Online Appendix for Dynamic Spatial General Equilibrium (Not for Publication)

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Online Appendix for “Dynamic Spatial General

Equilibrium” (Not for Publication)

Benny Kleinman^∗ Princeton University

Ernest Liu^† Princeton University

Stephen J. Redding^‡

Princeton University, NBER and CEPR

July 2021

A Introduction

In this online appendix, we report the detailed derivations for the results reported in the paper and further supplementary results. In SectionB, we report the derivations for our baseline model with a single traded sector from Section2of the paper. We characterize the existence and uniqueness of the general equilibrium of the model and present the proofs of the other propositions in the paper. In SectionC, we establish a number of isomorphisms, in which we show that our results hold throughout the class of trade models with a constant trade elasticity.

In SectionD, we introduce a number of extension of our baseline specification, as discussed in Section4of the paper. SubsectionD.1shows that our framework naturally accommodates shocks to trade and migration costs. SubsectionD.2 allows for agglomeration forces in production and residence and provides a characterization of the existence and uniqueness of the equilibrium in the presence of these agglomeration forces.

Subsection D.3 introduces multiple final goods sectors with region-specific capital. Section D.4incorporates multiple final goods sectors with region-sector-specific capital. SectionD.5fur- ther generalizes the analysis to allow for multiple final goods sectors with region-sector-specific capital and input-output linkages. SubsectionD.6incorporates trade deficits following the conventional approach of the quantitative international trade literature in treating these deficits as exogenous. SectionD.7allows capital to be used residentially (for housing) as well as commer- cially (in production).

In SectionE, we present the derivations for the extension of our baseline model with a single traded sector and single non-traded sector used for our baseline quantitative analysis in Section 5 of the paper. Section F reports additional empirical results that are discussed in the paper.

SectionG reports further empirical results for our specification check to examine the potential scope for non-linearities from Section6in the paper. SectionHreports further details about the data sources and definitions.

B Baseline Dynamic Spatial Model

In this section of the online appendix, we introduce our baseline dynamic spatial model, which features a model of trade between locations with a constant trade elasticity, a dynamic discrete choice model of migration with a constant migration elasticity, and an optimal consumption- investment decision for the accumulation of capital. We derive our main sufficient statistics results for the comparative statics of the spatial distribution of economic activity in steady-state and along the full transition path, using our four observed matrices of expenditure shares, income shares, outmigration shares and inmigration shares.

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To simplify the exposition, we model trade between locations as inArmington(1969), in which goods are differentiated by origin. In SectionCof this online appendix, we establish a number of isomorphisms, in which we show that our results hold throughout the class of models with a constant trade elasticity in Arkolakis, Costinot and Rodriguez-Clare (2012). For expositional clarity, we also focus in this section on shocks to productivities and amenities, but we show in SectionDof this online appendix that our approach also holds for shocks to trade and migration costs. As part of that section, we also show that our approach admits a large number of other extensions and generalizations, including agglomeration economies, multiple factors, multiple sectors, and input-output linkages, among others.

We consider an economy that consists of many locations indexed by i ∈ {1, . . . , N}. Time is discrete and is indexed by t. There are two types of infinitely-lived agents: workers and landlords.

Workers are endowed with one unit of labor that is supplied inelasticity and are geographically mobile subject to migration costs. Workers do not have access to an investment technology and hence live “hand to mouth,” as in Kaplan and Violante (2014). Landlords are geographically immobile and own the capital stock in their location. They make a forward-looking decision over consumption and investment in this local stock of capital. We assume that capital is geographically immobile once installed and depreciates gradually at a constant rate δ.

In SubsectionsB.1-B.6, we introduce our specifications of worker migration and landlord investment decisions. In SubsectionB.7, we provide a characterization of the existence and uniqueness of the deterministic steady-state equilibrium of the model. In SubsectionB.8, we derive our sufficient statistics for the first-order general equilibrium effect of shocks to productivities and amenities on the entire spatial distribution of economic activity in steady-state and along the transition path. In SubsectionB.9, we characterize the distributional consequences of shocks to productivity and amenities, as discussed in Section5.6of the paper.

In Subsection B.10, we report the derivations for the expression for expected utility in the paper. In SubsectionB.11, we provide the derivations for the expression for the migration choice probabilities in the paper.

B.1 Worker Migration Decisions

At the beginning of each period t, the economy inherits a mass of workers in each location i (`it), where the total labor endowment of the economy is given by ` = P^N_i=1`_it. Workers produce and consume in their current location during period t, before observing mobility shocks {gt} and subsequent location fundamentals {zgt+1, b_gt+1} for all possible locations g ∈ {1, . . . , N}, and deciding where to move for the next period t+1, given bilateral migration costs {κ_git}. Therefore, the value function for a worker in location i in period t (V^wit) is equal to the current flow of utility in that location plus the expected continuation value next period from the optimal choice

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of location:

V^wit = ln u^w_it+ max

{g}^N₁ βEt

V^wgt+1 − κ_git+ ρ_gt , (B.1) where we use the superscript w to denote workers; we assume logarithmic utility (ln u^w_it); β is the discount rate; Et[·] denotes an expectation taken over future location characteristics; the distribution for idiosyncratic mobility shocks; ρ controls the dispersion of idiosyncratic mobility shocks; and we assume κ_iit = 1and κ_nit> 1for n 6= i.

We make the conventional assumption that the idiosyncratic mobility shocks are drawn from an extreme value distribution:

F () = e^−e^(−−¯^γ), (B.2)

where γ is the Euler-Mascheroni constant.

Under this assumption, the value for a worker of living in location i at time t after taking expectation with respect to the idiosyncratic mobility shocks {_gt} (v^w_it ≡ E[V^wit]) can be re- written in the following form:

v_it^w = ln u^w_it + ρ ln

N

X

g=1

exp βEtv^w_gt+1 /κ_git1/ρ

, (B.3)

as shown in SubsectionB.10below, where the expectation in Etv_gt+1^w = EtE[V^wit]is taken over future fundamentals, {zis, b_is}^∞_s=t+1. The corresponding probability of migrating from location i to location g satisfies the following gravity equation:

D_igt = exp βEtv_gt+1^w /κgit

1/ρ

PN

m=1 exp βE^tv_mt+1^w /κmit

1/ρ, (B.4)

as shown in SubsectionB.11below.

B.2 Worker Consumption

Worker preferences are modeled as in the standard Armington model of trade. As workers do not have access to an investment technology, they choose their consumption of varieties each period to maximize their flow utility in the location in which they have chosen to live. Worker flow utility depends on local amenities (b_nt) and goods consumption (c^w_nt) and is assumed to take the logarithmic form:

ln u^w_nt = ln b_nt+ ln c^w_nt, (B.5)

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where c^w_ntis a consumption index for workers in location n defined over the consumption of the variety supplied by each location i (c^w_nit):

c^w_nt =

" _N X

i=1

(c^w_ni)^θ+1^θ

#^θ+1_θ

, θ = σ − 1, σ > 1, (B.6)

where σ > 1 is the constant elasticity of substitution (CES) between varieties and θ = σ −1 is the trade elasticity. Amenities (b_nt) capture characteristics of a location that make it a more attractive place to live regardless of goods consumption (e.g. climate and scenic views). In this section, we assume that amenities are exogenous, but in SectionDof this online appendix, we allow them to be endogenous to the surrounding concentration of economic activity through agglomeration or congestion forces.

The corresponding worker indirect utility each period depends on amenities (bnt), the wage (wnt) and the consumption goods price index (pit):

ln u^w_nt = ln b_nt+ ln w_nt− ln p_nt, (B.7) where the consumption goods price index (p_nt) in location n depends of the price of the variety sourced from each location i (pnit):

p_nt =

" _N X

i=1

p^−θ_nit

#^−1/θ

. (B.8)

Using the properties of constant elasticity of substitution (CES) preferences, the share of expenditure of importer n on the goods supplied by exporter i is:

Snit≡ (pnit)^−θ PN

m=1(p_nmt)^−θ. (B.9)

B.3 Production

Firms in each location use labor (`it) and capital (kit) to produce output (yit) of the variety supplied by that location. Production is assumed to occur under conditions of perfect competition and subject to the following constant returns to scale technology:

yit= zit

`_it µ

µ kit

1 − µ

1−µ

, 0 < µ < 1, (B.10)

where zit denotes productivity in location i at time t. As for amenities above, we assume in this section that productivity is exogenous, but in Section 4below we allow it to be endogenous to the surrounding concentration of economic activity through agglomeration forces.

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We assume that trade between locations is subject to iceberg variable costs of trade, such that τ_nit≥ 1units of a good must be shipped from location i in order for one unit to arrive in location n, where τnit > 1for n 6= i and τiit = 1. From profit maximization, the cost to a consumer in location n of sourcing the good produced by location i depends solely on iceberg trade costs and constant marginal costs:

p_nit= τ_nitp_iit = τ_nitw^µ_itr_it^1−µ

z_it , (B.11)

where p_iitis the “free on board” price of the good supplied by location i before trade costs.

From profit maximization and zero profits, total payments to each factor of production are a constant share of total revenue:

w_it`_it = µp_iity_it, (B.12)

r_itk_it= (1 − µ) p_iity_it. (B.13)

B.4 Landlord Consumption

Landlords in each location choose their consumption and investment in capital to maximize their intertemporal utility subject to their intertemporal budget constraint. Landlords’ intertemporal utility equals the present discounted value of their flow utility:

v_it^k = E^t

∞

X

s=0

β^t+s c^k_it+s1−1/ψ

1 − 1/ψ , (B.14)

where c^k_it is a consumption index defined over the consumption of the good supplied by each location (c^k_imt), as in equation (B.6) for workers above; ψ is the elasticity of intertemporal substitution.

We assume that the investment technology in each location uses the varieties from all locations with the same functional form as consumption. In particular, landlords in a given location can produce one unit of capital in that location using one unit of the consumption index in that location.¹ We assume that capital is geographically immobile once installed and depreciates at a constant rate δ. The intertemporal budget constraint for landlords in each location requires that total income from the existing stock of capital (ritk_it) equals the total value of their consumption (pitc^k_it) plus the total value of net investment (pit(k_it+1− (1 − δ) k_it)):

r_itk_it= p_it c^k_it+ k_it+1− (1 − δ) k_it . (B.15)

1Although this assumption that consumption and investment use goods in the same proportions is the standard specification, we obtain similar results in an alternative specification in which investment in each location uses only the good produced by that location.

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Combining landlords’ intertemporal utility (B.14) and budget constraint (B.15), the landlord’s intertemporal optimization problem is:

{^c^kt+s,kmaxt+s+1}^∞_s=0Et

∞

X

s=0

β^t+s c^k_it+s1−1/ψ

1 − 1/ψ , (B.16)

subject to pitc^k_it+ pit(kit+1− (1 − δ) kit) = ritkit.

Lemma. (Lemma1in the paper) We denote Rit≡ 1 − δ + r_it/p_itas the gross return on capital. The optimal consumption of location i’s landlords satisfies cit = ς_itR_itk_it, where ςitis defined recursively as

ς_it⁻¹ = 1 + β^ψ

E^t

R

ψ−1 ψ

it+1ς⁻

1 ψ

t+1

ψ

.

Landlord’s optimal saving and investment satisfies kit+1 = (1 − ς_it) R_itk_it.

Proof. For notational simplicity we drop the locational subscript. Consider a landlord facing lin- ear returns R_t on wealth k_t for all t. Let v (k_t; t) denote the value function at time t; we can rewrite the landlord’s consumption-saving problem recursively as:

v (kt; t) = max

{ct,kt+1}

c^1−1/ψ_t

1 − 1/ψ + βE^tv (kt+1; t + 1) s.t. ct+ kt+1= Rtkt,

where, with a slight abuse of notation, we denote landlord consumption as c instead of c^kfor the purpose of this proof. We guess-and-verify that there exists at, ςtsuch that v (k; t) = ^(a^t^R_1−1/ψ^t^k^t⁾^1−1/ψ, and that optimal ct= ς_tR_tk_t.

Under the conjecture, vk(k_t; t) = a^1−1/ψ_t R^1−1/ψ_t k^−1/ψ_t , we setup the Lagrangian as:

Lt = c^1−1/ψ_t

1 − 1/ψ + βE^tv (kt+1; t + 1) + ξt[Rtkt− ct− kt+1] . The first-order conditions imply:

{c_t} c^−1/ψ_t = ξ_t, {k_t} ξ_t+1 = βk_t+1^−1/ψEt

h

a^1−1/ψ_t+1 R^1−1/ψ_t+1 i . Hence:

c_t= β^−ψk_t+1Et

h

a^1−1/ψ_t+1 R_t+1^1−1/ψi−ψ

. (B.17)

The Envelope condition v_k(k_t; t) = ξ_tR_timplies

a^1−1/ψ_t R^1−1/ψ_t k^−1/ψ_t = c^−1/ψ_t R_t. (B.18)

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Substituting our guess that ct ≡ ς_tR_tk_tinto the Envelope condition (B.18), we obtain:

a^1−ψ_t = ς_t.

The budget constraint implies k_t+1 = (1 − ς_t) R_tk_t, and substituting this result into (B.17), we get:

ς_t= β^−ψEt

h

a^1−1/ψ_t+1 R^1−1/ψ_t+1 i^−ψ

(1 − ς_t)

⇐⇒ ς_t⁻¹ = 1 + β^ψEt

R

ψ−1 ψ

t+1ς_t+1^−1/ψ

ψ

, (B.19)

as desired.

Note that, in the special case of logarithmic flow utility (ψ = 1), landlord’s optimal consumption and saving rate is independent of future returns to capital, and ςt = (1 − β)for all t, as in Moll (2014).

B.5 Market Clearing

Goods market clearing implies that income in each location, which equals the sum of the income of workers and landlords, is equal to expenditure on the goods produced by that location:

(w_it`_it+ r_itk_it) =

N

X

n=1

S_nit(w_nt`_nt+ r_ntk_nt) . (B.20) Using the property that payments to capital and labor are constant shares of total revenue in equations (B.12) and (B.13), we can rewrite this goods market clearing condition as follows:

w_it`_it+1 − µ

µ w_it`_it=

N

X

n=1

S_nit

w_nt`_nt+ 1 − µ µ w_nt^h`^h_nt

,

1

µw_it`_it =

N

X

n=1

S_nit1

µw_nt`_nt,

w_it`_it =

N

X

n=1

S_nitw_nt`_nt. (B.21)

Capital market clearing implies that the rental rate for capital is determined by the requirement that landlords’ income from the ownership of capital equals payments for its use. Using the property that payments to capital and labor are constant shares of total revenue in equations (B.12) and (B.13), we can write this capital market clearing condition as:

r_it = 1 − µ µ

wit`it

k_it . (B.22)

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B.6 General Equilibrium

Given the state variables {`i0, ki0}, the general equilibrium of the economy is the stochastic process of allocations and prices such that firms in each location choose inputs to maximize profits, workers make consumption and migration decisions to maximize utility, landlords make consumption and investment decisions to maximize utility, and prices clear all markets, with the appropriate measurability constraint with respect to the realization of location fundamentals. For expositional clarity, we collect the equilibrium conditions and express them in terms of a sequence of four endogenous variables {`it, k_it, w_it, R_it, v_it}^∞_t=0. All other endogenous variables of the model can be recovered as a function of these variables.

Capital Returns and Accumulation: Using capital market clearing (B.22), the gross return on capital in each location i must satisfy:

R_it =

1 − δ +1 − µ µ

w_it`_it pitkit

/p_it

, where the price index (B.8) follows

pnt =





N

X

i=1

wit

1 − µ µ

1−µ

(`it/kit)^1−µτni/zi

!^−θ



−1/θ

. (B.23)

The law of motion for capital is

k_it+1 = (1 − ς_it)

1 − δ + 1 − µ µ

w_it`_it p_itk_it

k_it, (B.24)

where (1 − ςit)is the saving rate defined recursively as in Lemma1in the paper:

ς_it⁻¹= 1 + β^ψ

E^t

R

ψ−1 ψ

it+1ς⁻

1 ψ

i,t+1

ψ

.

Goods Market Clearing: Using the equilibrium pricing rule (B.11), the expenditure share (B.9) and capital market clearing (B.22), the goods market clearing condition (B.20) can be written as:

w_it`_it =

N

X

n=1

S_nitw_nt`_nt, (B.25)

Snit = w_it(`_it/k_it)^1−µτ_ni/z_i^−θ PN

m=1 wmt(`mt/kmt)^1−µτnm/zm

^−θ, Tint ≡ S_nitw_nt`_nt w_it`_it ,

where S_nitis the expenditure share of importer n on exporter i at time t; we have defined T_intas the corresponding income share of exporter i from importer n at time t; and note that the order of subscripts switches between the expenditure share (Snit) and the income share (Tint), because the first and second subscripts will correspond below to rows and columns of a matrix, respectively.

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Population Flow: Using the out-migration probabilities (B.4), the population flow condition for the evolution of the population distribution over time is given by:

`_gt+1 =

N

X

i=1

D_igt`_it, (B.26)

D_igt = exp βEtv_gt+1^w /κgit

1/ρ

PN

m=1 exp βEtv_mt+1^w /κ_mit1/ρ, E_git ≡ `_itD_igt

`_gt+1 ,

where Digtis the outmigration probability from location i to location g between time t and t + 1;

we have defined Egit as the corresponding inmigration probability to location g from location i between time t and t + 1; and again note that the order of subscripts switches between the outmigration probability (Digt) and the inmigration probability (Egit), because the first and second subscripts will correspond below to rows and columns of a matrix, respectively.

Worker Value Function: Using the worker indirect utility function (B.7) in the value function (B.3), the expected value from living in location n at time t can be written as:

v_nt^w = ln b_nt+ ln w_nt p_nt

+ ρ ln

N

X

g=1

exp βEtv^w_gt+1 /κgnt

1/ρ

. (B.27)

B.7 Existence and Uniqueness (Proof of Proposition 1 in the Paper)

We now use the system of equations for general equilibrium (B.24)-(B.27) to prove the existence and uniqueness of a deterministic steady-state equilibrium with time-invariant fundamentals {zi, b_i, τni, κni} and endogenous variables {v_i^∗, w_i^∗, R^∗_i, `^∗_i, k^∗_i}. Given these time-invariant fundamentals, we can drop the expectation over future fundamentals, such that Etv^w_gt+1 = v^w_gt+1.

B.7.1 Capital Labor Ratio

In steady-state, kit+1 = kit = k^∗_i, c^k_it+1 = c_it^k = c^k∗_i , and ςit+1= ςit = ς_i^∗, which implies:

1 − ς_i^∗ = β.

Using these results and the capital accumulation condition (B.24), we can solve for the steady- state capital-labor ratio:

k^∗_i = β

1 − δ +1 − µ µ

w^∗_i`^∗_i p^∗_ik_i^∗

k_i^∗,

k_i^∗ = β1 − µ µ

w^∗_i

p^∗_i `^∗_i + β (1 − δ) k_i^∗,

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k_i^∗ = β 1 − β (1 − δ)

1 − µ µ

w^∗_i p^∗_i `^∗_i, k^∗_i

`^∗_i = β 1 − β (1 − δ)

1 − µ µ

w^∗_i

p^∗_i . (B.28)

B.7.2 Price Index

Using this result for the steady-state capital-labor ratio, we can re-write the price index in equation (B.23) as follows:

(p^∗_n)^−θ =

N

X

i=1

w_i^∗ 1 − µ µ

1−µ

(`^∗_i/k_i^∗)^1−µτ_ni/z_i

!−θ

,

(p^∗_n)^−θ =

N

X

i=1

w_it^∗ 1 − µ µ

1−µ

(`^∗_i/k_i^∗)^1−µτ_ni/z_i

!−θ

,

(p^∗_n)^−θ =

N

X

i=1

w_i^∗ 1 − β (1 − δ) β

1−µ

(p^∗_i/w^∗_i)^1−µτ_ni/z_i

!−θ

,

(p^∗_n)^−θ=

N

X

i=1

1 − β (1 − δ) β

^{−θ(1−µ)}

(w_i^∗)^−θµ(p^∗_i)^{−θ(1−µ)}(τ_ni/z_i)^−θ,

(p^∗_n)^−θ =

N

X

i=1

ψτe_ni(w_i^∗)^−θµ(p^∗_i)^{−θ(1−µ)}, (B.29)

ψ ≡ 1 − β (1 − δ) β

−θ(1−µ)

, eτ_ni ≡ (τ_ni/z_i)^−θ.

B.7.3 Goods Market Clearing Condition

Using this result for the steady-state capital-labor ratio, we can also re-write the goods market clearing condition (B.25) as follows:

w_i^∗`^∗_i =

N

X

n=1

w_i^∗(`^∗_i/k_i^∗)^1−µτ_ni/z_i^−θ PN

m=1 w^∗_m(`^∗_m/k^∗_m)^1−µτ_nm/z_m^−θw^∗_n`^∗_n,

w^∗_i`^∗_i =

N

X

n=1

w_i^∗

1−µ µ

1−µ

(`^∗_i/k_i^∗)^1−µτ_ni/z_i

−θ

(p^∗_n)^−θ w_n^∗`^∗_n,

w^∗_i`^∗_i =

N

X

n=1

w^∗_i

1−β(1−δ) β

1−µ

(p^∗_i/w^∗_i)^1−µτ_ni/z_i

−θ

(p^∗_n)^−θ w^∗_n`^∗_n,

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w_i^∗`^∗_i =

N

X

n=1

(w^∗_i)^−θµ

1−β(1−δ) β

−θ(1−µ)

(p^∗_i)^{−θ(1−µ)}(τ_ni/z_i)^−θ (p^∗_n)^−θ w_n^∗`^∗_n,

`^∗_i (w_i^∗)^1+θµ(p^∗_i)^θ(1−µ) =

N

X

n=1

1 − β (1 − δ) β

^{−θ(1−µ)}

(p^∗_n)^θ(τ_ni/z_i)^−θw^∗_n`^∗_n,

`^∗_i (w^∗_i)^1+θµ(p^∗_i)^θ(1−µ)=

N

X

n=1

ψeτ_ni(p^∗_n)^θw^∗_n`^∗_n. (B.30)

B.7.4 Value Function

We now show that the value function (B.27) can be re-written as follows:

v_n^w∗= ln b_n+ ln w_n^∗ p^∗_n

+ ρ ln

N

X

g=1

exp βv_g^w∗ /κ_gn1/ρ

,

exp (v^w∗_n ) = b_n w^∗_n p^∗_n

" _N X

g=1

#^ρ ,

exp β ρv_n^w∗

= b^β/ρ_n w^∗_n p^∗_n

β/ρ" _N X

g=1

#^β ,

exp β ρv^w∗_n

= b^β/ρ_n w_n^∗ p^∗_n

β/ρ" _N X

g=1

κ^−1/ρ_gn exp β ρv_g^w∗

#^β ,

exp β ρv^w∗_n

= w_n^∗ p^∗_n

β/ρ" _N X

g=1

κgn/b^β_n^−1/ρ

exp β ρv_g^w∗

#^β ,

exp β ρv_n^w∗

= w^∗_n p^∗_n

β/ρ" _N X

g=1

eκ_gnexp β ρv_g^w∗

#^β

, eκ_gn ≡ κ_gn/b^β_n^−1/ρ ,

exp β ρv^w∗_n

= w_n^∗ p^∗_n

β/ρ

φ^β_n, φ_n≡

N

X

g=1

eκ_gnexp β ρv_g^w∗

. (B.31)

Using this solution in the definition of φnimmediately above, we have:

φ_n=

N

X

g=1

eκ_gn p^∗_g−β/ρ

w_g^∗β/ρ

φ^β_g. (B.32)

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B.7.5 Population Flow Condition

We now show that the population flow condition (B.26) can be re-written as follows:

`^∗_g =

N

X

i=1

exp βv_g^w∗ /κ_gi1/ρ

PN

m=1(exp (βv_m^w∗) /κ_mi)^1/ρ`^∗_i,

`^∗_g =

N

X

i=1

κ^−1/ρ_gi exp

β ρv^w∗_g PN

m=1κ^−1/ρ_mi exp

β ρv^w∗_m `

∗ i,

`^∗_g =

N

X

i=1

κ^−1/ρ_gi exp β ρv_g^w∗

" _N X

m=1

κ^−1/ρ_mi exp β ρv^w∗_m

#⁻¹

`^∗_i,

`^∗_g =

N

X

i=1

κgi/b^β_i

−1/ρ

exp β ρv_g^w∗

" _N X

m=1

κmi/b^β_i

−1/ρ

exp β ρv^w∗_m

#⁻¹

`^∗_i,

`^∗_g =

N

X

i=1

eκgiexp β ρv_g^w∗

" _N X

m=1

eκmiexp β ρv^w∗_m

#⁻¹

`^∗_i, eκgi≡ κgi/b^β_i

−1/ρ

,

`^∗_g =

N

X

i=1

eκ_giexp β ρv^w∗_g

φ⁻¹_i `^∗_i, φ_i ≡

N

X

m=1

κe_miexp β ρv^w∗_m

. Now using the value function result (B.31) above, we have:

`^∗_g =

N

X

i=1

eκ_gi

w_g^∗ p^∗_g

β/ρ

φ^β_gφ⁻¹_i `^∗_i,

p^∗_gβ/ρ

w_g^∗−β/ρ

`^∗_gφ^−β_g =

N

X

i=1

eκ_gi`^∗_iφ⁻¹_i . (B.33) B.7.6 System of Equations

Collecting together these results, the steady-state equilibrium of the model {p^∗_i, w^∗_i, `^∗_i, φ^∗_i}can be expressed as the solution to the following system of equations:

(p^∗_i)^−θ =

N

X

n=1

ψτein(p^∗_n)^{−θ(1−µ)}(w^∗_n)^−θµ, (B.34)

(p^∗_i)^θ(1−µ)(w_i^∗)^1+θµ`^∗_i =

N

X

n=1

ψτe_ni(p^∗_n)^θw_n^∗`^∗_n, (B.35)

(p^∗_i)^β/ρ(w^∗_i)^−β/ρ`^∗_i (φ^∗_i)^−β =

N

X

n=1

eκ_in`^∗_n(φ^∗_n)⁻¹, (B.36)

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φ^∗_i =

N

X

n=1

eκ_ni(p^∗_n)^−β/ρ(w_n^∗)^β/ρ(φ^∗_n)^β, (B.37) where we have the following definitions:

ψ ≡ 1 − β (1 − δ) β

−θ(1−µ)

, eτ_ni ≡ (τ_ni/z_i)^−θ,

φ^∗_i ≡

N

X

n=1

eκniexp β ρv_n^w∗

, eκin ≡ κin/b^β_n^−1/ρ .

Proposition. Existence and Uniqueness (Proposition1 in the paper). There exists a unique steady-state spatial distribution of economic activity {wi^∗, v_i^∗, `^∗_i, k_i^∗}given time-invariant location characteristics {zi, bi, τni, κni} that is independent of the economy’s initial conditions {`i0, ki0}.

Proof. The exponents on the variables on the left-hand side of the system of equations (B.34)- (B.37) can be represented as the following matrix:

Λ =







−θ 0 0 0

θ (1 − µ) (1 + θµ) 1 0

β/ρ −β/ρ 1 −β

0 0 0 1





 .

The exponents on the variables on the right-hand side of the system of equations (B.34)-(B.37) can be represented as the following matrix:

Γ =







−θ (1 − µ) −θµ 0 0

θ 1 1 0

0 0 1 −1

−β/ρ β/ρ 0 β





 .

Note that all elements of the kernelsτe_ni andeκ_in are strictly positive. Additionally, both Λ and and Γ are invertible. Let Θ denote the following composite matrix: Θ = ΓΛ⁻¹. From Theorem 1 in Allen, Arkolakis and Li (2020), there exists a unique steady-state equilibrium if the spectral radius of this composite matrix is less than one (ρ (Θ) < 1). The eigenvalues of this matrix are:





 ξ₁ ξ₂ ξ₃ ξ₄







=







1

β

ρ+(1+µθ) (1+µθ)

β

ρ+(1+µθ)

β 1 − µ





 ,

which are all strictly less than one.

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As the expenditure shares (S) and income shares (T ) are homogeneous of degree zero in factor prices, we require a choice of units or numeraire in order to solve for changes in wages.

We choose the total income of all locations as our numeraire: P^N_i=1w_it`_it = PN

i=1q_it = q_t = 1, which implies P^N_t=1q_i^∗d ln q^∗_i =PN

t=1q_i^∗^dq_q∗^∗ⁱ i

=PN

t=1 dq_i^∗ = 0. Similarly, the outmigration shares (D) and inmigration shares (E) are homogeneous of degree zero in the total population of all locations, which requires a choice of units to solve for population levels. We solve for population shares, imposing the requirement that the population shares sum to one: P^N_i=1`_it = ` = 1, which implies P^N_i=1`^∗_i d ln `^∗_i =PN

i=1`^∗_i ^d`_`∗^∗ⁱ

i =PN

i=1 d`^∗_i = 0.

B.8 Linearization

We now derive our main sufficient statistics results for the response of the spatial distribution of economic activity with respect to shocks to the economic environment. In SubsectionB.8.1, we totally differentiate the general equilibrium conditions of the model to obtain comparative statics. In SubsectionsB.8.2-B.8.3, we derive our sufficient statistics for changes in steady-state.

In SubsectionB.8.4, we derive our sufficient statistics for the entire transition path of the spatial distribution of economic activity. Throughout the following, we use bold math font to denote a vector (lowercase letters) or matrix (uppercase letters).

B.8.1 Comparative Statics

We now totally differentiate the conditions for general equilibrium to obtain comparative static expressions that we use in our sufficient statistics for changes in steady-state and the entire transition path.

Expenditure Shares Totally differentiating this expenditure share equation (B.9), we get:

dS_nit S_nit = θ

N

X

h=1

S_nhtdp_nht

p_nht − dp_nit p_nit

!

, (B.38)

d ln S_nit= θ

N

X

h=1

S_nhtd ln p_nht− d ln p_nit

! .

Prices Using the relationship between capital and labor payments (B.22), the pricing rule (B.11) can be re-written as follows:

p_nit=

τ_nitw_it

1−µ µ

1−µ

1 χit

1−µ

z_it ,

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where χitis the capital-labor ratio:

χ_it ≡ k_it

`_it. Totally differentiating this pricing rule, we have:

dp_nit

p_nit = dτ_nit

τ_nit + dw_it

w_it − (1 − µ) dχ_it

χ_it − dz_it z_it ,

d ln p_nit= d ln τ_nit+ d ln w_it− (1 − µ) d ln χ_it− d ln z_it. (B.39) Price Indices Totally differentiating the consumption goods price index in equation (B.8), we have:

dp_nt p_nt =

N

X

m=1

S_nmtdp_nmt

p_nmt , (B.40)

d ln p_nt =

N

X

m=1

S_nmtd ln p_nmt.

Real Income. Totally differentiating real income we have:

d ln w_it p_it

= d ln w_it− d ln p_it,

d ln w_it pit

= d ln w_it−

N

X

m=1

S_nmtd ln p_nmt,

d ln w_it pit

= d ln w_it−

N

X

m=1

S_nmt[ d ln τ_nmt+ d ln w_mt− (1 − µ) d ln χ_mt− d ln z_mt] , (B.41)

Migration Shares Totally differentiating the outmigration share in equation (B.4), we get:

dD_igt D_igt = 1

ρ

"

βEtdv_gt+1− dκ_git κ_git

−

N

X

h=1

D_iht

βEtdv_ht+1− dκ_hit κ_hit

#

, (B.42)

d ln D_igt = 1 ρ

"

(βEtdv_gt+1− d ln κ_git) −

N

X

h=1

D_iht(βEtdv_ht+1− d ln κ_hit)

# .

Goods Market Clearing Totally differentiating the goods market clearing condition (B.20), we have:

dw_it

w_it + d`_it

`_it =

N

X

n=1

S_nitw_nt`_nt w_it`_it

dw_nt

w_nt + d`_nt

`_nt + dS_nit S_nit

.

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Using our results for the derivatives of expenditure shares (B.38) and prices (B.39), we can rewrite this as:

dw_it

w_it + d`_it

`_it =

N

X

n=1

T_int dw_nt

w_nt + d`_nt

`_nt + θ X

h∈N

S_nhtdp_nht

p_nht − dp_nit p_nit

!!

,

T_int ≡ S_nitw_nt`_nt w_it`_it .

d ln w_it + d ln `_it

=







PN

n=1T_int( d ln w_nt+ d ln `_nt) +θPN

n=1

PN

m=1T_intS_nmt( d ln τ_nmt+ d ln w_mt− (1 − µ) d ln χ_mt− d ln z_mt)

−θPN

n=1T_int( d ln τ_nit+ d ln w_it− (1 − µ) d ln χit− d ln zit)





. (B.43)

Population Flow. Totally differentiating the population flow condition (B.26) we have:

d ln `_gt+1 =

N

X

i=1

E_git[ d ln `_it+ d ln D_igt] ,

d ln `gt+1=

N

X

i=1

Egit

"

d ln `it+1

ρ βE^tdvgt+1− d ln κgi−

N

X

m=1

Dimt(βE^tdvmt+1− d ln κmit)

!#

. (B.44)

Value Function. Note that the value function can be re-written using the following results:

v_it= ln w_it hPN

m=1p^−θ_imti−1/θ + ln b_it+ ρ ln

N

X

g=1

(exp (βEtv_gt+1) /κ_git)^1/ρ,

" _N X

m=1

p^−θ_imt

#^−1/θ

= p^−θ_iit S_iit

^−1/θ

, τiit = 1,

N

X

g=1

(exp (βEtv_gt+1) /κ_git)^1/ρ = (exp (βEtv_it+1) /κ_iit)^1/ρ

D_iit , κ_iit = 1, v_it = −1

θ ln S_iit+ ln w_it− ln p_iit+ ln b_it+ βEtv_it+1− ρ ln D_iit. (B.45) Totally differentiating this expression for the value function, we have:

dv_it= −1

θd ln S_iit+ d ln w_it− d ln p_iit+ d ln b_it+ βEtdv_it+1− ρ d ln D_iit, where

d ln Siit = −θ d ln piit+ θ

" _N X

m=1

Simtd ln pimt

# ,

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d ln D_iit = 1 ρ

"

βEtdv_it+1− d ln κ_iit−

N

X

m=1

D_imt(βEtv_mt+1− d ln κ_mit)

# . Using these results for d ln Siitand d ln Diit in the expression for dvitabove, we have:

dvit=

d ln wit−PN

m=1Simtd ln pimt

+ d ln b_it+PN

m=1D_imt(βEtdv_mt+1− d ln κ_mit)

,

where we have used d ln κiit = 0. Using the total derivative of the pricing rule (B.39), we can re-write this derivative of the value function as follows:

dvit=

"

d ln wit−PN

m=1Simt( d ln τnmt+ d ln wmt− (1 − µ) d ln χmt− d ln zmt) + d ln bit+PN

m=1Dimt(βE^tdvmt+1− d ln κmit)

#

. (B.46)

B.8.2 Steady-State Sufficient Statistics

Suppose that the economy starts from an initial steady-state with constant values of the endogenous variables: kit+1 = k_it= k^∗_i, `it+1 = `_it = `^∗_i, w_it+1^∗ = w_it^∗ = w^∗_i and v_it+1^∗ = v^∗_it = v_i^∗, where we use an asterisk to denote a steady-state value, and drop the time subscript for the remainder of this subsection, since we are concerned with steady-states. We consider small shocks to productivity ( d ln z) and amenities ( d ln b) in each location, holding constant the economy’s aggregate labor endowment ( d ln ` = 0), trade costs ( d ln τ = 0) and commuting costs ( d ln κ = 0).

Capital Accumulation. From the capital accumulation equation (B.24), the steady-state stock of capital solves:

(1 − β (1 − δ)) χ^∗_i = (1 − β (1 − δ))k^∗_i

`^∗_i = β1 − µ µ

w^∗_i p^∗_i . Totally differentiating, we have:

d ln χ^∗_i = d ln w^∗_i p^∗_i

.

Using the total derivative of real income (B.41) above, this becomes:

d ln χ^∗_i = d ln w^∗_i −

N

X

m=1

S_im^∗ [ d ln w^∗_m− (1 − µ) d ln χ^∗_m− d ln z_m] , where we have used and d ln τnm= 0. This relationship has the matrix representation:

d ln χ^∗ = d ln w^∗− S d ln w^∗+ (1 − µ) S d ln χ^∗+ S d ln z,

(I − (1 − µ) S) d ln χ^∗ = (I − S) d ln w^∗+ S d ln z. (B.47)

Online Appendix for Dynamic Spatial General Equilibrium (Not for Publication)