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C: the heterogeneous nonrational model with a lockdown

VII: Robustness

This section discusses the results of a series of robustness exercises in which I vary the key parameters and assumptions of the heterogeneous nonrational model without intervention. First, this section discusses the modelโ€™s robustness to variations in its most influential parameters, i.e., ๐œ‘, ๐œ‹,, ๐œŒ, ๐œ‹., ๐œ‹), and ๐œ‹'. Then, I analyze the modelโ€™s robustness to alternative assumptions for forming expectations on the future ratio of infected agents and susceptible agentsโ€™ future consumption and labor. Table 4 presents the results of these robustness exercises.

Consumption lowest point

Peak rate infected

Deceased rate Number deceased in the

Netherlands Productivity of infected people, ๐œ‘

0.2 (baseline) 0.4

โˆ’3.98%

โˆ’5.50%

2.43%

3.88%

0.32%

0.37%

46 625 52 333 Average mortality rate, ๐œ‹,

0.0062 (baseline) 0.01

โˆ’3.98%

โˆ’4.69%

2.43%

2.65%

0.32%

0.46%

46 625 66 667 Productivity of agents with long COVID, ๐œŒ

0.7

0.82 (baseline)

โˆ’4.11%

โˆ’3.98%

2.30%

2.43%

0.24%

0.32%

34 333 46 625 Long COVID recovery rate, ๐œ‹.

0.05

0.2 (baseline)

โˆ’6.89%

โˆ’3.98%

3.28%

2.43%

0.36%

0.32%

51 667 46 625 Share of infection due to consumption, work, and general contacts

0.1, 0.1, 0.8

1/6, 1/6, 2/3 (baseline)

โˆ’4.80%

โˆ’3.98%

2.87%

2.43%

0.35%

0.32%

51 000 46 625 Constant expectations for the number of infected

Bell-shaped (baseline) Constant

โˆ’3.98%

โˆ’4.13%

2.43%

2.76%

0.32%

0.29%

46 625 41 667 Homogeneous bell-shaped expectations for consumption and labor

Bell-shaped Constant (baseline)

โˆ’5.19%

โˆ’3.98%

3.18%

2.43%

0.30%

0.32%

43 333 46 625 Table 4: Robustness of the heterogeneous nonrational model without containment

๐œ‘ controls the productivity of infected agents. Figure 14 schematically summarizes the effect of ๐œ‘ on aggregate consumption.

Figure 14: Schematic representation of the effect of ฯ† on aggregate consumption

When ๐œ‘ increases, the number of infected increases for two reasons. First, susceptible agents cut back economic activity less because the virus becomes less harmful when ๐œ‘ increases. Second, susceptible agents are more likely to be infected during consumption when infected agentsโ€™ consumption increases. The increase in infected agents results in more deaths. The increased individual susceptible agentโ€™s and infected agentโ€™s consumption has an upward effect on aggregate consumption. The increased number of infected and increased death toll have a downward effect on aggregate consumption. Table 4 shows that, when ๐œ‘ equals 0.4, the number of deceased agents and the peak number of infected increases by 0.05% points and 1.45% points. Also, the trough of aggregate consumption becomes deeper by 1.52% points, indicating that the downward effects โ€“ mostly the increased number of infected โ€“ on consumption dominate the upward effects.

Average mortality ๐œ‹, determines the heterogeneous mortality rates. When it increases, the adverse effects of the virus become more powerful. As a result, susceptible agents cut back economic activity more, leading to fewer infections. The decrease in infections decreases the number of deceased agents at the end of the pandemic. The increase in the average mortality rate itself, however, increases the number of deceased agents.

When ๐œ‹, equals 0.01, the number of infected agents increases by 0.22% points in Table 4. This increase is likely the consequence of randomness in the simulations resulting from the stochasticity of the model. The increase in the number of deceased agents by 0.14% points results from the increase in the peak number of infected agents and the increase in the mortality rate. The aggregate consumption trough becomes deeper by 0.71% points. The increase in the number of infected and the increased death toll explain this deepening.

๐œŒ governs the productivity of agents with long COVID. It does not affect epidemiological dynamics since susceptible agents do not consider long COVID. When ๐œŒ decreases, the consumption of agents with long COVID falls, resulting in a deeper aggregate consumption trough. Table 4 shows that, when ๐œŒ equals 0.7, the number of infected agents and the number of deceased agents increase by 0.13%

points and 0.08% points because of randomness. Furthermore, aggregate consumption falls by 0.13%

points more at the pandemicโ€™s peak. The decrease in ๐œŒ explains this consumption fall. The decrease in the number of infected likely offsets this fall.

๐œ‹. controls how many weeks agents suffer from long COVID on average. As with ๐œŒ, ๐œ‹. does not affect the number of infected agents. Though, when ๐œ‹. decreases, agents are less productive for a longer time, negatively affecting aggregate consumption. When ๐œ‹. falls from 0.2 to 0.05, agents suffer from long COVID for twenty weeks on average instead of five. In the simulations, the number of infected agents and the number of deceased agents at the pandemicโ€™s peak increase by 0.85% points and 0.04% points, resulting from randomness. Aggregate consumption decreases by 2.91% points more

at the pandemicโ€™s peak, as shown in Table 4. This decrease results from the increase in the duration of long COVID symptoms and is likely exacerbated by the increase in the peak number of infected.

Furthermore, Table 4 reports on the relative importance of the virusโ€™s three transmission channels. Figure 15 schematically summarizes the effect of ๐œ‹) and ๐œ‹' on aggregate consumption.

The decrease in consumptionโ€™s and laborโ€™s shares of total infections has two effects. First, susceptible agents have a lower probability of being infected per unit of labor and consumption. As a result, aggregate consumption increases. Second, agents have less incentive to decrease economic activity. A relative increase in economic activity leads to more infected and deceased agents. Thus, the effect on the number of infected agents is ambiguous. If the number of infected agents increases, aggregate consumption likely decreases, and vice versa. Thus, the effect on aggregate consumption is ambiguous as well. In the simulation, decreasing the relative importance of economic decisions from a third to twenty percent of total infections increases the number of infected agents by 0.44% points and the number of deceased agents by 0.03% points. Furthermore, aggregate consumption falls by 0.82%

points more at the pandemicโ€™s peak, indicating that the downward effect on aggregate consumption dominates.

Robustness to expectations

This section also analyzes the effects of using different expectations for the ratio of infected agents, consumption, and labor. First, instead of expecting a bell-shaped curve for the ratio of infected, I simulate a version of the model in which agents expect that the ratio of infected in every future period ๐‘ก + ๐‘ฃ is the same as in period ๐‘ก.

When agents expect a constant ratio of infected agents, agents net underestimate the future ratio of infected for low ๐‘ก, while they start to net overestimate the future ratio of infected agents when the Figure 15: Schematic representation of the effect of ๐œ‹! and ๐œ‹" on aggregate consumption

pandemicโ€™s peak is nearby. When agents net underestimate the future ratio of infected, they overestimate ๐‘ˆ%,!2)6 and cut back economic activity too much. As a result, the peak number of infected decreases.

When agents net overestimate the future ratio of infected, they underestimate ๐‘ˆ%,!2)6 and cut back economic activity too little, leading to a higher peak number of infected.

Thus, under this alternative assumption, the growth of the number of infected starts slowly and then ramps up. Conversely, after the pandemicโ€™s peak, the number of infected first declines rapidly and then stabilizes. The effect on the peak number of infected, the number of deceased agents, and aggregate consumption is ambiguous.

Simulating the model with constant expectations, the peak number of infected increases by 0.33% points. This increase suggests that the upward effect on the number of infected caused by overestimating the future ratio of infected dominates the downward effect of underestimation. The aggregate consumption trough becomes deeper by 0.15% points, as shown in Table 4. The deepening of this trough is likely the result of the increased peak number of infected. Furthermore, Table 4 shows that the number of deceased agents decreases by 0.03% points. The dynamics of the number of infected agents may explain that, even though the peak number of infected agents has increased, the number of deceased agents has decreased. Figure 16 presents the number of infected in this simulation and the number of infected in the heterogeneous nonrational model.

The red curve in Figure 16 shows the number of infected agents in the heterogeneous nonrational model. The blue curve shows the number of infected agents when agents expect a constant number of infected agents. The blue curve shows that the growth of the number of infected starts slower than under bell-shaped epidemiological expectations and then ramps up. As a result of this dynamic, 56.2% of the

Figure 16: The number of infected in the heterogeneous nonrational model and under an alternative assumption for forming pandemic expectations

initial population remains susceptible instead of 56.0%, resulting in fewer deaths. A 0.2%-point difference in susceptible agents leads to 0.00124% points fewer deceased agents on average. Therefore, the narrow curve of the number of infected agents can explain why the number of deceased agents did not increase for a higher peak number of infected. Randomness likely explains the decrease in the number of deceased agents.

Second, I replace constant consumption and labor expectations with homogeneous bell-shaped consumption and labor expectations, based on the perfect foresight model. Figure 17 shows these consumption expectations relative to what materializes under these expectations for the 16-30 age group.

The red curve in Figure 17 shows the aggregate realized consumption of agents aged sixteen to thirty. The blue curve shows their consumption expectations. Figure 17 shows that agents expect a deeper, earlier aggregate consumption trough than what materializes. As a result, they severely underestimate their consumption in the first 47 weeks of the pandemic. Then, they slightly overestimate their consumption. For the final weeks of the pandemic, they underestimate their consumption again.

The bell-shaped expectations use aggregate consumption from the perfect foresight model and depend on the number of deceased agents. As a result, expectations do not recover to their initial level, explaining the underestimation at the end.

Agents net underestimate their consumption more than under constant consumption expectations. This underestimation leads to a weaker response in consumption and labor by the logic described in section six. In turn, this results in an increase of the number of infected agents by 0.75%

points, which causes a deeper consumption trough by 1.21% points, as shown in Table 4. Even though Figure 17: Realized consumption of the age group 16-30 relative to consumption expectations under an

alternative assumption for forming consumption expectations

the peak number of infected agents is higher, Table 4 shows that 0.02% points fewer agents die under these alternative expectations. By presenting consumption expectations relative to the aggregate consumption for the 67+ age group, Figure 18 illustrates why heterogeneity in the model may explain the decrease in deceased agents.

The red curve in Figure 18 shows the aggregate realized consumption of agents aged sixty-seven and above. The blue curve shows their consumption expectations. Because relatively many agents in this group die, aggregate consumption does not recover to its initial level. Though, realized consumption does recover to its initial level for individual old agents. Keeping this in mind, the ratio of the red areas relative to the blue area in Figure 17 is larger than the ratio of the red area relative to the blue area in Figure 18. This suggests that the young relatively overconsume more than the old. As a result, the young are relatively more likely to be infected under the alternative consumption and labor expectations than the old. As a result, fewer agents die.

Having discussed these parameter and expectations changes, economic and epidemiological dynamics appear consistent across configurations. For example, the peak number of infected agents appears to be the primary determinant of the depth of the aggregate consumption trough across configurations. Furthermore, the net underestimation of future consumption consistently leads to a suboptimally small reduction in economic activity. As a result of these consistencies, the qualitative conclusions are robust to various configurations of the model: the aggregate consumption trough lies between โˆ’3.98% and โˆ’6.89%, the peak number of infected lies between 2.30% and 3.88%, and the number of deceased agents between 0.24% and 0.46%.

Figure 18: Realized consumption of the age group 67+ relative to consumption expectations under an alternative assumption for forming consumption expectations

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