Exam 1 on:

Diversification through Trade

Francesco Caselli, Miklós Koren, Milan Lisicky & Silvana Tenreyro
The Quarterly Journal of Economics 135(1), 2020, 449-502
Set 24 September 2026 · 100 points
Reading note · PDF
Fable 5.1 (adjudication), GPT 6-Astra (verification), Opus 5.5 (setting the exam), GPT 6-Sol (setting the exam)

Table of Contents

1. The common wisdom and the premise it hides
2. Diversification in a one-good world
3. When partners’ shocks move together
4. From trade flows to shocks
5. Specialization, hedging and the residual that carries the covariance
6. Wired in series: input–output links and connector exposure
References

6 questions, 23 parts, 100 points. Focus: Connector economies in us–China fragmentation: the paper’s variance decomposition and portfolio logic (country vs sector shocks, covariance terms, input–output wiring, how the shock decomposition and counterfactuals are built and identified), stress-tested against correlated partner shocks, a common geopolitical factor and in-series supply-plus-demand exposure, and what the framework implies for a naive partner hhi versus effective diversification.. Hints are folded; open them only when stuck. Answer in any form, all at once or a few parts at a time, and hand the answers to /grader caselli2020DiversificationThroughTrade 1.

1. The common wisdom and the premise it hides

The paper sets itself against a long-standing view that openness to trade raises income volatility, and against a neighbouring macro literature on open-economy business cycles. Throughout, income volatility means the variance of detrended log real income, where real income is nominal value added deflated by a consumer (expenditure-based) price index.

a) The specialization view and its premise

a) · 3 points

State the common-wisdom argument that openness raises income volatility and whom the paper traces it to. Then name the empirical premise about the sources of shocks on which the argument rests, and the evidence the paper cites against that premise.

Hint

The argument runs through what trade does to the composition of production.

b) Contrast with international RBC

b) · 3 points

In the international real business cycle model of Backus, Kehoe, and Kydland (1992), gdp is more volatile in an open than in a closed economy. In two or three sentences, give their mechanism and say what feature of this paper’s environment reverses the sign.

Hint

Ask what crosses the border after a country gets a good or bad productivity draw.

c) Why deflate by consumer prices

c) · 3 points

The paper measures real income as nominal value added deflated by an expenditure-based (consumer) price index, not by the gdp deflator. Explain why this choice governs whether the diversification channel shows up in measured income at all, citing the benchmark result the paper invokes.

Hint

Where, in a Ricardian model, does a foreign productivity shock show up for a home household?

2. Diversification in a one-good world

Take the one-sector version of the model (Section II.C): an Eaton and Kortum (2002) economy in which each country $n$ produces varieties of a single good with equipped labour and the good itself as an intermediate, value-added share $\beta$ , Fréchet shape $\theta$ . Define augmented productivity (eq. 13, one sector)

Z_{nt}\equiv T_n\,A_{nt}^{\theta}\,L_{nt}^{\beta\theta},

where $T_n$ is the Fréchet location parameter, $A_{nt}$ the productivity shock and $L_{nt}$ equipped labour. Real income is $Y_{nt}=w_{nt}L_{nt}/P_{nt}$ . Hats denote log deviations from a deterministic trend.

Autarky ( $\kappa_{nm}=0$ for $n\neq m$ ): $Y_{nt}\propto Z_{nt}^{1/(\beta\theta)}$ , so $\operatorname{Var}(\hat Y_{nt})=\frac{1}{(\beta\theta)^2}\operatorname{Var}(\hat Z_{nt})$ (p. 462).
Costless trade ( $\kappa_{nm}=1$ for all $n,m$ ), derived in Appendix A.1:

Y_{nt}=(\xi B)^{1/\beta}\,Z_{nt}^{\frac{1}{1+\beta\theta}}\left(\sum_{m=1}^{N}Z_{mt}^{\frac{1}{1+\beta\theta}}\right)^{\frac{1}{\beta\theta}},

with $\xi,B$ constants. Let $\gamma_m\equiv \bar Z_m^{1/(1+\beta\theta)}\big/\sum_{i=1}^N\bar Z_i^{1/(1+\beta\theta)}$ be country $m$ ’s relative size evaluated at trend, so $\sum_m\gamma_m=1$ .

a) Log-linearising free-trade income

a) · 5 points

Log-linearise the costless-trade expression around trend to show that

\hat Y_{nt}=\frac{1}{\beta\theta}\Big[a_n\,\hat Z_{nt}+b\sum_{m\neq n}\gamma_m\hat Z_{mt}\Big]

and find $a_n$ and $b$ . Then show that the loadings inside the bracket, $a_n$ and $b\gamma_m$ for $m\neq n$ , sum to one.

Hint

Differentiate

\log\sum_m Z_m^{1/(1+\beta\theta)}

with respect to

\log Z_m

at trend: the derivative is a share.

b) Why partners’ luck is yours

b) · 3 points

In two or three sentences: (i) why does a positive productivity shock in a foreign country raise home real income under free trade, and (ii) why does a smaller country (lower

\gamma_n

) keep less of its own shock?

Hint

Real income is deflated by the price of what the home country buys.

c) Free trade beats autarky

c) · 5 points

Suppose all countries’ shocks have the same variance

\sigma

and are mutually uncorrelated. Derive

\operatorname{Var}(\hat Y_{nt})

under costless trade, prove that it is strictly below the autarky variance whenever

\gamma_n<1

, and write the ratio (free trade over autarky) as a Herfindahl index of the weights from part a).

Hint 1

With uncorrelated shocks only squared loadings survive.

Hint 2

You need

\sum_m\gamma_m^2\le 1

and

\gamma_n<1

d) How beta-theta governs the gain

d) · 6 points

Keep the assumptions of part c), treat the size weights

\gamma_m

as fixed, and write

x\equiv\beta\theta

and

S\equiv\sum_{m\neq n}\gamma_m^2

. (i) Find the limits of the variance ratio

R_n

x\to0

and

x\to\infty

. (ii) Sign

\partial R_n/\partial x

, giving a sufficient condition for it to be positive. (iii) Use the result to rationalise the direction of two robustness exercises:

\theta=2

versus

\theta=8

(Table IV) and the model with no input–output linkages,

\gamma^{kj}=0

and

\beta^j=1

(Table VII).

Hint 1

Differentiate

a_n^2+b^2S

with

a_n=(\gamma_n+x)/(1+x)

b=1/(1+x)

Hint 2

S\le(1-\gamma_n)^2

3. When partners’ shocks move together

Stay in the one-sector, costless-trade economy of Question 2, where

\beta\theta\,\hat Y_{nt}=\sum_{i=1}^N\omega_i\hat Z_{it},\qquad \omega_n=\frac{\gamma_n+\beta\theta}{1+\beta\theta},\quad \omega_m=\frac{\gamma_m}{1+\beta\theta}\ (m\neq n),\quad \sum_i\omega_i=1.

Let $\Sigma$ be the $N\times N$ covariance matrix of $(\hat Z_{1t},\dots,\hat Z_{Nt})$ and $H\equiv\sum_i\omega_i^2$ . The paper’s general free-trade formula (p. 462) reads, in the $\omega$ notation,

\operatorname{Var}(\hat Y_{nt})=\Big(\frac{1}{\beta\theta}\Big)^2\Big\{\omega_n^2\operatorname{Var}(\hat Z_{nt})+\sum_{m\neq n}\omega_m^2\operatorname{Var}(\hat Z_{mt})+2\,\omega_n\sum_{m\neq n}\omega_m\operatorname{Cov}(\hat Z_{mt},\hat Z_{nt})\Big\}.

The paper’s own caveat is that diversification wins only if partners’ shocks are not too large in variance or covariance (p. 450). Parts b) and c) take that caveat to a us–China fragmentation setting. Their answers are derivable from the model, but they are not in the paper.

a) The portfolio variance in full

a) · 5 points

Write

\operatorname{Var}(\hat Y_{nt})

for a general

\Sigma

as a quadratic form in

\omega

, and expand it into own-variance, home–partner and partner–partner terms. Compare with the displayed formula: which terms does it leave out, and under what assumption on

\Sigma

is the omission harmless?

Hint

Count the cross-products

\omega_i\omega_k\operatorname{Cov}(\hat Z_i,\hat Z_k)

for

i\neq k

by type.

b) A common factor cannot be traded away

b) · 6 points

Suppose every country, home included, loads one-for-one on a common factor:

\hat Z_{it}=f_t+e_{it}

, with

\operatorname{Var}(f)=\sigma_f^2

e_{it}

i.i.d. with variance

\sigma_e^2

, and

f\perp e

. Derive

\operatorname{Var}(\hat Y_{nt})

under autarky and under costless trade, and state the lower bound on free-trade volatility however diversified the weights become. Then say by how much the displayed formula of the setup would understate the free-trade variance in this case.

Hint 1

Use

\sum_i\omega_i=1

from Question 2a.

Hint 2

The factor enters with total loading

\sum_i\omega_i

c) More partners, more correlation

c) · 5 points

Now let all

N

shocks have variance

\sigma

and pairwise correlation

\rho\ge0

. Show that

\operatorname{Var}(\hat Y_{nt})=\frac{\sigma}{(\beta\theta)^2}\big[\rho+(1-\rho)H\big]

. Then consider a fragmentation episode that moves a connector economy from

(H_0,\rho_0)=(0.30,0.10)

(H_1,\rho_1)=(0.20,0.30)

: its effective exposure is more spread out, but shocks have become more correlated. Take this as a reduced-form change in the weights and in

\Sigma

. Does volatility rise or fall, and by how much? Give the general condition for a rise.

Hint

\big(\sum_i\omega_i\big)^2=1=H+\sum_{i\neq k}\omega_i\omega_k

4. From trade flows to shocks

Return to the full model: $N$ countries, $J$ sectors. The share of country $n$ ’s spending on sector- $j$ goods that goes to country $m$ is (eq. 6)

d^j_{nmt}=\frac{T^j_m\big(c^j_{mt}/(A^j_{mt}\kappa^j_{nmt})\big)^{-\theta}}{\sum_{i}T^j_i\big(c^j_{it}/(A^j_{it}\kappa^j_{nit})\big)^{-\theta}},\qquad c^j_{mt}\equiv B^j(w^j_{mt})^{\beta^j}\prod_k(P^k_{mt})^{\gamma^{kj}},

where $\kappa^j_{nmt}\in(0,1]$ is the fraction of a shipment from $m$ that arrives in $n$ (p. 456), $\kappa^j_{nnt}=1$ , and $\gamma^{kj}$ is the cost share of sector- $k$ inputs in sector $j$ . Augmented productivity is $Z^j_{nt}\equiv T^j_n(A^j_{nt})^\theta(L_{nt})^{\beta^j\theta}$ (eq. 13). In tradeable sectors it is recovered bilaterally from eq. (15), which in logs reads

\log Z^j_{nt}=\zeta^j_{mnt}-\theta\log P^j_{mt},\qquad \zeta^j_{mnt}=\log d^j_{mnt}-\theta\log\kappa^j_{mnt}+\theta\beta^j\log(w^j_{nt}L_{nt})+\theta\sum_k\gamma^{kj}\log P^k_{nt}+\text{const},

and is averaged over partners: $\log Z^j_{nt}=\frac1N\sum_m\big(\zeta^j_{mnt}-\theta\log P^j_{mt}\big)$ (p. 468). After band-pass detrending, the cyclical part is decomposed (eq. 16) as

\hat Z^j_{nt}=\lambda^j_t+\mu_{nt}+\epsilon^j_{nt},\quad \lambda^j_t=\frac1N\sum_n\hat Z^j_{nt},\quad \mu_{nt}=\sum_j\bar\alpha^j\big(\hat Z^j_{nt}-\lambda^j_t\big),\quad \epsilon^j_{nt}=\hat Z^j_{nt}-\lambda^j_t-\mu_{nt},

with $\bar\alpha^j$ the time-average expenditure shares, $\sum_j\bar\alpha^j=1$ . This is the “weighted average of shocks” described on p. 470; the printed fn. 19 carries an extra factor $J^{-1}$ in front of $\mu_{nt}$ , which would make $\mu$ one $J$ -th of that average. Take the weighted average as written here; part c) notes what the $J^{-1}$ would change.

In the counterfactuals, the baseline feeds in the historical paths of $\kappa$ and $Z$ . The comparison run holds every $\kappa$ at its 1972 level. The “diversification” runs set $\lambda$ and $\epsilon$ to zero.

a) Backing out trade costs

a) · 4 points

Derive eq. (14),

\dfrac{d^j_{nmt}d^j_{mnt}}{d^j_{mmt}d^j_{nnt}}=(\kappa^j_{nmt})^{2\theta}

, from the gravity equation, stating which assumptions you use and which observables enter. What does the left-hand side identify if you drop the symmetry assumption?

Hint

Form

d_{nm}/d_{nn}

: the denominator of the share cancels.

b) Where a one-sided tariff goes

b) · 5 points

Suppose that in some year the United States raises a tariff on imports from China in sector

j

, so that

\log\kappa^j_{US,CH}

falls by

\Delta>0

while

\kappa^j_{CH,US}

is unchanged. Treat the observed shares, wages and prices as data. Trace how the paper’s procedure books this change: what the estimated symmetric

\hat\kappa

records, and the resulting error in the inferred

\log Z^j

of China and of the United States through the bilateral step of eq. (15). What does this imply for the counterfactual that holds

\kappa

at 1972 levels?

Hint 1

\log\hat\kappa=\tfrac12(\log\kappa_{US,CH}+\log\kappa_{CH,US})

Hint 2

The error in

\log Z_n

from partner

m

-\theta(\log\hat\kappa_{mn}-\log\kappa_{mn})

c) Where a common factor lands

c) · 5 points

(i) Show that the restriction

\sum_n\mu_{nt}=0

holds by construction. (ii) Add a global shock

g_t

to every

\hat Z^j_{nt}

and say which of

\lambda,\mu,\epsilon

absorbs it. (iii) Now let

g_t

hit only a bloc

S

|S|

countries, for example the United States, China and a connector, with loading one in all sectors. Show how it splits between

\lambda

and

\mu

, inside and outside the bloc. What does this mean for what column (2) of Table I calls “diversification”?

Hint

\lambda^j_t

is a cross-country mean.

d) What is assumed about correlations

d) · 4 points

Which features of the cross-country covariance of shocks does the quantitative exercise take from the data, and which does it assume? Answer under three headings: the volatility computation itself, the ex ante labour allocation, and what the counterfactual holds fixed when trade costs change. End with one sentence on what this means for reading Table I as the effect of a us–China policy shock.

Hint

Ask what is fed into the model in each scenario and what the agent must forecast.

5. Specialization, hedging and the residual that carries the covariance

Equipped labour $L_{nt}$ is allocated across sectors, $L^j_{nt}$ with $\sum_jL^j_{nt}=L_{nt}$ , before the period’s shocks are realised, and cannot move across sectors afterwards; input sourcing happens after the shocks (pp. 455–458). Utility is $\log C_{nt}$ with budget $P_{nt}C_{nt}=\sum_jw^j_{nt}L^j_{nt}$ (eq. 4). Table I reports, for each country:

(1) the percentage change in the variance of detrended log real income from the observed fall in trade costs relative to trade costs held at 1972 levels, with all shocks;
(2) the same comparison with the sectoral shocks $\lambda$ and $\epsilon$ removed;
(3) = (1) − (2), which the notes to the table call the “difference between Columns (2) and (1)”.

Averages: −36.1%, −41.4% and +5.3%.

Appendix Table A.5 gives, for Mexico, the variances:

run	variance
baseline, all shocks	0.001050
baseline, no sectoral shocks	0.005229
1972 trade costs, all shocks	0.002430
1972 trade costs, no sectoral shocks	0.007485

Table I’s Mexico row reads −56.8%, −92.9%, 36.1%.

For parts d) and e), write detrended log income in any run as $\hat Y=\hat Y^{\mu}+\hat Y^{s}$ , where $\hat Y^{\mu}$ is the path of the run with country shocks only and $\hat Y^{s}\equiv\hat Y-\hat Y^{\mu}$ is defined as everything the sectoral shocks add. The identity $\operatorname{Var}(\hat Y)=\operatorname{Var}(\hat Y^\mu)+\operatorname{Var}(\hat Y^s)+2\operatorname{Cov}(\hat Y^\mu,\hat Y^s)$ is then exact, whatever the model’s non-linearities.

a) Timing creates both channels

a) · 3 points

In two or three sentences, explain how the timing of the sectoral labour allocation and of supplier choice is what lets the model contain both the specialization channel and the diversification channel, and what would be left if labour could be reallocated across sectors ex post.

Hint

One decision is taken before the draw, one after.

b) The ex ante labour allocation

b) · 4 points

Derive the first-order condition (12),

\dfrac{L^j_{nt}}{L_{nt}}=E_{t-1}\Big[\dfrac{w^j_{nt}L^j_{nt}}{\sum_kw^k_{nt}L^k_{nt}}\Big]

, from

\max_{\{L^j_{nt}\}}E_{t-1}\log\big(\sum_jw^j_{nt}L^j_{nt}/P_{nt}\big)

subject to

\sum_jL^j_{nt}=L_{nt}

, with the representative agent taking wages and prices as given.

Hint

Multiply each first-order condition by

L^j_{nt}

and sum over

j

to pin down the multiplier.

c) Why specialization can calm

c) · 3 points

Column (3) of Table I is positive for only 13 of 25 countries. In three or four sentences, explain how the model allows trade-induced specialization to lower volatility, and what role the ex ante allocation rule plays.

Hint

Look at what eq. (12) does with sectors that pay off when the rest of the economy does badly.

d) What column (3) actually contains

d) · 5 points

The paper says that “because of the additive properties of the variance” the specialization effect is column (1) minus column (2). Using the decomposition

\hat Y=\hat Y^\mu+\hat Y^s

of the setup, express column (3) in terms of variances and covariances of these components, and state two things besides ‘specializing into volatile sectors’ that it therefore contains. Why can an entry in column (2) exceed 100% in absolute value?

Hint

All three columns share one denominator; find it from the Mexico numbers.

e) Mexico’s hidden hedge

e) · 5 points

Using the Mexico variances in the setup: (i) reproduce the three Table I entries for Mexico, showing the denominator; (ii) bound the correlation between

\hat Y^\mu

and

\hat Y^s

in the baseline and in the 1972-cost run; (iii) say in one or two sentences what Mexico’s +36.1% “specialization” effect then mostly measures.

Hint

For fixed

\operatorname{Var}(\hat Y^\mu)

, the smallest achievable total variance given correlation

\rho

\operatorname{Var}(\hat Y^\mu)(1-\rho^2)

6. Wired in series: input–output links and connector exposure

In the full model, the sectoral price in country $n$ satisfies (from eqs. 6–7)

\log P^j_{nt}=-\frac1\theta\log\sum_mT^j_m\,e^{-\theta y^j_{nmt}}+\text{const},\qquad y^j_{nmt}\equiv\log\frac{c^j_{mt}}{A^j_{mt}\kappa^j_{nmt}},\qquad c^j_{nt}=B^j(w^j_{nt})^{\beta^j}\prod_k(P^k_{nt})^{\gamma^{kj}},

so that $\partial\log P^j_{nt}/\partial y^j_{nmt}=d^j_{nmt}$ . Sectoral revenue is $R^j_{nt}=\sum_md^j_{mnt}E^j_{mt}$ (eq. 9), expenditure is $E^j_{mt}=\alpha^j_tP_{mt}C_{mt}+\sum_k\gamma^{jk}R^k_{mt}$ (eq. 10), and value added is $w^j_{nt}L^j_{nt}=\beta^jR^j_{nt}$ (eq. 11). Producers pick the lowest-cost supplier after shocks are realised (p. 451). The paper argues that firms can switch suppliers within a year and that much of the benefit accrues at the intensive margin of an existing supplier portfolio (p. 458).

Consider a small connector economy $n$ whose sector $j$ buys input $k$ largely from China ( $d^k_{n,CH}$ large) and sells a share $s_{US}$ of its sector- $j$ revenue to the United States. Parts b)–d) go beyond the paper. Their answers must be derived from the model, with your own steps flagged.

a) Why the input matrix matters

a) · 3 points

Without input–output linkages (

\gamma^{kj}=0

\beta^j=1

, productivity recalibrated), the average effect of lower trade costs on volatility falls from −36.1% to −3.5% (Table VII). In three or four sentences, explain why the paper expects input–output links to matter for the trade–volatility relation, why the sign is ambiguous in principle, and which way the data come out.

Hint

Think of each sector’s input bundle as a portfolio of suppliers.

b) Supply and demand in series

b) · 6 points

Make a first-order, small-open-economy calculation: the connector’s sector-

j

labour

L^j_{nt}

is fixed, so its sectoral wage

w^j_{nt}

adjusts to satisfy eq. (11); all other countries’ wages, price indices and expenditures are held fixed; the connector’s sales to its own market are ignored; and its input price moves only by the direct term

\Delta\log P^k_n=d^k_{n,CH}\,u

. Derive the effect on

\log R^j_{nt}

, and hence on sector-

j

value added, of (a) a fall

u

in Chinese productivity in sector

k

\Delta\log A^k_{CH}=-u

, and (b) a fall

v

in us expenditure on sector

j

\Delta\log E^j_{US}=-v

, however it is generated. Then let both be driven by one geopolitical factor,

u=c_Cg

and

v=c_Ug

with

c_C,c_U>0

and

\operatorname{Var}(g)=\sigma_g^2

. Compare the variance of

\Delta\log R^j_{nt}

with the case where

u

and

v

are independent with the same variances. When would the factor instead hedge the connector? (For that step you may let the us price index respond.)

Hint 1

Chain:

A^k_{CH}\to P^k_n\to c^j_n\to d^j_{mn}\to R^j_n

, then close the loop with eq. (11): with

L^j_n

fixed,

\Delta\log w^j_n=\Delta\log R^j_n

Hint 2

With a destination’s price index fixed,

d^j_{mn}\propto (c^j_n)^{-\theta}

c) Re-sourcing gains are second order

c) · 5 points

Expand $\log P^j_{nt}$ to second order in the cost shocks $\Delta y^j_{nm}$ . Show that

\Delta\log P^j_n\approx\sum_md^j_{nm}\Delta y^j_{nm}-\frac\theta2\operatorname{Var}_d\big(\Delta y^j_{nm}\big),

where $\operatorname{Var}_d$ is the variance across suppliers weighted by the shares $d^j_{nm}$ . Interpret the two terms in light of the paper’s supplier-switching argument, and evaluate the second term when every supplier loads equally on a common factor.

Hint

Compute the Hessian of

-\frac1\theta\log\sum_mT_me^{-\theta y_m}

: differentiate

d_m

with respect to

y_k

d) Partner HHI versus effective diversification

d) · 4 points

A researcher proposes to track a connector’s diversification by the Herfindahl index of its trade partners’ shares. Using the model (and your answers above), list what the model says a correct measure of effective diversification must contain that partner hhi leaves out, and give a concrete scenario in which partner hhi falls while model-implied income volatility rises.

Hint

When, in Question 2c, did a Herfindahl index give the variance exactly?

References

Backus, David K., Patrick J. Kehoe, and Finn E. Kydland. 1992. “International Real Business Cycles.” Journal of Political Economy 100 (4): 745–775. doi:10.1086/261838. Reading note →

Eaton, Jonathan, and Samuel Kortum. 2002. “Technology, Geography, and Trade.” Econometrica 70 (5): 1741–1779. doi:10.1111/1468-0262.00352.