# Optimal Spatial Growth of Employment and Residences Written by Ralph Braid Journal of Urban Economics 24(1988) Presented by Jing Zhou.

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Optimal Spatial Growth of Employment and Residences Written by Ralph Braid Journal of Urban Economics 24(1988) Presented by Jing Zhou

1. Introduction 1.This paper examines the location patterns of employment and residences in an urban area characterized by irreversible land use commitments and smoothly growing population. 2. The main problem here is how to use the land efficiently. That is, how to divide the land between employment and residential use to get maximum economic return. Should we use land by complete integration, complete segregation or a mixture of different uses? 3. Assume all output is shipped to the center and there’s commuting costs for workers to go to employment areas. 4. The optimal solution should minimize the PDV of aggregate shipping costs plus commuting costs

2. Basic Assumptions of the Model 1. Population increases over time, and at time t, is N(t) 2.Each unit of output is produced with 1 unit of labor and a fix amount of land equal to 1/a1(C.R.S) 3. Output is maximized and equal to N(t). So at any given time, output is a constant 4. All of the output is shipped to the center of the urban area for export and shipping cost is K1 per mile. 5. Each resident has a completely inelastic demand for 1/a2 units of residential land and has commuting costs of K2 per mile. K2 should be no less than 0( no outcommuting)

3. Equilibrium for Residential and employment land use Assume the land can be only used for two purposes: employment and residential Let f(x)=fraction of land at distance x devoted to residential use 1-f(x)=fraction of land at distance x devoted to employment use b(t)=employment boundary(beyond boundary employment land development has not proceeded at time t) B(t)=residential boundary So we have (1) (2)

4.Shipping and Commuting Cost Total shipping costs of output to the center at time t are: (3) Total commuting costs are: (4) The first term on the right-hand side is total commuting costs if all consumers had to commute to the center and the second term corrects for the fact that employment locations are generally not at the center

5. Optimal Condition for Land Use 1. Since output is a constant at any given time. The optimal problem here is to minimize the TSC(t)+TCC(t) over time. This is equal to find the minimum PDV of TSC(t)+TCC(t) 2. PDV of TSC(t)+TCC(t) is(note 1+r is approximately ): (5) 3. SO the problem is to find f(x), b(t) and B(t) that can minimize J subject to (1) and (2)

5. Optimal Condition(2) After some first-condition and variance change, finally we get: Where Tb(x) is the employment development time at distance x and TB(x) is the residential development time at distance x (11) This is the optimal condition for land use. It is intuitive. Suppose one unit of land at distance x is switched from residential to employment use.The first term is the PDV of savings in shipping costs. The second term gives the PDV of dissavings in commuting costs. These two must be exactly equal to each other otherwise the first order condition is violated.

6. The Solution for The uniform Growth Path Suppose that population grows uniformly at a constant percentage rate n, so that (12) Assume the fraction devoted to employment use is f tentatively Then input(12) into (1) and (2), we get Where and are the employment boundary and residential boundary at time 0

6. The Solution for The uniform Growth Path(2) Input all the above into equation (11), finally,we will get The first equation gives us the ratio of the residential boundary to the employment boundary. The second equation gives us the fraction of land that is reserved for residential use at each distance from the center. It is independent of x.

6. The Solution for The uniform Growth Path(3) Now consider the relationship between k2*a2 and (k1- k2)*a1 1.If k2*a2=(k1-k2)*a1, B0=b0, f=a1/(a1+a2) At this situation, the residential boundary and employment boundary are identical, so there’s complete integration of employment and residences and no commuting occurs. 2. If K2*a2>(k1-k2)*a1, we get B0 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3423831/slides/slide_10.jpg", "name": "6.", "description": "The Solution for The uniform Growth Path(3) Now consider the relationship between k2*a2 and (k1- k2)*a1 1.If k2*a2=(k1-k2)*a1, B0=b0, f=a1/(a1+a2) At this situation, the residential boundary and employment boundary are identical, so there’s complete integration of employment and residences and no commuting occurs. 2. If K2*a2>(k1-k2)*a1, we get B0

6. The Solution for The uniform Growth Path(4) Consider when K2*a2<(k1-k2)*a1, the situation this paper focuses on, we get B0>b0, which means B(t)>b(t) And f { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/12/3423831/slides/slide_11.jpg", "name": "6.", "description": "The Solution for The uniform Growth Path(4) Consider when K2*a2<(k1-k2)*a1, the situation this paper focuses on, we get B0>b0, which means B(t)>b(t) And f

So, what the optimal land use should be at any given time? For an urban land, at any given time, it should have a employment boundary and a residential boundary. The employment boundary should be inside the residential boundary Land within the employment boundary is a mixture of employment use and residential use. Land between the employment and residential boundaries should be a mixture of vacant and residential use. Land outside the residential boundary is completely undeveloped.

7. Some comparative static analysis As k1/k2 increases, B0/b0 increases, f decreases. This is intuitive because when shipping cost is much greater than commuting cost, expanding the employment boundary is inevitable in order to save total shipping cost. Also people will reserve a larger part of land for future employment development to lower the total shipping costs As r increases, B0/b0 decreases, f increases. As r approaches infinity, B0/B1 approaches 1, we get completely integrated. This is true because very high interest rate mitigates against withholding any land from current development since current costs are weighted much more heavily than future costs As n increases, B0/b0 increases and f decreases. This is true because people need more land for future population increase.

8. Conclusion and future work Conclusion: The residential area expands outward over time, but a constant fraction of the land is always reserved for employment development because of the existence of the shipping costs Future extension work: 1. A model of growing urban area with two income groups. The result is similar( Braid, 1989) 2. Allow residential land to be converted to employment land at some costs

Comments This paper provides a convincing mathematical explanation for a urban development model. But there’s no data evidence provided. This paper is based on the assumption of C.R.S. But it is true that in real life, many industries show the I.R.S. So the benefit of converting residential land to employment should not only include the saving on shipping costs but also the I.R.S. benefits Another thing is the social costs, more specially, the pollution. When considering the fraction devoted to employment use, the increase in social costs should also be considered. In other word, even at C.R.S condition, output may be be equal to N(t)

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