Competitive input markets and allocative efficiency

Question

Suppose two firms are operating in a competitive input and output markets. If the two firms' production functions are $(k_1l_1)^{1/3}$ and $(1+k_1)^{-1} (l_2)^{1/2}$ where $k_1, l_1, l_2$ denote the capital input of the first firm and the labour inputs of two firms respectively, show that merging the two firms will improve the input allocative efficiency.

1. What is meant by a competitive input market? Does it mean that the input prices are fixed (or the firms are input-price-takers)?
2. If the firms are already operating in a competitive market, shouldn't they already be at the maximum allocative efficiency point? Does merging change anything?

Answer 1

1. What is meant by a competitive input market? Does it mean that the input prices are fixed (or the firms are input-price-takers)?

Yes. To be precise the price-takers behave as if the prices were fixed and their decisions had no effect on them; e.g. firm 1 will consider wage to be $w$ and not $w(l_1)$ when deciding $l_1$.

2. If the firms are already operating in a competitive market, shouldn't they already be at the maximum allocative efficiency point? Does merging change anything?

There are several ways a competitive market can deliver a suboptimal outcome. One of those cases is if externalities are present, if someone's production/consumption decision affects the profits/welfare of others 'outside of normal market channels'. Here firm 1's decision on $k_1$ will affect the profits of firm 2 through firm 2's production function. Firm 1 does not consider this in any way when it is maximizing its profit, since the input price of $k_1$ does not convey any information on how detrimental a marginal increase in $k_1$ would be to firm 2, and so we (probably) get market failure/a suboptimal outcome.