A note on the optimal design of an office building

Abstract. This study examines the economics of the optimal footprint area, atrium area and height of an office building. We extend the work of Doiron, Shilling and Sirmans (1992) by incorporating realistic revenue and cost functions and reverting to the sufficient conditions of optimality.

Introduction

This note develops a model for selecting key parameters of office building design. Doiron, Shilling and Sirmans (1992) (DSS) present a microeconomic model of optimal footprint area, atrium area and building height and investigate the results of the analysis empirically. DSS contains several peculiarities that this constructive note aims to correct. In order to optimally design an office building we draw upon (i) the research of Gat (1995), who demonstrates the convex nature of the cost function of a developer, (ii) the research of Brennan, Cannaday and Colwell (1984), who estimate a rent function, and (iii) the sufficiency conditions of an optimum (Chiang, 1984).

This study is organized as follows: section two identifies peculiarities in the optimal design of the model of DSS; the third section offers an alternative model along with a numerical example, while section four concludes this note.

Peculiarities in the DSS Model

The analysis of DSS is flawed in that it fails to incorporate realistic cost and revenue functions. Furthermore, the sufficient condition DSS impose on their rent function may lead to a silly solution.

First, the cost function selected by DSS is unreasonable. This is due to the fact that they assume:

(i) a fixed unit cost (p) of the overall space (qh) instead of usable space (q-a)h, where q is the footprint area, h is the height and a is the atrium area (i.e., development costs are unaffected by the size of the atrium). Thus, they are rebuilding the floor of the atrium (a) at each floor of the building (i.e., their concept of an atrium must be an open public space on each floor rather than a space that is open through all the floors).

(ii) a flat cost per unit of footprint (q) instead of a concave function arising due to constructing the outside walls of the building. DSS implicitly assume the impossible: that wall length is proportional to floor area.

(iii) total cost as being a constant proportion of height rather than a convex function as could be inferred from the research of Gat (1995).1

It is possible to avoid the above problematic assumptions of the mathematical characteristics of the bid rent curve and the cost function (imposed by DSS) if the profit function evaluated is a concave function with respect to q, a and h. At the optimum, the first-order necessary conditions (FONCs) need to be satisfied. That is, the partial derivative of the profit with respect to each of the endogenous variables (q,a,h) is zero. This implies that marginal revenue equals marginal cost with respect to each of the endogenous variables. In addition, there must be concavity (i.e., the second-order sufficient condition requires a negative definite Hessian matrix or negative eigenvalues of the Hessian matrix) (Chiang, 1984).

An Alternative Model

The first step here is to model the rent function appropriately. We draw upon the work of Brennan et al. (1984), who study office rent in the Chicago CBD. They determine that rental revenue function could be explained by the area included in a particular lease transaction and vertical height of the structure among other variables. The area paid for but not usable by tenants (similar to an atrium space) is also found to be significant. Other studies that support the finding that height and floor area (i.e., size) of office buildings have significant impact on its value include Mills (1992) and Shilton and Zaccaria (1994). One concave function that thus comes to mind is:

Summary

The focus of this constructive note is to correct the errors in the design of office buildings made by Doiron et al. in their 1992 study. If one avoids their errors in selection of a rent function, incorporates a realistic cost function and follows standard optimization principles, then one can model the major parameters in the design of office buildings. The parameters chosen in the alternative model are roughly realistic and the resultant optimal design is within the realm of possibility.

Notes

1Gat (1995) postulates a convex cost function of the structure. He quotes that "In the construction of a building on a given site, as more floor space is added to the site, vertical stacking of floors takes place and causes marginal cost to rise." The DSS cost function, pqh, is neither concave nor convex.

2The domain of ai's is determined by requiring that the profit function (7rD) be concave as described below in the text.

3This can be generalized by writing the Hessian in symbols. Unfortunately, it is very lengthy and cumbersome. A copy of it can be obtained by writing to the authors.

References

Brennan, T. P., R. E. Cannaday and P. F Colwell, Office Rent in the Chicago CBD, AREUEA Journal, 1984,12:3, 2430.

Chiang, A. C., Fundamental Methods of Mathematical Economics, 307-58, New York: McGraw

Hill, third edition 1984.

Doiron, J. C., J. D. Shilling and C. F Sirmans, Do Market Rents Reflect the Value of Special Building Features? The Case of Office Atriums, Journal of Real Estate Research, 1992, 7:2, 147-56.

Gat, D., Optional Development of a Building Site, Journal of Real Estate Finance and Economics, 1995, ll:1, 77-84.

Mills, E. S., Office Rent Determinants in the Chicago Area, Journal of the American Real Estate and Urban Economics Association, 1992, 20:2, 273-87. Shilton, L. and A. Zaccaria, The Avenue Effect, Landmark Externalities, and Cubic Transformation: Manhattan Office Space, Journal of Real Estate Finance and Economics, 1994, 8:2, 151-65.

Peter F Colwell*

M. Shahid Ebrahim**

*Department of Finance, University of Illinois at Urbana-Champaign, 340 Commerce West, 1206 South Sixth Street, Champaign, Illinois 61820.

**Department of Management Studies, University of Brunei Darussalam, Bandar Seri Begawan 2028, Brunei Darussalam.

Date Revised-August 1997; Accepted-December 1997.

The authors thank Gwendolyn Ball, Horacio Porta, Jerry Uhl, and Samer Wattar for thier helpful comments.

Copyright American Real Estate Society 1997
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