DSE 2018 Question 33: Endogenous growth and technical progress

Setup, from Q32 of the paper:

Consider an economy with no capital. The final good is produced with a linear labour technology $Y_t = A_t(1 - \lambda_t)L_0$, where $L_0$ is the constant labour force, $1 - \lambda_t$ is the share in final-goods production and $A_t$ is the technology index, which evolves as $\frac{dA}{dt} = \lambda_t L_0 A_t$ with $A_0 > 0$. Suppose the relative price of new technology (in terms of the final good) is unity, and labour is allocated so that the value of marginal product of labour is identical across the two sectors. In this case, the equilibrium value of $\lambda_t$ will be

Consider the economy of the previous problem, with final-good technology $Y_t = A_t(1 - \lambda_t)L_0$ and technology evolving as $\frac{dA}{dt} = \lambda_t L_0 A_t$. For any exogenously given value $\lambda_t = \lambda_0$, the long-run balanced growth rate in this economy is