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Urban Economics Working Paper: Transportation Infrastructure, Productivity Shocks, and City Evolution

Note: This transcription covers 15 scanned pages (pages 3, 4, 7, 8, 10, 12, 14, 15, 17, 19, 25, 26, 29, 30, 31) of a working paper draft with handwritten annotations/comments. Pages not scanned are omitted. The paper appears to be a monocentric urban model studying how city-specific productivity and transportation shocks affect urban configuration, land rents, and property prices.

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talization rate of the land changes over time, and is different in different locations within the city.

To summarize, this paper extends the monocentric urban model in a number of important ways. Most notably, we are the first to provide a stochastic dynamic model that captures the transmission of productivity shocks through an urban property market. We are not, however, the first to consider dynamic urban models. For example, Berliant and Wang (2005) reviews an expanding literature that considers the link between agglomeration spillovers, capital accumulation and urban growth.^2 In contrast to most of the prior literature, which focuses on the determinants of urban growth, our focus is on how the structure of cities affects the evolution of property prices and rents. There is clearly a relation between the growth rate of an urban economy and the growth rate of land rents and prices, however, as we show, the economic growth rate is just one of several determinants of price changes.

We are also not the first to study the relation between transportation infrastructure and urban design. The impacts of automobile and highways are studied in Downs (1992), Dunphy (1997), Glaeser and Kahn (2004), Nechyba and Walsh (2004), Baum-Snow (2007a), Ahlfeldt and Wendland (2011), Garcia-Lopez (2012), and Duranton and Turner (2012). In particular, Baum-Snow (2007b) uses a monocentric city model to show theoretically that the construction of new highways in a city cause the population to spread out along the highways. The impacts of rail transit are studied in Baum-Snow and Kahn (2005), Anderson (2007) and Gonzalez-Navarro and Turner (2016). Baum-Snow et al. (2016) empirically studies the effects of both road and rail construction in China. LeRoy and Sonstelie (1983) uses a standard Alonso-Muth model to explain the suburbanization and gentrification of city centers.

A key difference between the existing literature and our contribution is that we focus on city-specific changes in transportation infrastructure and productivity, holding the transportation infrastructure and productivity in other cities fixed. While a universal improvement in transportation technology reduces density and flattens the land value and population gradients, as shown in the above references, city-specific transportation improvements lead to higher density and, in most cases, steeper price and population gradients. This is because the city-specific improvements attract workers from other cities, reinforcing the agglomeration effect and increasing land values.

We examine how the city-specific productivity shocks affect city evolution. This relates our paper to a recent strand of literature that emphasize the importance disaggregated shocks which are city- or region-specific to a large extent. Firm level shocks are studied in Gabaix (2011) which argues that idiosyncratic shocks to large firms can generate non-trivial aggregate

^2 Duranton and Puga (2014) provides a very thorough review of this and other literature that examines the determinants of urban growth. Most of this literature considers static models, but provides insights about the determinants of growth by examining various comparative statics.

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fluctuations. Caliendo et al. (2014)

Given a productivity increase, our theoretical framework predicts that cities with better capacity in terms of accommodating in-migration of workers respond with a large expansion of population and a mild increase of wage and rent, while cities with limited capacity respond with a mild expansion of population and a large increase of wage and rent. This is consistent with the empirical findings in Glaeser et al. (2006), although city capacity is measured by the elasticity of housing supply in Glaeser et al. (2006) while in our paper it is reflected in the transportation infrastructure and the flexibility of residential land supply.

Within a monocentric city framework, our paper studies how the trade-off between agglomeration and congestion/commuting depends on the transport technology. The main message is that better transport technology is conducive to sustained growth -- it weakens the negative externality from congestion and commuting, and strengthens agglomeration. A related work regarding sustained urban growth is Berliant and Wang (2008) that relates the "perpetual" steady-state growth to the endogenous rise of sub-centers of cities. Following the economic growth literature, Berliant and Wang (2008) emphasizes capital accumulation and agglomeration is a function of aggregate capital in a city. In contrast our model highlights that city population is a key determinant of transport cost. Therefore, we model agglomeration as a function of population.

The rest of the paper is organized as follows. Section 2 reviews the literature and highlights our contribution. Section 3 lays out the model and defines the equilibrium. Section 4 characterizes urban configuration dynamics through a set of propositions. Section 5 presents numerical results that show the full dynamics. Section 6 concludes.

NOT SURE IF WE WANT TO DISCUSS THE CONTROVERSY ABOUT RAIL TRANSIT HERE The effects of rail transit on employment and social welfare are controversial. The development of rail transit has been under considerable debate in the policy community.^3 Affuso et al. (2003) argue that road improvements have substantially higher returns than railway schemes. Winston and Maheshri (2007) further argue that rail construction reduces social welfare and hence not desirable.

Other papers potential consequence of ignoring durable housing: "Urban Decline and Durable Housing", the asymmetry between urban decline and urban growth due to durable housing -- homes can be built quickly, but disappear slowly.

shock propagation: "NETWORKS AND THE MACROECONOMY: AN EMPIRICAL EXPLORATION", shows empirically that firm level or industry level shocks are propagated through economic networks and geographic networks.

^3 See a recent summary in Litman (2014)

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The distance j_i is simply a non-linear transformation of the location index i, so without loss of generality, we use j to denote both distance and location, with j = 0 representing the CBD where the distance is zero.

We assume that transportation costs represent lost time and are thus linear in wages. Specifically, for workers living in location j to earn the net reservation wage of W, they need to earn a wage of w = W \times e^{f(j,N,\tau)}. In other words, the net wage of workers living in location j is

W(j) = w \times e^{-f(j,N,\tau)} \qquad (2)

The cost function f(j, N, \tau) satisfies

j = location defined by distance to the CBD
N = city population
\tau = transport technology

Different transport technologies are characterized by different transport cost functions. For example, we assume that a car-based transport technology is more sensitive to population increases than a rail-based technology because highways are more prone to congestion than rails. We make the following assumptions regarding the alternative transportation technologies:

Assumptions Needs assumptions about differentiability

f(0, N, \tau) = 0, i.e., transport cost is zero for the workers living in the CBD. This is consistent with the definition of distance in equation (1).
\partial f / \partial N > 0, i.e., there is a positive congestion effect, such that the cost increases with city population.
\partial f / \partial j > 0, i.e., the transport cost gradient, defined as the change of cost with distance to CBD, is positive.
\partial^2 f / \partial j \partial N = 0, i.e. the congestion effect is not location-specific.^4

Transport cost gradient is of key importance in our analysis, for simplicity of notation we use f'(j|N, \tau) to denote transport cost gradient at location j, given population N and technology \tau.

^4 We make this assumption for transparency of the main mechanisms to be developed. Most of the results hold true if we allow the congestion effect to increase with distance from the CBD (\partial f^2 / \partial j \partial N \geq 0), which is shown in the technical appendix.

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2.2 The Workers

The model assumes a mass of identical workers who each provide exactly one unit of labor and allocate their wages to land rent, transportation costs, and consumption goods. For notational convenience, we will define a net wage, which is the wage minus the workers cost of commuting to work. Since the workers are identical and have the same external opportunities they each receive the same exogenous reservation level of utility regardless of where they live.

2.2.1 The Worker's Optimization Problem

Let h and c be residential land and consumption goods respectively, a worker who lives at location j solves the following optimization problem.

\max_{c,h} = u(c, h) s.t. c + p_r(j)h = W(j) \qquad (3)

where p_r(j) is the rental rate of residential land in location j, and W(j) is the net wage as defined in equation (2).

It is straightforward to show that the optimal allocation between land and consumption goods satisfies,

p_r(j) = \frac{\partial u(c,h)/\partial h}{\partial u(c,h)/\partial c} \qquad (4)

The right side of the above equation is the marginal rate of substitution between land and the consumption good. Given the assumed Cobb-Douglas utility function, i.e., u(c, h) = c^{1-\theta} h^{\theta}, equation (4) becomes

p_r(j) = \frac{\theta}{1-\theta} \frac{c}{h} \qquad (5)

From equation (5), we get c = \frac{1-\theta}{\theta} p(d)h. Substituting this into the budget constraint (equation (3)) yields the optimal consumption good choice,

c = (1-\theta)W(j) \qquad (6)

and land demand function

h = \theta \frac{W(j)}{p_r(j)} \qquad (7)

Workers also choose where to live, but since they all receive the same reservation utility, the location choice is irrelevant for individual workers in equilibrium. Indeed, the rental price of land at each location is determined within the equilibrium to make all workers indifferent.

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rent as a function of the number of workers living in given location. Combining equation (9) and (11), the relation between land rent and population in location j is

p_r(j) = B_0^{-\frac{\theta}{1-\theta}} \theta^{\frac{1}{1-\theta}} n(j)^{\frac{1}{1-\theta}} \qquad (12)

Thus, residential land rent is an increasing function of population in each location driven by the net wage.

2.2.3 Rent Gradient

The residential rent gradient describes the land rent as a function of the distance from the CBD, i.e.,

p'_r(j) = -\frac{B_0}{\theta} W(j)^{1/\theta} f'(j|N, \tau) \qquad (13)

The rent gradient depends on the transport cost gradient f'(j|N, \tau). All else equal, a lower transport cost gradient generates lower rents at each distance from the CBD. However, in our model, since reservation utility is assumed to be fixed, lower transport costs must be offset by something that keeps reservation utility constant. For example, if a city is endowed with a new technology that exogenously lowers its transportation costs, its population will increase, which endogenously increases the transport cost gradient because we assume \partial^2 f / \partial j \partial N \geq 0.

2.3 The Firms

There exists a unit measure of identical firms that use land in the citys CBD along with labor to produce the consumption good. We assume a constant returns to scale Cobb-Douglas production function,

F(\ell, n) = A \ell^{\sigma} n^{1-\sigma} \qquad (14)

where \ell_t and n_t are land and labor respectively. A is the total factor productivity (TFP) of this city relative to other cities.^6

Let p_c be the rental rate of commercial land, the firm's optimization problem is

\max_{\ell, n} F(\ell, n) - wn - p_c \ell

subject to equation (14). From the first-order condition, we obtain the usual allocation rule of a Cobb-Douglas production function:

\frac{n}{\ell} = \frac{1-\sigma}{\sigma} \frac{p_c}{w} \qquad (15)

^6 A is endogenous in the dynamic setting due to the positive externality of agglomeration.

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is constant the individuals at the periphery consume the same mix of land and the consumption good regardless of the productivity and transport technology. As we note below, this will not be the case when the firm has a fixed boundary.

Fixed Boundaries When the city's boundary is fixed, the land price at the periphery is determined by the bid-rent function.

2.5 General Equilibrium

The model has three interdependent markets that must clear simultaneously: the commercial land market, the residential land market and the labor market. In this setting, firms make zero profit and households receive their reservation utility. As such, as in Fujita (1989), Wheaton (1998) and Rossi-Hansberg (2004), the social optimum occurs when aggregate rent is maximized.

2.5.1 Land Market Equilibrium

Equilibrium in the residential land market implies a relation between wage and population as shown in the curves labeled "Land Eqlm" in figure 1. A higher wage attracts a larger population to maintain land market equilibrium, leading to an upward sloping land market equilibrium curve. Intuitively, a higher wage leads to a higher residential bid-rent, which causes more agricultural land in the periphery to be converted to urban land. The higher rents lead to lower land demand per worker (equation 10).

**I find this a bit confusing ?since we are considering the interaction between 3 endogenous variables. To me it is easier to talk in terms of shifts in the exogenous productive variable. An increase in productivity increases wages, which in turn attracts more workers, which in turn increases rents, causing agricultural land to convert to urban land.

We will refer to this relation between wage and population as the land market equilibrium condition. To formally present the condition, we start with equation (11) that shows the number of workers that are accommodated in location j given the effective wage W(j). Aggregating workers in each location leads to:

N = \int_0^{J(w)} n(j) dj = S_r \frac{B_0}{\theta} w^{(1-\theta)/\theta} dj + \int_1^{J(w)} \frac{B_0}{\theta} W(j)^{(1-\theta)/\theta} dj

= S_r \frac{B_0}{\theta} w^{(1-\theta)/\theta} dj + \frac{B_0}{\theta} w^{\frac{1-\theta}{\theta}} \int_1^{J(w)} e^{-\frac{1-\theta}{\theta} f(j,N,\tau)} dj \qquad (18)

Equation (18) is the relation between wage and population when residential land market clears, i.e. all the residential land available given wage w is occupied by workers. Note that the

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This labor demand equation can be rewritten into

w = (1-\sigma) A \left(\frac{S_c}{N}\right)^{\sigma}, \qquad (20)

where we have replaced N^* with the equilibrium total employment N. Since workers have perfect mobility and the reservation utility is identical across cities, equation is also the labor market equilibrium condition when the CBD is segmented.

** Im a little bit confused about terminology. We can specify a labor demand function, which specifies the amount of labor the firm will hire as a function of the wage and the commercial rent. This function is going to be the same in both the segmented and non-segmented CBDs. Of course, the equilibrium rents and wages will be different depending on the segmentation. I think this is what you say below. So would it be better to first specify that labor demand function and then separately look at the two cases?

Non-segmented CBD In this case, equation 2.5.2 is still the land demand function, but S_c now changes with wage rate, and there is an additional condition for labor market to clear: the equality of commercial and residential land rents. Using the residential bid-rent function (equation 9) and the commercial bid-rent function (16), p_r = p_c implies

\left[\frac{A\sigma^{\sigma}(1-\sigma)^{1-\sigma}}{w^{1-\sigma}}\right]^{1/\sigma} = B_0 w^{1/\theta} \qquad (21)

where we used the condition f(j, N, \tau) = 0 for j = 0.

Denote \Psi = \frac{\sigma^{\sigma}(1-\sigma)^{1-\sigma}}{B_0^{\sigma}}, Equation (21) can be re-written into

w = (\Psi A)^{\theta/(\sigma+\theta-\theta\sigma)} \qquad (22)

Equation (22) represents the labor market equilibrium curve in the case of non-separable land markets. Given a productivity, there is only one wage that satisfies p_r = p_c. The slight rise of wage above this fixed level leads to a higher residential land rent and lower commercial land rent, which causes the entire commercial land to be converted into residential use. Therefore the labor market equilibrium curve is flat.

2.5.3 Equilibrium Wage, Population and Commercial Land

The equilibrium wage, population and size of commercial land are determined when both the land market and the labor market clear. Equilibrium wage and population are determined by the intersection of the land market equilibrium curve and labor market equilibrium curve. The equilibrium is illustrated in figure (1), where land and labor market equilibrium curves are labeled "Land Eqlm" and "Labor Eqlm", respectively. The left panel is the graphical representation of equations (18) and (22), and the right panel of equations (18) and (19).

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Figure 1: Labor Market and Land Market Equilibria

CBD not segmented              CBD segmented

    |  Land Eqlm                  |  Land Eqlm
    | /                           | /
    |/                            |/
w --|--------- Labor Eqlm    w --|--------\
    |                             |         \ Labor Eqlm
    |                             |
    +----------→ N                +----------→ N

Note: This figures shows the equilibrium wage and population in cities with non-segmented CBD (left panel) and segmented CBD (right panel). The "Land Eqlm" curve represents land market equilibrium and the N^d curve represents labor demand of the firms.

Given the equilibrium wage and population, land rents (both commercial and residential) are easily calculated from the bid-rent functions. In the case of non-segmented CBD, the size of commercial land (S_c) is solved from equation (2.5.2).

2.5.4 Uniqueness of Equilibrium

In each of the four case shown in table 1, the endogenous variables are uniquely determined, hence the model has an unique equilibrium.

To show the uniqueness, we start from the case of a city with a segmented CBD and fixed boundaries. The model has the following endogenous variables: (w, N, p_r, p_c) that are solved from equations (9), 16, (18), and (19), namely the residential bid-rent function, commercial bid-rent function, land market equilibrium condition and labor demand equation. Clearly the latter two equations uniquely determine wage and population, and land rents are uniquely determined by bid-rent functions. Therefore, the equilibrium is unique.

Turning to the case of non-segmented CBD and fixed boundaries, we have one more endogenous variable: the size of commercial land, and one more equilibrium condition: p_c = p_r.^7 Therefore we have a system of five equations and five unknowns. As shown in figure 1, wage and population are uniquely determined by the equilibrium conditions in the land and labor markets. The remaining three endogenous variables are uniquely determined by wage and population. When the city's boundaries can expand, we also need to solve for the city boundary J which is uniquely determined by wage for any given transportation technology, as shown in

^7 The size of residential land in the CBD is simply the difference between size of the CBD and size of the commercial land.

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may be higher. It should also be noted that since the workers are richer, they may consume more land, which would increase rents.

Consequently, the bid-rent function of residential land needs to be modified. Landlords no longer simply charge a rent that guarantees a constant reservation utility, but they can charge higher rents (i.e. deliver lower utility) when the city receives negative productivity shocks, and charge lower rents for positive productivity shocks. This effectively dampens the volatility of population in two ways. Taking the higher rent in the face of a negative shock as an example, first, the high rent maintains sufficient land provision to workers, so that the land market equilibrium curve does not shift leftward. Second, in the case of non-segmented land markets, commercial land also remains relatively high so that firms do not substitute out labor for land. Since the city-level productivity depends on the lagged population, the lower population volatility implies lower volatilities of productivity, wage, and other dimensions of urban configuration.

3.1 Worker's problem

For simplicity, we consider only productivity shocks in one city where we refer to existing workers as "incumbents" and workers outside the city as "outsiders". The one-period indirect utility of incumbents is u(\tilde{A}), which is a function of the productivity shock, and the one-period utility of the outsiders is u^*, which is assumed to be constant (i.e., the reservation utility in the basic model). A worker takes the present and discount sum of the future utilities into account when making the moving decisions. Specifically, given a productivity shock \tilde{A}, an incumbent compares between the values of staying and moving, denoted V^{stay}(\tilde{A}) and V^{move}(\tilde{A}) respectively. Therefore the value of an incumbent given \tilde{A} is:

V(\tilde{A}) = \max\{V^{stay}(\tilde{A}), V^{move}(\tilde{A})\}. \qquad (23)

Similarly, an outsider compares the values of staying and moving, denoted W^{stay}(\tilde{A}) and W^{move}(\tilde{A}) respectively. The value of an outsider given the productivity shock \tilde{A} is

W(\tilde{A}) = \max\{W^{stay}(\tilde{A}), W^{move}(\tilde{A})\} \qquad (24)

where \tilde{A} is NOT the productivity shock of the cities where the outsiders live, because we assume no shock happens to cities of outsiders.

The functions of V^{stay}(\tilde{A}), V^{move}(\tilde{A}), W^{stay}(\tilde{A}), and W^{move}(\tilde{A}) have the following recursive representations:

V^{stay}(\tilde{A}) = u(\tilde{A}) + \beta EV(\tilde{A}') \qquad (25) V^{move}(\tilde{A}) = (1-\delta)u^* + \beta EW(\tilde{A}') \qquad (26) W^{stay}(\tilde{A}) = u^* + \beta EW(\tilde{A}') \qquad (27) W^{move}(\tilde{A}) = (1-\delta)u(\tilde{A}) + \beta EV(\tilde{A}') \qquad (28)

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Therefore, we have

V(\tilde{A}) = \max\{V^{stay}(\tilde{A}), V^{move}(\tilde{A})\} = V^{stay}(\tilde{A}) = u(\tilde{A}) + \beta EV(\tilde{A}')

Taking expectation with respect to the productivity shock yields

EV(\tilde{A}) = Eu(\tilde{A}) + \beta EV(\tilde{A}'). \qquad (29)

Since the productivity shocks are i.i.d, EV(\tilde{A}) = EV(\tilde{A}'). Equation (29) becomes

EV(\tilde{A}') = \frac{1}{1-\beta} Eu(\tilde{A}). \qquad (30)

Similarly, W^{stay}(\tilde{A}) \geq W^{move}(\tilde{A}) should hold for any \tilde{A}. Otherwise every outsiders would choose to move, which is inconsistent with the labor demand function and the residential land market equilibrium. Therefore, W(\tilde{A}) = \max\{W^{stay}(\tilde{A}), W^{move}(\tilde{A})\} = W^{stay}(\tilde{A}) = u^* + \beta EW(\tilde{A}). Taking expectation with respect to \tilde{A} leads to

EW(\tilde{A}') = \frac{1}{1-\beta} u^*. \qquad (31)

3.3 One-period Utility as a Function of Productivity Shock

Using the equilibrium condition EV(\tilde{A}) = EW(\tilde{A}), equation (30)-(31) imply

Eu(\tilde{A}) = u^*, \qquad (32)

i.e. the one-period indirect utility of incumbents equals the reservation utility of outsiders on average.

Now we study how u(\tilde{A}) is determined for each individual \tilde{A}. First of all, everything else equal, u(\tilde{A}) should be non-decreasing with \tilde{A}. Intuitively, labor demand falls when a negative \tilde{A} occurs. If the utility level rises when \tilde{A} is negative, then labor supply cannot fall, and it is impossible for the labor market to clear.

Given that utility is non-decreasing in productivity, the response of the economy to a negative productivity shock is illustrated in figure 2. The original equilibrium is point E_1 where the curves of land and labor market equilibria intersect. A negative productivity shock shifts the "Labor Eqlm" curve leftward, reaching point E_2 with the wage and population pair falls from (w_1, N_1) to (w_2, N_2). This is the new equilibrium in the baseline model where the reservation utility is fixed. However, when the utility level falls with productivity, the "Land Eqlm" curve shifts to the right because the lower utility leads to a higher bid-rent, which not only causes

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in cities with fixed boundaries. Consequently the firms are not able to have land rent lowered hence not able to compensate workers for the congestion. On the other hand, if land markets are separable, owners of commercial land are more willing to reduce rent, hence firms are able to compensate for the congestion. Clearly, the situation of compensated congestion should have higher elasticities of wage and population relative to the case of un-compensated congestion. Since the compensation comes from lowered commercial land rent, the \zeta_{p_c} is smaller in the case of separable land markets (the case of compensated congestion).

Segmented CBD Versus Nonsegmented CBD (In Open Cities)

Proposition 4 Consider open cities, let \zeta^* be the elasticity in a city with segmented CBD, and \zeta be the elasticity in a city with non-segmented CBD, then:

\zeta_N^* \geq \zeta_N, \zeta_w^* \geq \zeta_w, \zeta_{P_r}^* \geq \zeta_{P_r}, and \zeta_{p_c}^* \leq \zeta_{p_c} for cities with a high gradient of transport cost and a strong congestion effect (e.g. car-based cities)
the inequalities are all reversed for cities with a low gradient of transport cost and a weak congestion effect (rail-based cities).

Proof is in Appendix D

The proposition has important implications regarding whether residential and commercial land markets should be separated. On the one hand, for rail-based cities (with variable boundaries), non-separation of land markets policy is suggested as it is more conducive to growth. Intuitively, rail-based cities are able to expand their peripheries, so allowing the conversion of residential land near the CBD into commercial use effectively lowers commercial land rent, causing more growth in population and wage. On the other hand, for car-based cities, expansion in the peripheries is economically less viable, so residential land rent near the CBD is relatively more expensive. Consequently, under the non-separation policy owners of commercial land are unwilling to reduce rent to compensate for the congestion. Therefore for case-based cities our suggestion is to separate the commercial and residential land markets.

4.1.2 Rent Growth and Cap Rate

Proposition 5 The cross derivative \frac{\partial^2 p_r(j)}{\partial j \partial A} is less than zero and its absolute value increases with the congestion effect (\frac{\partial f}{\partial N}), implying:

a positive (negative) productivity shock causes steeper (flatter) land gradient;
a positive (negative) productivity shock causes more rise (fall) of land rent near the CBD and less rise (fall) in far out areas.

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Furthermore, the above two effects are stronger in cities with stronger congestion effect.

Proof is given in Appendix E. Thus rent of land near the CBD is more responsive to productivity shocks. Given a positive productivity shock, land near the CBD has a higher growth of rent compared with farther-out land, particularly for cities with strong congestion effect.

The growth rate of land rent is closely related to the capitalization rate of land and property. Everything else equal, higher growth rate implies lower capitalization rate. We consider the simplest case where land price is the discount sum of future rent which grows at a constant factor G_p(j). Using P(j)_0 to denote land price in location j at time 0, then

P_0(j) = \sum_{t=0}^{\infty} \beta^t p_{r_t}(j), = \sum_{t=0}^{\infty} \beta^t p_{r_0}(j) G_p(j)^t, = p_{r_0}(j) \sum_{t=0}^{\infty} (\beta G_p(j))^t, = \frac{p_{r_0}(j)}{1 - \beta G_p(j)}

Therefore the capitalization rate at location j is

CapRate = \frac{p_{r_0}(j)}{P_0(j)}, = 1 - \beta G_p(j),

i.e., the capitalization rate decreases with the growth rate of rent.

Within our framework, the changes of rent are caused by productivity shock. If a city receives positive productivity shocks and thus population and wage increase for a lengthy period of time, then land near the CBD consistently has higher growth rate of rent than land in farther-out locations according to Proposition 5. It is well recognized that the rise and fall of cities are highly persistent.^10 In light of this, when a city rises land near the CBD has lower capitalization rate. More specifically, we have the following corollary to Proposition 5.

Corollary If productivity shocks cause sufficiently persistent rise and fall of cities, then:

When a city rises, the capitalization rate of residential land increases with the distance to the CBD. The rate of increase is larger for cities with larger \frac{\partial f}{\partial N}.
When a city falls, the capitalization rate of residential land decreases with the distance to the CBD. The rate of decrease is larger for cities with larger \frac{\partial f}{\partial N}.

^10 See Davis et al. (2016) and the references therein.

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Figure 4: Changes to Urban Configurations Due to Productivity Increase

[Six panels of graphs showing changes over time (x-axis: Time, 0-15) for car-based and rail-based cities:]

Productivity: Both car and rail increase from ~1.08 to ~1.095-1.10 over time. Car (solid line) and rail (dashed line) follow similar paths.
Population (x10^4): Both increase. Rail rises more steeply (to ~9.5) than car (to ~8.5).
Wage: Both increase from ~1.0. Car rises more steeply (to ~1.035) than rail (to ~1.015).
Land Rental Rate (Commercial): Both increase from ~2300. Car rises more (to ~2500+) than rail.
Land Rental Rate (CBD Residential): Both increase from ~2300. Car rises more than rail.
Total Land Rent (x10^4): Both increase. Rail rises more steeply (to ~2.7) than car (to ~2.0).

Note: This figure shows the changes over time in urban configurations due to a 1% increase in productivity under the scenario of flexible CBD and flexible city boundary, for both car-based and raise-based cities. Relative to car-based cities, rail-based cities feature higher fixed transportation cost, but lower gradient of transportation cost with respect to distance. In addition, rail-based cities have smaller congestion effect.

Figure 5: Transport Cost Before and After Productivity Increase

[Two panels:]

Transport Cost (y-axis: 0.03-0.09, x-axis: Distance from CBD in km^2, 1-25): Shows transport cost as a function of distance. Car has steeper gradient than rail. After productivity increase (new SS), both shift upward slightly.
Rent Gradient (y-axis: 1900-2500, x-axis: Distance from the CBD in km, 1-20): Shows rent declining with distance. Car (solid) has steeper decline than rail (dashed). New steady state lines show upward shift. Legend: car, car (new SS), rail, rail (new SS).

Note: This figure shows the changes of transport cost due to a 1% increase in productivity and after the city converges to the new steady state equilibrium, under the scenario of flexible CBD and flexible city boundary.

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4.2.3 Volatilities

Table 2: Percentage Changes Due to a 1% Increase In Productivity

Transport	A	wage	N	P	Q	P	q	Size of CBD	Radius of City

open city; non-segmented CBD
Car	2.236	2.212	49.98	50.43	53.30	7.57	7.565	42.52	22.76
Rail	2.987	2.935	91.46	91.92	97.12	10.20	10.20	78.88	38.67

open city; segmented CBD
Car	2.111	1.941	44.021	44.452	46.82	6.62	46.82	0	20.03
Rail	2.746	2.478	76.993	77.525	81.38	8.50	81.38	0	32.97

closed city; non-segmented CBD
Car	1.070	1.058	2.329	3.343	3.412	3.572	3.572	-0.154	0
Rail	1.071	1.060	2.383	3.421	3.469	3.577	3.577	-0.105	0

closed city; segmented CBD
Car	1.070	1.059	2.329	3.345	3.413	3.574	3.413	0	0
Rail	1.071	1.061	2.384	3.423	3.469	3.579	3.469	0	0

Note: Given a one percent increase in productivity, the city rises and converges to a new steady state equilibrium. This table shows the total changes (in percentage) in productivity, wage, population, total residential land rent, total commercial land rent, rental rate of residential land, rental rate of commercial land, size of the CBD and radius of the city.

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Figure 6: City Volatility and Moving Cost

[Six panels showing simulation paths over time (x-axis: 0-150) arranged in two columns: delta=0% (left) and delta=1% (right)]

Top row - Wage:

delta=0%: Wage fluctuates between ~0.87-1.05, with car (solid) and rail (dashed) lines. Car shows a dramatic decline around time 100.
delta=1%: Wage fluctuates between ~0.95-1.05, more stable. Both car and rail remain closer together.

Middle row - Population (x10^6):

delta=0%: Population fluctuates between 0-14 million. Car shows volatile swings with a collapse near time 100. Rail shows similar but less extreme pattern.
delta=1%: Population is more stable, fluctuating between ~5-10 million. Rail (dashed) stays higher and more stable than car.

Bottom row - CBD Rent:

delta=0%: CBD rent fluctuates between ~1500-3000. Car shows volatile behavior with a collapse near time 100.
delta=1%: CBD rent is more stable, fluctuating between ~2300-2800. Both car and rail show moderate fluctuations.

Note: This figure shows simulation paths of wage, population and CBD rent with and without moving cost.

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