Introductory Theoretical Illustration

As a means of illustrating the paper’s intuition, consider the following stylised model. Suppose that an economy operates in continuous time, and is represented by N perfectly competitive firms, each using a Cobb-Douglas pro-

duction technology with labour and other firms’ products (intermediate goods) as inputs yi(t) = qi(t) ζi li(t)1−ν N Y j=1 yij(t)αij !ν (3.2) where yi is the i-th firm’s output, qi(t) is its productivity, li(t) and yij(t) are,

respectively, the amounts of labour and other firms’ products it employs in period t; ν _∈ [0; 1) is the total share of intermediate products in the i-th firm’s

sales, and αij ∈ [0; 1] is j-th firm’s share within that share; coefficient ζi ≡

(1₋ν)1−νQN

j=1(ναij)

tion, we simplify the exposition by assuming that all intermediate inputs are used with the same intensity, so that ∀i, j = 1_÷N, αij = α = _N1. Together all αijs

can be represented jointly by a technological matrixA_≡α

₁

1

1

is an N _×N matrix of ones.

In light of (3.2), and since the firms are perfect competitors, one can straightforwardly show that the vector of equilibrium prices’ natural logarithmsp∗(t)

is described by the matrix expression as follows12

p∗(t) = lnw(t)e_{N −}(E_{N −}νA)−1q(t) = lnw(t)e_{N −} E_N + ν 1−ν A q(t) (3.3) wheree_N andE_N are, respectively, theN_×1vector of ones and theN_×N identity

matrix; q(t) is the vector of the natural logarithms of firms’ productivities.

Assume that each firm’s product is consumed by the representative house- hold with symmetric Cobb-Douglas preferences

C(t) =N N Y i=1 ci(t) 1 N (3.4)

whereci(t)is the consumption of thei-th firm’s product. The expression is multi-

plied byN for normalisation purposes. The household comprisesL(t)individuals,

each supplying inelastically one unit of labour.

The household’s optimising behaviour entails that aggregate spending on consumption (which is equivalent to the economy’s GDP, given the absence of capital investment) amounts to P(t)C(t), where P(t) ≡ QN

i=1pi(t)

_N1

is the economy’s price index. In what follows, we use the household’s optimal con- sumption bundle as the numeraire, which entails that P(t) = 1 ∀t.

Combining the solution for the system of equilibrium prices (3.3) with the definition of the aggregate price index P(t)yields the following expression for the

natural logarithm of the equilibrium wage rate lnP∗(t) = 0 = lnw(t)₋ 1 Ne T N E_N + ν 1−νA q(t)_⇔ ⇔lnw(t) = 1 1₋ν N X i=1 lnqi(t) N (3.5)

where(_·)T _{is the transposition operator.}

As labour is the only primary production factor in the economy, total labour compensation constitutes the economy’s value added, so that w(t)L(t) is

the total output, andw(t) =y(t)is GDP per capita. When combined with (3.5),

this suggests the following expression for the economy’s growth rate at t

˙ (lnw(t)) = w˙(t) w(t) ≡g(t) = 1 1−ν N X i=1 gi(t) N (3.6)

where gi(t) ≡ q_q˙i_i(_(tt)₎ is the growth rate of firm i’s productivity. The term ₁₋1_ν

in the rightmost expression in (3.6) reflects the impact of the interconnection between firms on the growth rate. The key feature of the term is that it is strictly greater than one, and increases in ν. This implication comes directly from the

amplification effects inherent in production networks and encapsulated in the idea of Hulten’s multiplier introduced by Hulten (1978): an increase in firm i’s

productivity translates into that of its output, which leads to the expansion for all firms downstream from i.13 This in turn induces second- and higher-order

effects for all firms further downstream.14 _{Thereby, the resulting overall growth}

in the economy exceeds the productivity growth rate of any single industry. This result is further generalised in Section 3.3.2.

13_{Potentially, a firm’s higher productivity can have an impact on its upstream counteragents}

as well, but this effect is reflected only when the elasticity of substitution between a firm’s inputs is different from unity, i.e. when production technology (3.2) is CES instead of Cobb-Douglas – see Baqaee (2016) for a discussion.

14_{The same mechanism is explored by Acemoglu et al. (2012) and Acemoglu et al. (2017)}

in the context of micro-shock propagation, and by Jones (2011) for the amplification of misallocation-driven distortions.

Suppose that attall firms’ productivities are characterised by vectorq(t).

During a period starting at t, an episode of technological growth occurs, within

which the i0-th firm sees an increase in its productivity at the rate of gi0(t) =η. Given (3.6), the growth rate of the economy’s output per capita g(t) during the

episode is

g(t) =g = 1 1−ν

η

N (3.7)

Note that in the case when intersectoral linkage is absent in the economy – that is, when ν = 0 (so that industries use only primary production factors represented

by labour) – expression (3.6) transforms as follows

ˆ g(t) = ˆg = η N ⇔g = ˆ g 1−ν (3.8)

Comparing expressions (3.7) and (3.8) confirms the intuition of (3.6): the pres- ence of interlinkage serves as an amplifier of economic growth, as the growth rate of an economy with interconnections g always exceeds that of its counterpart

without a linkage gˆ. In the next section, we explore the amplifying properties of

production networks in the context of a richer framework by introducing endogen- ous productivity choices and generalising the structure of intersectoral production networks.

3.3.2 Extended Theoretical Framework