Is inequality harmful for innovation and growth? Price versus market size effects

Abstract

We introduce non-homothetic preferences into an R&D based growth model to study how demand forces shape the impact of inequality on innovation and growth. Inequality affects the incentive to innovate via a price effect and a market size effect. When innovators have a large productivity advantage over traditional producers a higher extent of inequality tends to increase innovators’ prices and mark-ups. When this productivity gap is small, however, a redistribution from the rich to the poor increases market sizes and speeds up growth.

Appendix

1.1 Proof of Lemma 1

Proof

• a. Differentiating z(θ, t) with respect to t, using $\dot {N}(\theta ,t)/N(\theta ,t)=g,$ yields $\dot {z}(\theta ,t)/z(\theta ,t)=r-\rho -g$. Guess that g = r−ρ, so z(θ) is stationary. In that case household θ purchases all goods with prices lower than or equal p(θ) at all times and pays average price $\bar {p}(\theta ),$constant over time, and has expenditures $\bar {p}(\theta )N(\theta ,t)$. Notice further that wages evolve according to $w(t)=bN(\tau )e^{-\left (r-g\right ) (t-\tau )}$ and that V(τ) = b F N(τ) (since the value of each firm is π/r = b F). This allows us to rewrite the household’s lifetime budget constraint as $\bar {p}(\theta )N(\theta ,t)=N(t)\theta b(1+\rho F/L)$, confirming our guess that N(θ, t) and N(t) grow pari passu.

• b. The budget constraint of the poorest consumer is $p(\underline {\theta })N(\underline {\theta } ,t)=N(t)\underline {\theta } b(1+\rho F/L)$. We calculate $p(\underline {\theta } )$ using $p(\underline {\theta } )-b/a=$ $[1-G(\hat {\theta } )][1-b/a]$ (all firms make the same profit), (2) and r = g + ρ. This yields $p(\underline {\theta } )=(g+\rho )bF/L+b/a$. Substituting into the budget constraint and solving for N(θ, t) yields the first claim of part b). The budget constraint of household $\theta \in (\underline {\theta } ,\hat {\theta })$ is $p(\underline {\theta } )N(\underline {\theta },t)+\int _{\underline {\theta } }^{\theta } p(\xi )dN(\xi ,t)=N(t)\theta b(1+\rho F/L)$. Differentiating with respect to θ yields $p(\theta )\left [ dN(\theta ,t)/d\theta \right ] =N(t)b(1+\rho F/L)$. Solving for d N(θ, t)/d θ and integrating yields $N(t)b(1+\rho F/L)\int _{\underline {\theta } }^{\theta } (1/p(\xi ))d\xi +N(\underline {\theta } ,t)$. Calculating $p(\xi )=b\left [ (g+\rho )aF+(1-G(\xi ))L\right ] /\left [ a(1-G(\xi ))L\right ]$ from Eq. 2 and substituting into the previous equation yields the second claim of part b). By definition, household $\hat {\theta }$ purchases all goods produced by monopolistic firms but no goods produced by the competitive fringe. A household $\theta >\hat {\theta }$ spends $N(t)\hat {\theta }b(1+\rho F/L)$ for the N(t) monopolistic goods and $N(t)(\theta -\hat {\theta })b(1+\rho F/L)$ for the remaining N(θ, t)−N(t) goods produced by the competitive fringe. This yields the third claim of part b).

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1.2 Proof of Proposition 1

Proof

• a. The integrand in Eq. 4 is a concave function of G(∙). If G(∙) undergoes a second order stochastically dominated transfer, where $\int _{0}^{\hat {\theta }}G(\theta )d\theta$ remains unchanged, the value of the integral in Eq. 4 must increase due to Jensen’s inequality. Hence, the consumption curve (4) shifts up at $\theta =\hat {\theta }$. Further, with $G(\hat { \theta })$ unchanged, the zero profit constraint does not change at $\theta = \hat {\theta }$, therefore the equilibrium growth rate rises.

• b. The integrand in Eq. 4 takes lower values at the values of θ involved in the transfer. Hence the value of the integral in Eq. 4 decreases meaning that less purchasing power is left in the hands of households with $\theta <\hat {\theta }$. Around $\theta =\hat { \theta },$the consumption curve shifts down and the zero profit constraint remains unaffected, the growth rate decreases. c. Neither (2) nor (4) are affected for $\theta \leq \hat {\theta }$.

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