URBAN FIRM LOCATION AND LAND USE UNDER CERTAINTY AND UNDER PRODUCT-PRICE AND LAND-RENT RISK

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1 URBAN FIRM OCATION AND AND USE UNDER CERTAINTY AND UNDER PRODUCT-PRICE AND AND-RENT RISK Joseph S. DeSalvo Department of Economics University of South Florida Tampa 40 E. Fowler Ave. CMC 34 Tampa, F ouis Eechoudt Ieseg School of Management (ille) 3, Rue de la Digue F59800 ille France February, 03. INTRODUCTION This paper introduces ris into location and land-use choices of a profit-maximizing urban firm. Although ris has been introduced into the location theories of Von Thünen, Weber, and Hotelling (Asami and Isard, 989), it has not to our nowledge been incorporated into a firm s choice of intraurban location and land use. We address this issue with a simple model of an urban firm, which chooses location (as distance from the CBD) and land input within an urban area. This model has antecedents in Aloa (974), Moomaw (980), and Cooe (983) but is both simpler (in that it includes only one input, land) and more general (in that it assumes general functional forms and profit maximization). We start with choices under certainty, providing comparative static results on location and land use for changes in product price and land rent. We find location and land use directly related to product price and inversely related to land rent. We then turn to choice of location and land use under product-price and land-rent ris. Both product-price and land-rent ris lead the firm to choose less land and a location closer to the CBD than under certainty. The appendix extends the model to a mean-preserving increase in ris.. THE MODE UNDER CERTAINTY We model a firm that chooses location and land to produce either housing or a product that must be shipped to the CBD for sale or transshipment. Since land is the only input to production, it can be thought of as a composite commodity in that the other inputs are used in fixed proportions with land.

2 Definitions and Assumptions The production function is Q = Q(), where Q is output and is land, and where Q > 0. The price of output is p(), where is distance from the CBD and where p < 0. The assumptions on price allow for the possibility that the firm is a housing-producing firm or a firm that produces a product that must be shipped to the CBD for final sale or transshipment. Housing prices fall with distance from the CBD because of the desirability of access to the CBD. For firms transporting a product to the CBD, the price function taes the form p() = p 0 T(), where p 0 is the price received at the CBD, assumed to be determined on a national or global maret, and T() > 0 is the transport cost per unit of output delivered to the CBD. Because transport costs increase with distance, T > 0, which renders p < 0. Finally, the rental rate of land is r(), where r < 0. The rental rate of land falls with distance from the CBD because of the desirability of access to the CBD. Objective Function, First-Order Conditions, and Second-Order Conditions Given the profit function π = pq ( ) ( ) r, ( ) the first-order conditions for a profit maximum are The second-order conditions are π = pq r= 0 pq= r π = pq r = 0 pq = r. π π π < 0 and D = > 0, where π π π = p Q r, π = π = p Q r, and π = pq. Since p and r fall with distance from the CBD, the first condition says the optimal location is found where the decline in marginal revenue with distance, p Q, equals the decline in land cost with distance, r. The second condition is the familiar result that optimal land use is found where the marginal revenue product of land, pq, equals its marginal factor cost, r. Equilibrium location and Alphabetic subscripts denote differentiation with respect to the subscripted variable. Another motive for an urban firm to be oriented toward the CBD is agglomeration economies. It is typically assumed that such economies attenuate with distance from the CBD, which, as suggested by Mills and Hamilton (994, p. 8), should steepen the p and r functions, drawing firms closer to the CBD. Although we do not explicitly model agglomeration, an increase in agglomeration economies should produce the same comparative static effects as a decrease in the product-price function or an increase in the transport-cost function.

3 land use are illustrated in Fig.. The second-order conditions ensure the slopes of the curves are as drawn, namely, pq r < 0 pq> r and pq < 0. $/ $/ ( ) pq 0 r( 0 ) r 0 p( 0 )Q 0 0 Fig. : Equilibrium ocation and and Use Comparative Static Analysis Proof of a Useful Result. For purposes of the comparative static analysis, we need to show that p Q r > 0. From the first-order condition, p Q r = 0, we get r = (p Q)/. Substituting this into the expression, p Q r, gives p Q (p Q)/. Then pq > Q < pq 0 as Q 0 since p < 0. < > But for the firm to earn positive economic profit at the optimum, Q < Q/, so p Q r > 0, which was to be proved. First-Order Conditions with Shift Parameters. Since p and r are functions of, we introduce shift parameters to capture exogenous changes in these functions. The first-order conditions with shift parameters are π = p ()Q(,θ ) r (,θ ) = 0 and π = p()q (,θ ) r(,θ ) = 0 3

4 Total Differentiation of First-Order Conditions. pq r pq r p d θ Q r θ dθ p Q r pq d = p Q r dθ θ θ Effects of a Change in Price on ocation and and Use. et p θ > 0 represent a parallel upward shift in the price function. We assume p θ = 0, so that we are only dealing with level effects. 3 Then, ( pq r) pθ Q = > 0 θ D θ ( p Q ) r p Q θ = > An increase in the price level causes the firm to move farther from the CBD and to use more land (and, thus, to produce more output). These results are illustrated in Fig.. The condition that p Q r > 0 ensures that the p Q curve shifts upward (downward) with an increase (decrease) in by less than does the r curve, while the same condition ensures that the pq curve shifts downward (upward) with an increase (decrease) in by less than does the r curve. D 0. $/ $/ r( 0 ) r r p Q( ) p Q( 0 ) 0 (0) ( 0 θ ) p () ( θ ) p r( ) Q,, Q 0 0 Fig. : The Effect of a Product-Price Increase on Equilibrium ocation and and Use 3 It can be shown that if p θ > 0, i.e., the slope of the price function increases (the function pivots upward), then we get the same comparative static result as for a pure level increase. If, however, ambiguous. 4 p θ < 0, then the effect is

5 Effects of a Change in Rent on ocation and and Use. et r θ > 0represent a parallel upward shift in the land-rent function. We assume Then, effects. 4 ( ) θ θ = < r θ = 0, so that we are only dealing with level ( p Q ) = < pq r r D D r r An increase in land rent causes the firm to move closer to the CBD and to use less land (and, thus, to produce less output). Fig. 3 illustrates these results. θ θ 0 0. $/ $/ p Q( 0 ) p Q( ) r 0 r () (, ) (0) r( 0, θ ) r θ p( 0 )Q p( )Q 0 0 Fig. 3: The Effect of a and-rent Increase on Equilibrium ocation and and Use 3. THE MODE UNDER RISK To deal with ris, we assume U = U(π) is a von Neumann-Morgenstern utility function displaying ris aversion, i.e., U > 0, U < 0. We treat ris about each exogenous variable separately, starting with product-price ris. 4 As for the price function, it can be shown that if r θ > 0, i.e., the slope of the rent function increases (the function pivots upward), then we get the same comparative static result as for a pure level increase. If, however, then the effect is ambiguous. 5 r θ < 0,

6 Product-Price Ris The Objective Function and First-Order Conditions. The objective function (( ( ) + ε ) ( ) ( ) ) E U p Q r is maximized with respect to and, where ε is a random variable with mean zero. The first-order conditions are ( ) E = E U pq r = 0 pq = r (( ε) ) ( ) [ ] [ ε] E = EU p+ Q r = 0 pq r EU + QEU = 0. The second-order conditions will not be presented as we shall not provide a comparative static analysis of this model. All of the comparative static results are ambiguous without further assumptions because as profit changes, ris aversion changes. We can, nevertheless, draw conclusions about how the firm s choices differ under certainty and ris, but first we must show that location and land use are directly related, which we have already seen from the comparative static results under certainty. ocation and and Use Are Directly Related. The first first-order condition yields the same result as does the model under certainty. Therefore, the shape of the utility function and the presence of a random term do not affect the first-order condition. The second first-order condition does not produce the same condition as in the certainty case. To determine how ris affects the firm s choice of location and land use, we show that location and land use (and, hence, output) are directly related. Totally differentiating p ()Q() r () = 0 yields d pq r = > 0. d p Q r The Effects of Product-Price Ris on ocation and and Use. Under ris, less land (and, hence, a closer location) is chosen than under certainty. To show this, we divide the second term after the implication symbol of the second first-order condition by E[UN], getting ( ) 6 [ ] [ ] EUε pq r + Q = 0. EU The numerator of the second term on the left-hand side of the equation is the covariance between U and ε. To see this, note that the covariance is defined as ( ε) = [ ε] [ ] [ ε] CovU, EU EU E,

7 but E[ε] = 0, as assumed. The covariance, EUε [ ], is negative, for as ε rises, price, profit, and the level of utility all rise, while U falls. Since Q and EU [ ] are positive, then the term, QEU [ ε ] / EU [ ], is negative. Hence, under certainty, if a value of is chosen when pq = r, then under ris a smaller will be chosen because [ ε ] [ ] EU p+ Q = r. EU The product-price ris has reduced the marginal revenue product of land and caused the firm to use less land. We have considered here the effect of the introduction of a zero-mean ris around the expected unit price of output. In Appendix, we discuss the impact of a mean-preserving increase in this ris. Since this notion is more general, more restrictions must be imposed on the utility function to generate a result similar to the one obtained so far. and-rent Ris The Objective Function and First-Order Conditions. In this case, the objective function is ( ( ) ( ) ( ( ) + ε ) ) E U pq r, where ε is a random variable with mean zero. Maximizing with respect to and produces the first-order conditions ( ) E = E U pq r = 0 pq r = 0 ( ε ) ( ) [ ] [ ε] E = E U pq ( r + ) = 0 pq r E U E U = 0. The Effects of and-rent Ris on ocation and and Use. As before, location and land use (and, hence, output) are directly related. Under land-rent ris, less land (and, hence, a closer location) is chosen than under certainty. To see this, divide the second term after the implication symbol of the second first-order condition by EU [ ], getting ( pq r) 7 [ ε ] [ ] EU = 0. EU The numerator of the second term on the left-hand side of the equation is the covariance between U EUε, is positive, for as ε rises, rent rises and ε because, as before, E[ε] = 0. The covariance, [ ]

8 causing profit to fall. Hence, the level of utility falls, while marginal utility rises. Since EU [ ] is positive, the term, EU [ ε ] / EU [ ], is positive. Hence, under certainty, if a value of is chosen when pq = r, then under ris a smaller will be chosen because pq [ ε ] [ ]. EU = r + EU The land-rent ris has raised the marginal factor cost of land. 4. SUMMARY AND THOUGHTS ABOUT FUTURE RESEARCH In a simple model of location and land use by the urban firm, we found that under certainty, both location (as distance from the CBD) and land use were directly related to product price and inversely related to land rent. Under both product-price and land-rent ris, less land and a closer location are chosen than under certainty. We do not thin these results have heretofore appeared in the literature on the urban firm, but there are a couple of things we would lie to do in future research. We would lie to supply comparative static analyses of the model under ris. We have experimented with specific functional forms in an attempt to find such results but have nothing to present at this time. We would also lie to generalize the model to include two variable inputs, land and structure (capital). This, of course, complicates the analysis, but it would allow the theory to replicate the reality of declining structural density (structure/land) with distance from the CBD as firms trade off structure for land as land rent falls with distance from the CBD and to see how ris affects that relation. Finally, we would lie to analyze delayed ris, i.e., when and have to be chosen before g is revealed. After g is revealed, then capital structure is chosen. 8

9 REFERENCES Aloa, Nurudeen An Approach to Intraurban ocation Theory, Economic Geography, 59 (), Asami, Yasushi and Walter Isard Imperfect Information, Ris, and Optimal Sampling in ocation Theory: An Initial Reexamination of Hotelling, Weber, and Von Thünen, Journal of Regional Science, 9 (4), Cooe, Timothy Testing a Model of Intraurban Firm ocation, Journal of Urban Economics, 3 (3), Eechoudt, ouis R. and Harris Schlesinger Changes in Ris and the Demand for Saving, Journal of Monetary Economics, 55, Mills, Edwin S. and Bruce W. Hamilton Urban Economics, 5th ed. New Yor: Harper Collins College Publishers. Moomaw, Ronald Urban Firm ocation: Comparative Statics and Empirical Evidence, Southern Economic Journal, 47 (), Rothschild, M., and J. Stiglitz Increasing Ris: I. A Definition, Journal of Economic Theory,,

10 Appendix as Returning to the first-order condition for on page 6, we first write it in an equivalent manner { ( ( ) ε) ( ) ( ) ( ( ) ε) ( ) ( ) } E = E U p + Q * r * p + Q * r = 0, where * stands for the optimal when the firm faces the ris g on its output price and where we specify at which point U is evaluated. Consider now that the random unit price becomes p( ) + θ, where is a mean-preserving increase in ris of g in the sense of Rothschild and Stiglitz (970). Following a standard argument in the economics of ris, this change induces the choice of a lower value of (and hence of ) if and only if { ( ( ) θ) ( ) ( ) ( ( ) θ) ( ) ( ) } E U p + Q * r * p + Q * r < 0 (A.) Using a well-nown result in the stochastic dominance literature, this condition is satisfied if the following function is concave in ( ( ) θ) ( *) ( ) * ( ( ) θ) ( *) ( ) U p + Q r p + Q r (A.) Differentiating (A.) twice with respect to one obtains the result that the increase in price ris reduces (and ) if U Q + > U Q { p( ) θ Q r( ) } (A.3) This condition, which involves the degree of partial prudence, is more restrictive than the one obtained for the introduction of a zero-mean ris, which refers only to ris aversion. Condition (A.3) is very similar to those obtained when one analyzes the impact of a risier rate of return in savings models (see, e.g., Eechoudt and Schlesinger, 008). This should not be surprising since the ris appears in a multiplicative, not additive, way in (A.), in that it multiplies Q(*), as does the risy rate of return in savings problems. Of course, a similar approach could be used to find the impact of a mean-preserving increase in land-rent ris. Because this ris also has a multiplicative structure (it multiplies ), the same line of reasoning applies. Consequently, we do not provide the result here. 0