Co-authored with Chase Coleman

In this lecture, we review the paper Globalization and Synchronization of Innovation Cycles by Kiminori Matsuyama, Laura Gardini and Iryna Sushko.

This model helps us understand several interesting stylized facts about the world economy.

One of these is synchronized business cycles across different countries.

Most existing models that generate synchronized business cycles do so by assumption, since they tie output in each country to a common shock.

They also fail to explain certain features of the data, such as the fact that the degree of synchronization tends to increase with trade ties.

By contrast, in the model we consider in this lecture, synchronization is both endogenous and increasing with the extent of trade integration.

In particular, as trade costs fall and international competition increases, innovation incentives become aligned and countries synchronize their innovation cycles.

The model builds on work by Judd [Jud85], Deneckner and Judd [DJ92] and Helpman and Krugman [HK85] by developing a two country model with trade and innovation.

On the technical side, the paper introduces the concept of coupled oscillators to economic modeling.

As we will see, coupled oscillators arise endogenously within the model.

Below we review the model and replicate some of the results on synchronization of innovation across countries.

It is helpful to begin with an overview of the mechanism.

As discussed above, two countries produce and trade with each other.

In each country, firms innovate, producing new varieties of goods and, in doing so, receiving temporary monopoly power.

Imitators follow and, after one period of monopoly, what had previously been new varieties now enter competitive production.

Firms have incentives to innovate and produce new goods when the mass of varieties of goods currently in production is relatively low.

In addition, there are strategic complementarities in the timing of innovation.

Firms have incentives to innovate in the same period, so as to avoid competing with substitutes that are competitively produced.

This leads to temporal clustering in innovations in each country.

After a burst of innovation, the mass of goods currently in production increases.

However, goods also become obsolete, so that not all survive from period to period.

This mechanism generates a cycle, where the mass of varieties increases through simultaneous innovation and then falls through obsolescence.

In the absence of trade, the timing of innovation cycles in each country is decoupled.

This will be the case when trade costs are prohibitively high.

If trade costs fall, then goods produced in each country penetrate each other’s markets.

As illustrated below, this leads to synchonization of business cycles across the two countries.

Let’s write down the model more formally.

(The treatment is relatively terse since full details can be found in the original paper)

Time is discrete with $ t = 0, 1, \dots $.

There are two countries indexed by $ j $ or $ k $.

In each country, a representative household inelastically supplies $ L_j $ units of labor at wage rate $ w_{j, t} $.

Without loss of generality, it is assumed that $ L_{1} \geq L_{2} $.

Households consume a single nontradeable final good which is produced competitively.

Its production involves combining two types of tradeable intermediate inputs via

$$ Y_{k, t} = C_{k, t} = \left( \frac{X^o_{k, t}}{1 - \alpha} \right)^{1-\alpha} \left( \frac{X_{k, t}}{\alpha} \right)^{\alpha} $$Here $ X^o_{k, t} $ is a homogeneous input which can be produced from labor using a linear, one-for-one technology.

It is freely tradeable, competitively supplied, and homogeneous across countries.

By choosing the price of this good as numeraire and assuming both countries find it optimal to always produce the homogeneous good, we can set $ w_{1, t} = w_{2, t} = 1 $.

The good $ X_{k, t} $ is a composite, built from many differentiated goods via

$$ X_{k, t}^{1 - \frac{1}{\sigma}} = \int_{\Omega_t} \left[ x_{k, t}(\nu) \right]^{1 - \frac{1}{\sigma}} d \nu $$Here $ x_{k, t}(\nu) $ is the total amount of a differentiated good $ \nu \in \Omega_t $ that is produced.

The parameter $ \sigma > 1 $ is the direct partial elasticity of substitution between a pair of varieties and $ \Omega_t $ is the set of varieties available in period $ t $.

We can split the varieties into those which are supplied competitively and those supplied monopolistically; that is, $ \Omega_t = \Omega_t^c + \Omega_t^m $.

Demand for differentiated inputs is

$$ x_{k, t}(\nu) = \left( \frac{p_{k, t}(\nu)}{P_{k, t}} \right)^{-\sigma} \frac{\alpha L_k}{P_{k, t}} $$Here

- $ p_{k, t}(\nu) $ is the price of the variety $ \nu $ and
- $ P_{k, t} $ is the price index for differentiated inputs in $ k $, defined by

The price of a variety also depends on the origin, $ j $, and destination, $ k $, of the goods because shipping varieties between countries incurs an iceberg trade cost $ \tau_{j,k} $.

Thus the effective price in country $ k $ of a variety $ \nu $ produced in country $ j $ becomes $ p_{k, t}(\nu) = \tau_{j,k} \, p_{j, t}(\nu) $.

Using these expressions, we can derive the total demand for each variety, which is

$$ D_{j, t}(\nu) = \sum_k \tau_{j, k} x_{k, t}(\nu) = \alpha A_{j, t}(p_{j, t}(\nu))^{-\sigma} $$where

$$ A_{j, t} := \sum_k \frac{\rho_{j, k} L_{k}}{(P_{k, t})^{1 - \sigma}} \quad \text{and} \quad \rho_{j, k} = (\tau_{j, k})^{1 - \sigma} \leq 1 $$It is assumed that $ \tau_{1,1} = \tau_{2,2} = 1 $ and $ \tau_{1,2} = \tau_{2,1} = \tau $ for some $ \tau > 1 $, so that

$$ \rho_{1,2} = \rho_{2,1} = \rho := \tau^{1 - \sigma} < 1 $$The value $ \rho \in [0, 1) $ is a proxy for the degree of globalization.

Producing one unit of each differentiated variety requires $ \psi $ units of labor, so the marginal cost is equal to $ \psi $ for $ \nu \in \Omega_{j, t} $.

Additionally, all competitive varieties will have the same price (because of equal marginal cost), which means that, for all $ \nu \in \Omega^c $,

$$ p_{j, t}(\nu) = p_{j, t}^c := \psi \quad \text{and} \quad D_{j, t} = y_{j, t}^c := \alpha A_{j, t} (p_{j, t}^c)^{-\sigma} $$Monopolists will have the same marked-up price, so, for all $ \nu \in \Omega^m $ ,

$$ p_{j, t}(\nu) = p_{j, t}^m := \frac{\psi }{1 - \frac{1}{\sigma}} \quad \text{and} \quad D_{j, t} = y_{j, t}^m := \alpha A_{j, t} (p_{j, t}^m)^{-\sigma} $$Define

$$ \theta := \frac{p_{j, t}^c}{p_{j, t}^m} \frac{y_{j, t}^c}{y_{j, t}^m} = \left(1 - \frac{1}{\sigma} \right)^{1-\sigma} $$Using the preceding definitions and some algebra, the price indices can now be rewritten as

$$ \left(\frac{P_{k,t}}{\psi}\right)^{1-\sigma} = M_{k,t} + \rho M_{j,t} \quad \text{where} \quad M_{j,t} := N_{j,t}^c + \frac{N_{j,t}^m}{ \theta} $$The symbols $ N_{j, t}^c $ and $ N_{j, t}^m $ will denote the measures of $ \Omega^c $ and $ \Omega^m $ respectively.

To introduce a new variety, a firm must hire $ f $ units of labor per variety in each country.

Monopolist profits must be less than or equal to zero in expectation, so

$$ N_{j,t}^m \geq 0, \quad \pi_{j, t}^m := (p_{j, t}^m - \psi) y_{j, t}^m - f \leq 0 \quad \text{and} \quad \pi_{j, t}^m N_{j,t}^m = 0 $$With further manipulations, this becomes

$$ N_{j,t}^m = \theta(M_{j,t} - N_{j,t}^c) \geq 0, \quad \frac{1}{\sigma} \left[ \frac{\alpha L_j}{\theta(M_{j,t} + \rho M_{k,t})} + \frac{\alpha L_k}{\theta(M_{j,t} + M_{k,t} / \rho)} \right] \leq f $$With $ \delta $ as the exogenous probability of a variety becoming obsolete, the dynamic equation for the measure of firms becomes

$$ N_{j, t+1}^c = \delta (N_{j, t}^c + N_{j, t}^m) = \delta (N_{j, t}^c + \theta(M_{j, t} - N_{j, t}^c)) $$We will work with a normalized measure of varieties

$$ n_{j, t} := \frac{\theta \sigma f N_{j, t}^c}{\alpha (L_1 + L_2)}, \quad i_{j, t} := \frac{\theta \sigma f N_{j, t}^m}{\alpha (L_1 + L_2)}, \quad m_{j, t} := \frac{\theta \sigma f M_{j, t}}{\alpha (L_1 + L_2)} = n_{j, t} + \frac{i_{j, t}}{\theta} $$We also use $ s_j := \frac{L_j}{L_1 + L_2} $ to be the share of labor employed in country $ j $.

We can use these definitions and the preceding expressions to obtain a law of motion for $ n_t := (n_{1, t}, n_{2, t}) $.

In particular, given an initial condition, $ n_0 = (n_{1, 0}, n_{2, 0}) \in \mathbb{R}_{+}^{2} $, the equilibrium trajectory, $ \{ n_t \}_{t=0}^{\infty} = \{ (n_{1, t}, n_{2, t}) \}_{t=0}^{\infty} $, is obtained by iterating on $ n_{t+1} = F(n_t) $ where $ F : \mathbb{R}_{+}^{2} \rightarrow \mathbb{R}_{+}^{2} $ is given by

$$ \begin{aligned} F(n_t) &= \begin{cases} \big( \delta (\theta s_1(\rho) + (1-\theta) n_{1, t}), \delta (\theta s_2(\rho) + (1-\theta) n_{2, t}) \big) \; & \text{for } n_t \in D_{LL} \\ \big( \delta n_{1, t}, \delta n_{2, t} \big) \; &\text{for } n_t \in D_{HH} \\ \big( \delta n_{1, t}, \delta (\theta h_2(n_{1, t}) + (1-\theta) n_{2, t}) \big) &\text{for } n_t \in D_{HL} \\ \big( \delta (\theta h_1(n_{2, t}) + (1-\theta) n_{1, t}, \delta n_{2, t}) \big) &\text{for } n_t \in D_{LH} \end{cases} \end{aligned} $$Here

$$ \begin{aligned} D_{LL} & := \{ (n_1, n_2) \in \mathbb{R}_{+}^{2} | n_j \leq s_j(\rho) \} \\ D_{HH} & := \{ (n_1, n_2) \in \mathbb{R}_{+}^{2} | n_j \geq h_j(\rho) \} \\ D_{HL} & := \{ (n_1, n_2) \in \mathbb{R}_{+}^{2} | n_1 \geq s_1(\rho) \text{ and } n_2 \leq h_2(n_1) \} \\ D_{LH} & := \{ (n_1, n_2) \in \mathbb{R}_{+}^{2} | n_1 \leq h_1(n_2) \text{ and } n_2 \geq s_2(\rho) \} \end{aligned} $$while

$$ s_1(\rho) = 1 - s_2(\rho) = \min \left\{ \frac{s_1 - \rho s_2}{1 - \rho}, 1 \right\} $$and $ h_j(n_k) $ is defined implicitly by the equation

$$ 1 = \frac{s_j}{h_j(n_k) + \rho n_k} + \frac{s_k}{h_j(n_k) + n_k / \rho} $$Rewriting the equation above gives us a quadratic equation in terms of $ h_j(n_k) $.

Since we know $ h_j(n_k) > 0 $ then we can just solve the quadratic equation and return the positive root.

This gives us

$$ h_j(n_k)^2 + \left( (\rho + \frac{1}{\rho}) n_k - s_j - s_k \right) h_j(n_k) + (n_k^2 - \frac{s_j n_k}{\rho} - s_k n_k \rho) = 0 $$Let’s try simulating some of these trajectories.

We will focus in particular on whether or not innovation cycles synchronize across the two countries.

As we will see, this depends on initial conditions.

For some parameterizations, synchronization will occur for “most” initial conditions, while for others synchronization will be rare.

Here’s the main body of code.

In [1]:

```
using InstantiateFromURL
# optionally add arguments to force installation: instantiate = true, precompile = true
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.8.0")
```

In [2]:

```
using LinearAlgebra, Statistics
using Plots, Parameters
gr(fmt = :png);
```

In [3]:

```
function h_j(j, nk, s1, s2, θ, δ, ρ)
# Find out who's h we are evaluating
if j == 1
sj = s1
sk = s2
else
sj = s2
sk = s1
end
# Coefficients on the quadratic a x^2 + b x + c = 0
a = 1.0
b = ((ρ + 1 / ρ) * nk - sj - sk)
c = (nk * nk - (sj * nk) / ρ - sk * ρ * nk)
# Positive solution of quadratic form
root = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)
return root
end
DLL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =
(n1 ≤ s1_ρ) && (n2 ≤ s2_ρ)
DHH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =
(n1 ≥ h_j(1, n2, s1, s2, θ, δ, ρ)) && (n2 ≥ h_j(2, n1, s1, s2, θ, δ, ρ))
DHL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =
(n1 ≥ s1_ρ) && (n2 ≤ h_j(2, n1, s1, s2, θ, δ, ρ))
DLH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =
(n1 ≤ h_j(1, n2, s1, s2, θ, δ, ρ)) && (n2 ≥ s2_ρ)
function one_step(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
# Depending on where we are, evaluate the right branch
if DLL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
n1_tp1 = δ * (θ * s1_ρ + (1 - θ) * n1)
n2_tp1 = δ * (θ * s2_ρ + (1 - θ) * n2)
elseif DHH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
n1_tp1 = δ * n1
n2_tp1 = δ * n2
elseif DHL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
n1_tp1 = δ * n1
n2_tp1 = δ * (θ * h_j(2, n1, s1, s2, θ, δ, ρ) + (1 - θ) * n2)
elseif DLH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
n1_tp1 = δ * (θ * h_j(1, n2, s1, s2, θ, δ, ρ) + (1 - θ) * n1)
n2_tp1 = δ * n2
end
return n1_tp1, n2_tp1
end
new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =
one_step(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
function pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ,
maxiter, npers)
# Initialize the status of synchronization
synchronized = false
pers_2_sync = maxiter
iters = 0
nsync = 0
while (~synchronized) && (iters < maxiter)
# Increment the number of iterations and get next values
iters += 1
n1_t, n2_t = new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
# Check whether same in this period
if abs(n1_t - n2_t) < 1e-8
nsync += 1
# If not, then reset the nsync counter
else
nsync = 0
end
# If we have been in sync for npers then stop and countries
# became synchronized nsync periods ago
if nsync > npers
synchronized = true
pers_2_sync = iters - nsync
end
n1_0, n2_0 = n1_t, n2_t
end
return synchronized, pers_2_sync
end
function create_attraction_basis(s1_ρ, s2_ρ, s1, s2, θ, δ, ρ,
maxiter, npers, npts)
# Create unit range with npts
synchronized, pers_2_sync = false, 0
unit_range = range(0.0, 1.0, length = npts)
# Allocate space to store time to sync
time_2_sync = zeros(npts, npts)
# Iterate over initial conditions
for (i, n1_0) in enumerate(unit_range)
for (j, n2_0) in enumerate(unit_range)
synchronized, pers_2_sync = pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ,
s1, s2, θ, δ, ρ,
maxiter, npers)
time_2_sync[i, j] = pers_2_sync
end
end
return time_2_sync
end
# model
function MSGSync(s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2)
# Store other cutoffs and parameters we use
s2 = 1 - s1
s1_ρ = min((s1 - ρ * s2) / (1 - ρ), 1)
s2_ρ = 1 - s1_ρ
return (s1 = s1, s2 = s2, s1_ρ = s1_ρ, s2_ρ = s2_ρ, θ = θ, δ = δ, ρ = ρ)
end
function simulate_n(model, n1_0, n2_0, T)
# Unpack parameters
@unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model
# Allocate space
n1 = zeros(T)
n2 = zeros(T)
# Simulate for T periods
for t in 1:T
# Get next values
n1[t], n2[t] = n1_0, n2_0
n1_0, n2_0 = new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)
end
return n1, n2
end
function pers_till_sync(model, n1_0, n2_0,
maxiter = 500, npers = 3)
# Unpack parameters
@unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model
return pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2,
θ, δ, ρ, maxiter, npers)
end
function create_attraction_basis(model;
maxiter = 250,
npers = 3,
npts = 50)
# Unpack parameters
@unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model
ab = create_attraction_basis(s1_ρ, s2_ρ, s1, s2, θ, δ,
ρ, maxiter, npers, npts)
return ab
end
```

Out[3]:

We write a short function below that exploits the preceding code and plots two time series.

Each time series gives the dynamics for the two countries.

The time series share parameters but differ in their initial condition.

Here’s the function

In [4]:

```
function plot_timeseries(n1_0, n2_0, s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2)
model = MSGSync(s1, θ, δ, ρ)
n1, n2 = simulate_n(model, n1_0, n2_0, 25)
return [n1 n2]
end
# Create figures
data_ns = plot_timeseries(0.15, 0.35)
data_s = plot_timeseries(0.4, 0.3)
plot(data_ns, title = "Not Synchronized", legend = false)
```

Out[4]:

In [5]:

```
plot(data_s, title = "Synchronized", legend = false)
```

Out[5]:

In the first case, innovation in the two countries does not synchronize.

In the second case different initial conditions are chosen, and the cycles become synchronized.

Next let’s study the initial conditions that lead to synchronized cycles more systematically.

We generate time series from a large collection of different initial conditions and mark those conditions with different colors according to whether synchronization occurs or not.

The next display shows exactly this for four different parameterizations (one for each subfigure).

Dark colors indicate synchronization, while light colors indicate failure to synchronize.

As you can see, larger values of $ \rho $ translate to more synchronization.

You are asked to replicate this figure in the exercises.

Replicate the figure shown above by coloring initial conditions according to whether or not synchronization occurs from those conditions.

In [6]:

```
function plot_attraction_basis(s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2; npts = 250)
# Create attraction basis
unitrange = range(0, 1, length = npts)
model = MSGSync(s1, θ, δ, ρ)
ab = create_attraction_basis(model,npts=npts)
plt = Plots.heatmap(ab, legend = false)
end
```

Out[6]:

In [7]:

```
params = [[0.5, 2.5, 0.7, 0.2],
[0.5, 2.5, 0.7, 0.4],
[0.5, 2.5, 0.7, 0.6],
[0.5, 2.5, 0.7, 0.8]]
plots = (plot_attraction_basis(p...) for p in params)
plot(plots..., size = (1000, 1000))
```

Out[7]: