increases (in both positive and negative direction of the swings). In the case of business cycles, one can introduce bounds such that we get persistent, though bounded oscillations. (In the case of growth introducing such nonlinearity is, of course self-contradictory.) In the case of discrete time, the present author introduced a method of ”relative dynamics” which made it possible to study growing systems. This is tricky in case of continuous time as in a continuous orbit such relative variables would result in division by zero.
CUBIC NONLINARITY As an example of nonlinearity, consider the multiplier-accelerator model in continous time according to Phillips (1954) augmented by iterregional trade. However, let us make the principle of acceleration nonlinear through subtracting a cubic term from the leading linear term. 𝜕2𝑌 𝜕𝑌 𝜕𝑌 2 + 1 − 𝑐 𝑌 − 𝑚𝛻 𝑌 = 𝑎 + 𝑐 − 2 − 𝑎 𝜕𝑡 2 𝜕𝑡 𝜕𝑡
3
2 2 𝜕 𝑌 𝜕 𝑌 where 𝛻 2 𝑌 = 2 + 2 𝜕𝑥1 𝜕𝑥2
Just one solution is a pyramid shaped wave with flat sides and constant rates of change. The wave number can take any value, but as the system is nonlinear, the solutions with different wave numbers cannot be combined. In one dimesion (oscillation in overhead lines due to wind) such solutions have been proved to be attractive.
𝜕2𝑌 𝜕𝑌 2 Solution: 2 = 0, 𝛻 𝑌 = 0, = 𝜕𝑡 𝜕𝑡
𝑎+𝑐−2 − 𝑎 0 +
𝑎+𝑐−2 𝑎
CUBIC NONLINARITY
One possible solution to the nonlinear accelerator model with trade in space.
TOPOLOGY MARKET AREAS Hexagons Christaller (1933), Lösch (1940)
OPTIMAL COMPACTNESS Average distance from centre of a unit area n-gon to all points: 𝜋 𝜋 𝜋 1 + cot lntan + 𝑛 4 2𝑛 𝜋 𝜋 3 𝑛cos 𝑛 sin 𝑛 = 0.4036 (n=3), 0.3826 (n=4), and 0.3761 (n=6). Hexagon most economical, but savings of transportation costs are 1.43 % from squares to hexagons. What about relocation costs? Is there a better reason for hexagonal shapes than optimality?
WARNING: In physical experiment of compressing spheres (peas or lead shot), always rhombic dodecahedra turn up – never the more compact shapes Lord Kelvin proved to be better. Friction? Why should firms take huge relocation costs to save 1.43 % in transportation?
STRUCTURAL STABILITY 2+2=1+3 2+1=0+3 2+0=?+3
two surfaces intersct along curve in space a third surface intersects this curve in a point no way for a fourth surface to intersect a point
Hexagons intersect three by three, squares four by four. On a map of states, we only see four country incidence where the map was drawn by an administrator with pen and ruler, never where national state boundaries as results of wars an peace treaties meet.
Launhardt (1885) ”funnels” and market areas
MORE TOPOLOGY FLOWS OF TRADE Beckmann (1952) model
STRUCTURAL STABILITY AND CATASTROPHE
𝜕𝜆 𝑑𝑥1 1 𝜕𝑥1 recall ODE 𝑑𝑠 = 𝑑𝑥2 𝑘 𝜕𝜆 𝑑𝑠 𝜕𝑥2
this is a system for determining the trade flow portrait as the gradient to the price potential
1) by the theorem for existence and uniqueness of solotions to ODE, there is one and only one solution curve through each point, unless both RHS vanish at singular points. 2) these singular points and their incident trajectories give structure to the phase portrait 3) such a graph of arcs and nodes can be divided in elementary trinagles 4) Peixoto’s Theorem (1977) says that for a structurally stable flow (i) there are a finite number of singular points (ii) which are stable nodes SN, unstable nodes UN, or saddle points SP, and that there are no (heteroclinic) saddle connections. 5) consequently there cannot be two saddle points as corners in the triangles, and, of course, not two nodes of the same type, as all sides of a triangle are connected. SP
SN
UN
How can the elementary triangles be organized in regular tessellation elements? 1) 6 by 6 as triangles, or 2) 8 by 8 as squares, or 3) 12 by 12 as hexagons
The triangles and hexagons result in the same tessellation shape, with twice as many traingles as hexagons or the other way around. Note that the pure hexagonal tessellation is unstable. The square tessellation is different, with equal numbers of different nodes.
Once we have the tessellation skeleton, one just needs to fill out with the regular flow trajctories, and the orthogonal constant price contours. The square case as an example
PARAMETER SPACE As there are two stable regular tessellations, one may want to work out the transitions between them. This can be done using the elliptic umblic catastrophe. Thom (1972), in local form
x13 3x1 x12 ax1 bx2 c( x12 x22 )
PHASE SPACE
Clobal bifurcation examples can be worked out using trigonometric functions, of which Thom’s normal form is a Taylor expansion. Note that all these characterisations are topological, they apply to all distorted cases obtained by stretching without cutting or gluing. Further, note that the examples use regular patterns; of course they can be combined.
