I explore how a supply shock to one industry affects both industries. I begin by focusing on the “within” effect.
Proposition 2 (∂φ∗i
∂ai > 0). Better ideas in a industry imply that more ideas are
produced, output increases and average quality increases within the industry.
Figure 2.5 shows the impact of a positive supply shock hitting industry 1.
As ideas become more productive in one industry, the output of each idea in- creases in that industry. The result is that the industry’s output becomes more abundant, which lowers the price of the good and raises the cutoff. It is less obvious how the number of implemented ideas changes, because there are two countervailing effects. First, increasing the cutoff results in fewer ideas be- ing developed. However, there are now more ideas above the new threshold as well. Given the distributional assumptions, the net result is an increase in the number of implemented ideas.
Figure 2.5: Changes in Cutoffs from an increase ina1
Idea Quality
Final Good Producer
Ideas in Industry 1
Ideas in Industry 2
Densit
y
Densit
y
Idea Quality
Number of Implemented a1 a1 φ1∗φ∗1 a2 φ∗2φ∗2 Ideas Number of Implemented IdeasNow consider the impact of the shock on the other industry, which can be thought of as the “cross” effect.
Proposition 3 (∂φ∗i
∂aj < 0). Better ideas one industry imply that more ideas are
produced, output increases and average quality declines in the other industry.
As more of the final good is produced, the demand for the other intermediate good increases. Inventors respond by implementing more ideas, which can only be of lower quality.
Corollary 1. Any supply shock results in the number of implemented ideas in one industry being negatively correlated with the average quality of ideas in the other industry.
Corollary 2. A positive supply shock increases the aggregate number of imple- mented ideas.
Proposition 2 and 3 together imply thatN rises whena1 increases.
Corollary 3. Any supply shock result in the average quality of each industry moving in opposite directions.
Proposition 2 and 3 imply that the cross-effect and within-effect are oppo- site in direction. As a result, the average quality of all implemented ideas depends on whether the cross-effect is greater than the within effect. In the next proposition, I show that average quality depends on the relative quality of ideas between industries.
Innovation Supply and Implemented Idea Quality: A U Relationship Proposition 4 (∂a∂Q
i < 0,
∂Q
∂aj > 0 if ai < aj). The average quality of the imple-
mented ideas declines (increases) if ideas get better in the worse (better) industry.
The proof can be found in Appendix A.2.
Figure 2.6 demonstrates how the average quality of implemented ideas changes when there is an increase in supply. To understand Proposition 4, consider the following example.
Example 1. Final Good Production is Cobb-Douglas (= 1) withθ = 0.5
Fact 1. Aggregate quality is a weighted average of the average quality of each industry.
Figure 2.6: Number and Quality of the Aggregate Implemented Ideas
a1-a2
0
N
Q
Proof. The price is inversely related to the quality of the marginal idea in that industry. Using (2.6) and (2.7), the factor shares are proportional to the number of implemented ideas:
1
φi Xi =
k
k−1N1 (2.11)
With a Cobb-Douglas production, the expenditure share of each good is con- stant. The implication is that the average quality of all implemented ideas is determined by the cutoffs, because (2.10) reduces to
Q= Q1+Q2
2 .
Recall Corollary 3, which implies that any change in quality is always op- posite in direction. For proposition 4 to hold, the following must be true.
Fact 2. The cross-effect is smaller than the within-effect on quality when the shock occurs in the higher-quality industry.
Proof. The expenditure share for each good is 1
2.Using (2.11), it must be that 1 2 = 1 φ1X1 Y = kN1 (k−1)N1 √ Q1Q2 = k (k−1)√Q1Q2 . (2.12)
From (2.12) it obvious that any change to the average quality of ideas in one industry is opposite in direction but proportionally equal in magnitude. From (2.8) it must be that the cutoffs move in proportionally equal and opposite di- rections. As a result, the effect of a shock to any industry is always larger in level for the higher-quality industry.
a high-quality industry, suppose that a1 < a2. That is, industry 1 has rela- tively lower-quality ideas to implement. Recall, complementarities imply that
N1 = N2. Prior to a shock, it then must be that industry 1 implements rela- tively lower-quality ideas compared to industry 2 (φ1 < φ2). A positive shock to industry 2, lowers the cutoff in industry 1 to because it increases the price of good1. However, it is very costly to produce more X1 (φ1
1 >
1
φ2) because the
cutoff only generates a small increase in output. As a result, the cross-effect is smaller when the shock occurs in the high-quality industry.