The creative class, its preferences, and unbalanced growth in an urban economy

Abstract

We theoretically analyze unbalanced growth in an urban economy arising from the preferences of the creative class concerning the relative desirability of agricultural, manufacturing, and service goods. We study two cases. In the first case, the production functions for the three categories of goods are identical. Our analysis leads to four results. First, we compute the equilibrium physical to creative capital ratios and the relationships between the neutral productivity shifters and the output prices. Second, we show that agricultural and services consumption are a constant multiple of manufacturing consumption. Third, we note that under certain conditions, an equilibrium in which all sectors of our urban economy grow at a constant rate does not exist. Fourth, we show that a constant growth path (CGP) equilibrium exists in which, across the three sectors, the pattern of consumption changes and there is a reallocation of creative capital. In the second case, the production functions for the three categories of goods are dissimilar. In this more realistic setting, we study generalizations of the previously described four results.

Appendix

Proof of Theorem 1

We begin by claiming that if the condition α ^A/B ^A = α ^S/B ^S is not satisfied then a CGP equilibrium in our urban economy does not exist. To this end, consider a CGP equilibrium in which c ^M(t) grows at the constant rate β _c after some time T. From the consumption Euler equation (20), we deduce that $k\left( t \right)=k^{\ast} =\left( {f^l} \right)^{-1}\left\{ {\left( {\theta \beta_c +\rho } \right)/B^M} \right\}$ is constant ∀ t ≥ T. Then the constraint (21) at any time t ≥ T can be expressed as

$$ B^Mf\left( {k^{\ast} } \right)-\left( {\zeta +\beta } \right)k^{\ast} =\frac{c^M\left( T \right)e^{\left( {\beta_c -\beta } \right)\left( {t-T} \right)}}{\eta^M}+\frac{e^{-\beta \left( {t-T} \right)}}{X\left( T \right)}B^M\left( {\frac{\alpha^A}{B^A}-\frac{\alpha^S}{B^S}} \right). $$

(39)

If β _c  > β then the RHS of Eq. 39 exceeds the LHS for sufficiently large t and this leads to a contradiction. On the other hand if β _c  < β then the RHS goes to 0 but the LHS is positive which again leads to a contradiction. From this it is clear that we must have β _c  = β. In this case, $\exp \left\{ {-\beta \left( {t-T} \right)} \right\}\left( {\alpha^A/B^A-\alpha^S/B^{S} } \right)$ is constant only if α ^A/B ^A = α ^S/B ^S = 0, that is, only if the condition α ^A/B ^A = α ^S/B ^S in Theorem 1 holds. Therefore, we conclude that this last condition is necessary for a CGP equilibrium in our urban economy to exist.

Having demonstrated necessity, we now claim that the Theorem 1 condition is sufficient for a unique CGP to exist. To this end, suppose that the Theorem 1 condition holds. Let the normalized manufacturing consumption be $\overline c^M\left( t \right)=c^M\left( t \right)/X\left( t \right)$. Then, the differential equation system given by Eqs. 20 and 21 can be written in normalized variables as

$$ \dot{{k}}\left( t \right)+\frac{\overline c^M\left( t \right)}{\eta ^M}=B^Mf\left\{ {k\left( t \right)} \right\}-\left( {\zeta +\beta } \right)k\left( t \right) $$

(40)

and

$$ \frac{d\overline c^M\left( t \right)/dt}{\overline c^M\left( t \right)}=\frac{1}{\theta }\left\{ {B^Mf^{\prime}\left\{ {k\left( t \right)} \right\}-\rho } \right\}-\beta . $$

(41)

The above system (40)–(41) has a unique steady state in which $\overline c ^M\left( t \right)=\left( {c^M} \right)^{\ast}$ and k(t) = k ^* are constant and are implicitly defined by

$$ f^{\prime}\left( {k^{\ast} } \right)=\left( {\rho +\theta \beta } \right)/B^M\;\mbox{and}\;\left( {\overline c^M} \right)^\ast =\eta^M\left\{ {B^Mf\left( {k^{\ast}} \right)-\left( {\zeta +\beta } \right)k^{\ast} } \right\}. $$

(42)

In addition, for any k(0), there exists a saddle path $\left\{ {\overline c ^M\left( t \right),k\left( t \right)} \right\}$, that converges to the steady state. Observe that the corresponding {c ^M(t), k(t)} _t also satisfies the transversality condition in Eq. 22 because we have $r^{\ast} -\zeta -\beta =\rho -\zeta +\left( {1-\theta } \right)\beta >0$ from assumption 4 in Acemoglu (2009, p. 310). From this it follows that the equilibrium path is saddle path stable in the normalized variables and c ^M(t) and K(t)/K _c (t) grow asymptotically at rate β. In addition, if k(0) = k ^* then the equilibrium is at the steady state and c ^M(t) and K(t)/K _c (t) grow exactly at rate β. Put differently, a CGP equilibrium exists.□

Proof of Theorem 2

We begin by combining the demand and the supply side equations for our urban economy and then derive the two conditions stated in Theorem 2. Suppose that there exists a CGP equilibrium in which consumption of the manufactured good c ^M(t) grows at the constant rate β _c and the physical capital to effective creative capital ratio k(t) ≡ k ^* is constant. Then from the consumption Euler Eq. (36), the return to physical capital r(t) ≡ r ^* is also constant. This last fact and Eq. 32 together tell us that k ^M(t) ≡ (k ^M)^* and w ^M(t) ≡ (w ^M)^* are also constant. Because c ^M(t) grows at rate β _c , the resource constraint (35) tells us that $\beta_{c}=\beta^{M}$ and that the expression $\alpha^Ap^A\left( t \right)-\alpha^Sp^S\left( t \right)$ is either zero for all time t or grows at the rate β ^M. Now, using Eqs. 30 and 31, we can write the last expression above as

$$ \begin{array}{rll} \alpha^Ap^A\left( t \right)-\alpha^Sp^S\left( t \right)&=&\alpha^A\chi ^A\left( {r^\ast } \right)^{\omega^A}\left( {w_\ast^M } \right)^{1-\omega ^A}\left\{ {\frac{B^M\left( t \right)}{B^A\left( t \right)}} \right\}^{1-\omega^A}\\&&- \ \alpha^S\chi^S\left( {r^\ast } \right)^{\omega ^S}\left( {w_\ast^M } \right)^{1-\omega^S}\left\{ {\frac{B^M\left( t \right)}{B^S\left( t \right)}} \right\}^{1-\omega^S}. \end{array} $$

(43)

Since the growth rates of the terms $\{ {B^M( t )/B^A( t )} \}^{1-\omega^A}$ and $\{ B^M( t )/B^S( t ) \}^{1-\omega^S}$ are strictly less than β ^M, the expression $\alpha^Ap^A( t )-\alpha ^Sp^S\left( t \right)$ cannot grow at the rate β ^M and this tells us that this growth rate must be zero for all time t. This last finding has two implications. First, Eq. 43 implies that the terms $\{ {B^M( t )/B^A( t )} \}^{1-\omega^A}$ and $\{ {B^M( t )/B^S( t )} \}^{1-\omega^S}$ must grow at the same rate. Put differently, the growth rates of the creative capital augmenting productivity terms must satisfy the restriction that

$$ \beta^M=\frac{\beta^A\left( {1-\omega^A} \right)-\beta^S\left( {1-\omega ^S} \right)}{\omega^S-\omega^A}. $$

(44)

Second, the initial agricultural and service good prices must satisfy $\alpha^Ap^A\left( 0 \right)-\alpha^Sp^S\left( 0 \right)$. Substituting for the ratio $w_\ast^M /r^\ast$ from Eq. 29 into Eq. 43, the previous requirement can be written as

$$ \frac{\alpha^A\chi^A}{\alpha^S\chi^S}=\left\{ {\frac{1-\omega^M}{\omega ^M}\left( {k^M} \right)^\ast } \right\}^{\omega^A-\omega^S}\left\{ {\frac{B^S\left( 0 \right)^{1-\omega^S}B^M\left( 0 \right)^{\omega ^S-\omega^A}}{B^A\left( 0 \right)^{1-\omega^A}}} \right\}. $$

(45)

Note that Eqs. 44 and 45 are the two conditions specified in the statement of Theorem 2 in the body of this paper.

To characterize the ratio $\left( {k^M} \right)^\ast$ endogenously in terms of the exogenous parameters, observe that the Euler equation implies that r ^∗ = ρ + θβ ^M. Using this last result in Eq. 32, we see that $\left( {k^M} \right)^\ast$ satisfies $\left( {k^M} \right)^\ast =\left\{ {\omega^M/\left( {\rho +\theta B^M} \right)} \right\}^{1/\left( {1-\omega ^M} \right)}$. Substituting this last expression for $\left( {k^M} \right)^\ast$ in Eq. 45, the second condition on preferences that we seek can be written as

$$ \frac{\alpha^A\chi^A}{\alpha^S\chi^S}=\left[ {\frac{1-\omega^M}{\omega ^M}\left\{ {\frac{\omega^M}{\rho +\theta \beta^M}} \right\}^{1/\left( {1-\omega^M} \right)}} \right]^{\omega^A-\omega^S}\left\{ {\frac{B^S\left( 0 \right)^{1-\omega^S}B^M\left( 0 \right)^{\omega ^S-\omega^A}}{B^A\left( 0 \right)^{1-\omega^A}}} \right\}. $$

(46)

From the above discussion we conclude that if a CGP equilibrium is to exist then the parameters of the model described in Section  4 will need to satisfy the two requirements given in Eqs. 44 and 46. Conversely, when these two requirements hold, it can be seen that the results of Section 3 generalize to this dissimilar production functions case and there exists a CGP equilibrium in which consumption of the manufacturing good c ^M(t) grows at the constant rate β ^M.□