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Cultural workers and the character of cities

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Abstract

This paper examines the location choice of cultural producers when cities differ in housing supply and income demographics. We develop a two-region spatial model containing three types of industries: a constant returns to scale traditional sector, a modern sector with external scale economies and a monopolistically competitive cultural sector. The model is initially analyzed when workers supply labor inelastically to their respective industry. Both integration and segregation are never a stable equilibrium, and the conditions for the stability of both concentrated and partially interior equilibrium are solved for. The model is extended to allow for cultural producers to divide their time between cultural production and moonlighting in the traditional sector, in order to smooth their income. Under this extension, there is an equilibrium where a share of cultural producers live isolated from a larger integrated market. We also identify an equilibrium where one region is able to sustain full-time cultural producers, while in the opposite region cultural producers must moonlight in the traditional sector. Under partially interior equilibria, the number of varieties of the cultural good is always larger in the region with the greater supply of housing.

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Notes

  1. This assumption does not impact the qualitative results of the model. However in practice these shares likely vary. For example, using Florida (2011) the creative class comprised around one-third of the working population in 2000. In the model presented here, the share represented by the creative class would be divided between cultural producers and modern workers with the remaining share employed in the traditional sector.

  2. To see this note that a stable equilibrium requires \(\sigma >1+\delta /\gamma \). The difference between the distribution of cultural producers without moonlighting in (40) and with moonlighting is the first term on the right in (65), \(2\delta /\sigma (1-\beta )\), which is decreasing in \(\sigma \). Therefore, we only need to consider the smallest possible value for \(\sigma \) to verify whether the term is less than unity. Using \(\sigma =1+\delta /\gamma \) and \(1-\beta =\alpha +\gamma +\delta \) , we then have

    $$\begin{aligned} \sigma (1-\beta )=(1+\delta /\gamma )(\alpha +\delta +\gamma )>2 \delta \end{aligned}$$

    indicating that the multiple is less than 1.

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Acknowledgements

I would like to thank an editor of this journal, Janet E. Kohlhase, and two anonymous referees for helpful comments that helped improve the quality of the manuscript. In addition, I would also like to thank John Carruthers for helpful suggestions at the 2017 WRSA meeting in Santa Fe and Richard Arnott for comments on an early draft of the paper. Finally, I would like to thank the Western Regional Science Association. This manuscript received the Springer Award for best paper by an early career scholar at the 56st Annual WRSA Meeting, Santa Fe, NM 2017.

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Appendix

Appendix

1.1 The instability of an integrated equilibrium

We show that when \(\sigma >1+\delta /\gamma \) an integrated equilibrium is unstable. To begin, we consider an equilibrium in which \(\dot{\lambda }_j=0 \,\,\forall j\). This implies that \(P_1=P_2\), \(w_{c1}=w_{c2}\) and \(A_{m1}=A_{m2}\) and the distribution is given by

$$\begin{aligned} \lambda _{c1}^I= \left( \frac{\frac{\beta \sigma }{\sigma (1-\beta )-\delta }+\left( \frac{H_2}{H_1}\right) ^{\psi }}{\frac{\beta \sigma }{\sigma (1-\beta )-\delta }+1}\right) \frac{H_1^ \psi }{H_1^ \psi +H_2^ \psi }, \end{aligned}$$
(70)
$$\begin{aligned} \lambda _{t1}^I= \frac{\frac{1}{2}\frac{\beta \sigma }{\sigma (1-\beta )-\delta }(H_1^\psi -H_2^\psi )+H_2^\psi }{H_1^\psi +H_2^\psi }, \end{aligned}$$
(71)
$$\begin{aligned} \lambda _{m1}^I= \frac{1}{2}, \end{aligned}$$
(72)
$$\begin{aligned} \psi\equiv & {} \frac{\gamma (\sigma -1)}{\gamma (\sigma -1)-\delta }, \end{aligned}$$
(73)

where I is a mnemonic for integration. The Jacobian matrix of first partial derivatives for Eqs. (21)–(23) and the corresponding signs are given by

$$ \begin{aligned} J=\begin{bmatrix}\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{m1}}&\quad \frac{ \partial \dot{\lambda }_m}{\partial \lambda _{t1}}&\quad \frac{ \partial \dot{\lambda }_m}{\partial \lambda _{c1}}\\ \frac{\partial \dot{\lambda }_t}{\partial \lambda _{m1}}&\quad \frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}&\quad \frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}\\\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{m1}}&\quad \frac{ \partial \dot{\lambda }_c}{\partial \lambda _{t1}}&\quad \frac{\partial \dot{\lambda }_c}{\partial \lambda _{c1}}\end{bmatrix},\qquad \begin{bmatrix}? & \quad - &\quad + \\ - &\quad - &\quad + \\ +&\quad + &\quad -\end{bmatrix} \end{aligned}$$
(74)

The ambiguity of \(\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{m1}}\) arises from the competing forces of the external economies which both raise the nominal wage and increase rents. Stability of the equilibrium requires all eigenvalues to be negative when evaluated at the equilibrium values. Given that the determinant of a matrix is equal to the product of its eigenvalues, a necessary (though not sufficient) condition for the stability of the Jacobian is that the determinant then be negative. We now show that the determinant is positive. It is useful to calculate the determinant using the bottom row of J and note that \(V_{mi}=w_{mi} V_{ti}\) and that \(w_{m1}=p_m A_{m1}=p_m A_{m2}=w_{m2}\) when \(\lambda _{m1}=1/2\). Furthermore \(\frac{\partial w_{m1}}{\partial \lambda _{m1}}|_{\lambda _{m1} =1/2}=\frac{\partial w_{m2}}{\partial \lambda _{m1}}|_{\lambda _{m1}=1/2}\), and \(V_{j1}=V_{j2}=V_j\), in equilibrium. Denoting the determinant of J by \(\Delta \) yields

$$\begin{aligned} \Delta =\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{m1}} A-\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{t1}}B+\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{c1}}C, \end{aligned}$$
(75)

where

$$\begin{aligned} A&=\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{t1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}-\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{c1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}\nonumber \\&=w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}-w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}\nonumber \\&=0, \end{aligned}$$
(76)
$$\begin{aligned} B&=\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{m1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}-\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{c1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\nonumber \\&=\left( 2\frac{\partial w_{m1}}{\partial {\lambda _{m1}}}|_{\lambda _{m1}=1/2} V_t +w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\right) \frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}-w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\nonumber \\&=2 \frac{\partial w_{m1}}{\partial {\lambda _{m1}}}|_{\lambda _{m1}=1/2} V_t \frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}\nonumber \\&=\frac{4}{3}V_{m}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}, \end{aligned}$$
(77)
$$\begin{aligned} C&=\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{m1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}-\frac{ \partial \dot{\lambda }_m}{\partial \lambda _{t1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\nonumber \\&=\left( 2\frac{\partial w_{m1}}{\partial {\lambda _{m1}}}|_{\lambda _{m1}=1/2} V_t +w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\right) \frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}-w_{m1}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{m1}}\nonumber \\&=\frac{4}{3}V_{m}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}. \end{aligned}$$
(78)

We can then rewrite the determinant as

$$\begin{aligned} \Delta&=\frac{4}{3}V_m\left( -\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{t1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{c1}}+\frac{ \partial \dot{\lambda }_c}{\partial \lambda _{c1}}\frac{ \partial \dot{\lambda }_t}{\partial \lambda _{t1}}\right) \nonumber \\&=\frac{4}{3}V_m\left( -(1-\gamma )\left( \frac{1}{\lambda _m1 w_{m1}+\lambda _{t1}}+\frac{1}{(1-\lambda _{m1})w_{m2}+\lambda _{t1}}\right) V_c\right) \left( \frac{\delta }{\sigma -1}\left( \frac{1}{\lambda _{c1}}+\frac{1}{1-\lambda _{c1}}\right) V_t\right) \nonumber \\&\quad +\left( \frac{1+\delta -\sigma }{1-\sigma }\left( \frac{1}{\lambda _{c1}}+\frac{1}{1-\lambda _{c1}}\right) V_c\right) \left( \gamma \left( \frac{1}{\lambda _m1 w_{m1}+\lambda _{t1}}+\frac{1}{(1-\lambda _{m1})w_{m2}+\lambda _{t1}}\right) V_t\right) \nonumber \\&=\frac{4}{3}V_mV_tV_c\left( \frac{1}{\lambda _{m1} w_{m1}+\lambda _{t1}}+\frac{1}{(1-\lambda _{m1})w_{m2}+\lambda _{t1}}\right) \left( \frac{1}{\lambda _{c1}}+\frac{1}{1-\lambda _{c1}}\right) \nonumber \\&\quad \times \left( \frac{\delta (1-\gamma )}{1-\sigma }+\frac{\gamma (1+\delta -\sigma )}{1-\sigma }\right) \end{aligned}$$
(79)

The sign of \(\Delta \) is determined by the sign of the final term in brackets, which can readily shown to be positive when \(\sigma >1+\delta /\gamma \).

1.2 Simulations for partial integration of modern workers

This section provides simulations of the population distributions and the associated eigenvalues of the Jacobian matrix for the equations of motion under the partial integration of modern workers.

Fig. 2
figure 2

Simulations of the equilibrium population distributions of modern and cultural workers and the associated eigenvalues for the Jacobian of the utility gaps. Note: \(\beta =.5\), \(H_2=1\), \(\gamma =.25\), \(\delta =.1\), \(A_m=1\)

Figure 2a, b provides the population distributions. Figure 2c, d simulates the eigenvalues of the Jacobian matrix from (22) and (23) as the parameters \(H_1\) and \(\epsilon \) are increased. With regard to \(H_1\), the eigenvalues are negative over the region where both \(\lambda _{c1}>0\) and \(\lambda _{m1}>0\). When \(\lambda _{m1}=0\) the stability of the equilibrium is laid out in the condition for partial segregation. We see that for low values of \(\epsilon \) the eigenvalues are negative but as scale economies increase one eigenvalue becomes positive just as the number of modern workers in region 1 goes to 0.

1.3 Stability of a partially integrated equilibrium of traditional workers under moonlighting

Under this configuration, we will consider the case that \(\theta _1=1\), \(\theta _2\in (0,1)\) and \(\lambda _{m1}=1\). Therefore, the wage for cultural producers in region 1 and the price for the modern good are determined by the market clearing conditions in (14) and (16). Noting that \(w_{c2}=1\) we can then write the wage for modern workers and cultural producers as

$$\begin{aligned} p_mA_{m1}= \frac{\beta (\sigma -\delta )}{\sigma (1-\beta )-\delta }\left( 1+(1 -\lambda _{c1})+\frac{\delta }{\sigma -\delta }\lambda _{t1}\right) , \end{aligned}$$
(80)
$$\begin{aligned} w_{c1}= \frac{\delta }{\sigma (1-\beta )-\delta }\left( \frac{\beta (1 +(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}{\lambda _{c1}}\right) . \end{aligned}$$
(81)

The equation of motion for traditional and cultural workers is then given by

$$\begin{aligned} \dot{\lambda }_c=&\frac{\eta }{p_m^\beta }\left( A_{c1}\left( \frac{\left( \beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}\right) ^{1-\gamma }H_1^\gamma }{\lambda _{c1}^{\frac{\sigma -1-\delta }{\sigma -1}}}\right) \right. \nonumber \\&\quad \left. -A_{c2}\left( \frac{H_2^\gamma }{\left( (1-\lambda _{c1})+(1-\lambda _{t1})\right) ^{\frac{\gamma (\sigma -1)-\delta }{\sigma -1}}}\right) \right) , \end{aligned}$$
(82)
$$\begin{aligned} \dot{\lambda }_t&=\frac{\eta }{p_m^\beta }\left( A_{t1}\left( \frac{H_1^\gamma }{\left( \beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}\right) ^{\gamma }\lambda _{c1}^{\frac{\delta }{1-\sigma }}}\right) \right. \nonumber \\&\quad \left. -A_{t2}\left( \frac{H_2^\gamma }{\left( (1-\lambda _{c1})+(1-\lambda _{t1})\right) ^{\frac{\gamma (\sigma -1)-\delta }{\sigma -1}}}\right) \right) , \end{aligned}$$
(83)

where \( A_{ij}\) are a bundle of constants that are independent of \(\lambda _{t1}\) and \(\lambda _{c1}\). It is straightforward to verify that \(\frac{\partial \dot{\lambda }_c}{\partial \lambda _{c1}}<0\) and \(\frac{\partial \dot{\lambda }_t}{\partial \lambda _{t1}}<0\) when \(\sigma >1+\delta /\gamma \), so that the trace of the Jacobian is negative. We now show that the determinant of the Jacobian is positive, implying that both eigenvalues are negative and that the equilibrium is stable. Denote the determinant of the Jacobian as \(\Delta ^{\mathrm{PITM}}\) where PITM denotes partial integration of traditional workers with moonlighting. We then have

$$\begin{aligned} \Delta ^{\mathrm{PITM}}&=\frac{\partial \dot{\lambda }_c}{\partial \lambda _{c1}}\frac{\partial \dot{\lambda }_t}{\partial \lambda _{t1}}-\frac{\partial \dot{\lambda }_c}{\partial \lambda _{t1}}\frac{\partial \dot{\lambda }_t}{\partial \lambda _{c1}}\nonumber \\&=\left( \frac{\beta (1-\gamma )}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}+\frac{\sigma -1-\delta }{\sigma -1}\frac{1}{\lambda _{c1}}+\frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\right) V_c\nonumber \\&\quad \times \left( \frac{\gamma (1-\beta )}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}} +\frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\right) V_t\nonumber \\&\quad -\left( \frac{(1-\beta )(1-\gamma )}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}-\frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\right) V_c\nonumber \\&\quad \times \left( \frac{\delta }{\sigma -1}\frac{1}{\lambda _{c1}}+\frac{\beta \gamma }{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}-\frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\right) V_t\nonumber \\&=V_tV_c\left( \frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\left( \frac{\beta (1-\gamma )}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}+\frac{\sigma -1-\delta }{\sigma -1}\frac{1}{\lambda _{c1}}\right. \right. \nonumber \\&\quad \left. +\frac{\delta }{\sigma -1}\frac{1}{\lambda _{c1}}+\frac{(1-\beta )\gamma }{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}\right) +\frac{\gamma (\sigma -1)-\delta }{\sigma -1}\frac{1}{(1-\lambda _{c1})+(1-\lambda _{t1})}\nonumber \\&\quad \times \left( \frac{\gamma (1-\beta )+\beta (1-\gamma )}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}\right) \nonumber \\&\quad \left. + \frac{(1-\beta )}{\sigma -1}\frac{1}{\lambda _{c1}}\frac{1}{\beta (1+(1-\lambda _{c1}))+(1-\beta )\lambda _{t1}}\left( \gamma (\sigma -1)-\delta \right) \right) . \end{aligned}$$
(84)

If the final term in brackets is positive, the determinant is positive, which holds when \(\sigma >1+\delta /\gamma \), indicating that the equilibrium is stable.

1.4 Instability of equilibrium for partial integration of modern workers with labor choice

In this section, we verify the instability of the equilibrium in which both modern and cultural workers are divided between regions, while traditional workers are concentrated in a single region. Suppose that \(\lambda _{t1}=1\). Given that when cultural workers split their time between industries they receive the same utility as traditional workers. Then in equilibrium it must be that case that \(\dot{\lambda }_t=0\) if \(\dot{\lambda }_c=0\), which implies that the price index is equivalent in each region. This further implies modern workers are equally divided between regions in an interior equilibrium so that \(\lambda _{mi}=1/2\). Define regional income as

$$\begin{aligned} I_i\equiv \lambda _{mi }p_mA_{mi}+\lambda _{ci}+\lambda _{ti}. \end{aligned}$$
(85)

We can then write the equations of motion for cultural and modern workers as

$$\begin{aligned} \dot{\lambda }_m&=\kappa _m^C\left( \frac{(1+\lambda _{m1})^\epsilon }{I_1^{ \frac{1-\sigma +\delta }{1-\sigma }}}-\frac{(1+(1-\lambda _{m1}))^\epsilon }{I_2^{\frac{1-\sigma +\delta }{1-\sigma }}}\right) , \end{aligned}$$
(86)
$$\begin{aligned} \dot{\lambda }_c&=\kappa _c^C\left( \frac{1}{I_1^{\frac{1-\sigma +\delta }{1-\sigma }}}-\frac{1}{I_2^{\frac{1-\sigma +\delta }{1-\sigma }}}\right) . \end{aligned}$$
(87)

Denote the determinant of the Jacobian \(\Delta ^{PIMM}\) where PIMM denotes partial integration of modern workers under moonlighting. In equilibrium \(V_{m1}=V_{m2}=V_m\) and \(V_{c1}=V_{c2}=V_c\). Furthermore denote,

$$\begin{aligned} \frac{d \lambda _{mi}p_m A_{mi}}{d \lambda _{m1}}=p_mA_{m1}\frac{\partial \lambda _{mi}}{\partial \lambda _{m1}}+ \lambda _{mi}A_{m1}\frac{\partial p_m}{\partial \lambda _{m1}}+\lambda _{mi}p_m\frac{\partial A_{m1}}{\partial \lambda _{m1}}, \end{aligned}$$

where \(\frac{\partial p_m}{\partial \lambda _{m1}}|_{\lambda _{m1}=1/2}=0\). We then have

$$\begin{aligned} \Delta ^{PIMM}&=\frac{\partial \dot{V}_m}{\partial \lambda _{m1}}\frac{\partial \dot{V}_c}{\partial \lambda _{c1}}-\frac{\partial \dot{V}_m}{\partial \lambda _{c1}}\frac{\partial \dot{V}_c}{\partial \lambda _{m1}}\nonumber \\&=-\left( \frac{4\epsilon }{3}-\frac{1-\sigma +\delta }{1-\sigma }\left( \frac{\frac{d\lambda _{m1}p_m A{m1}}{d\lambda _{m1}}}{I_1}+\frac{\frac{d\lambda _{m2}p_m A_{m2}}{d\lambda _{m1}}}{I_2}\right) \right) V_m \times \left( \frac{1-\sigma +\delta }{1-\sigma }\left( \frac{1}{I_1}+\frac{1}{I_2}\right) \right) V_c\nonumber \\&\quad -\left( \frac{1-\sigma +\delta }{1-\sigma }\left( \frac{1}{I_1}+\frac{1}{I_2}\right) \right) V_m\times \left( \frac{1-\sigma +\delta }{1-\sigma }\left( \frac{\frac{d\lambda _{m1}p_m A{m1}}{d\lambda _{m1}}}{I_1}+\frac{\frac{d\lambda _{m2}p_m A_{m2}}{d\lambda _{m1}}}{I_2}\right) \right) V_c\nonumber \\&=-\frac{4\epsilon }{3}\left( \frac{1-\sigma +\delta }{1-\sigma }\left( \frac{1}{I_1}+\frac{1}{I_2}\right) \right) V_mV_c <0. \end{aligned}$$
(88)

The result implies that one eigenvalue must be positive; therefore, any such equilibrium is unstable.

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Lopez, J.C.G. Cultural workers and the character of cities. Ann Reg Sci 62, 211–246 (2019). https://doi.org/10.1007/s00168-018-0891-4

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