How would a firm increase price in such a scenario? I have the following question: If a firm with a lower marginal cost wants to increase prices in a homogenous goods industry in which demand is perfectly inelastic and supply is in excess, what should it do ?.
I'm thinking along the following lines: In a homogenous goods industry, the only differentiating element is the price of the product. By setting a lower price near to its marginal cost, the firm can grab the entire demand for the product ( the demand wont increase but the demand that otherwise went to other firms could be supplied by this firm, assuming no capacity constraints). After the excess supply from inefficient firms are removed from the market, the firm can then reduce its quantity and therefore reach a higher equilibrium price. Is this reasoning conceptually correct ?.
I also have this follow-up question, if the overall industry demand is inelastic what can I infer about the firm level demand ?. Can any inference be made from the question on the degree of competition in the industry ?. Thank you!.
It could work (it would depend on model parameters) if you would assume that it is impossible for firm to enter the market once firm exists. This is actually quite complex and long problem so I will provide only partial solution.
This would be a sequential Bertrand competition problem (since you assume that firms only can compete on prices). Here in first round company A (the one with lower marginal costs $c_A&lt;c_B$) would have a choice - either set its price $p_A=p_B$, split the market between itself and second firm B and just get the extra profit thanks to its marginal costs being lower and do this in both period 1 and period 2.
So you would have strategy of 'tolerating B' (T) giving profits:
$$T: \pi_{A,t=1}+\delta \pi_{A,t=2}= (p_A -c_A) \frac{q(p)}{2} +  \delta (p_A -c_A )\frac{q(p)}{2}$$
where $p_A =p_B =c_B &gt; c_A$ and $\delta$ is a discount factor for future profits.
Then you could have second strategy of 'contesting B' (C). In this strategy firm A would  undercut firm B in the first stage allowing it to get monopoly profits in the second stage so the strategy profit would look like:
$$C: \pi_{A,t=1}+\delta \pi_{A,t=2} = (p_{A,t=1} -c_A) q(p_{A,t=1}) +  \delta (p_{A,t=2} -c_A)q(p_{A,t=2})$$
Where now parameters would have to be such that $p_{A,t=1}&lt;p_A$ (i.e. the first period price has to be lower in strategy C compared to strategy T) and then the second period price $p_{A,t=2}$ would be monopoly price in that period. Whether, strategy C or T makes sense will depend exactly on parameters of this model, depending on your assumptions on discount rates, marginal costs differential between the firms and consumer demand you could end up with either T or C being dominant strategy. Also, this requires assumption that if firm B gets loss in the first round it will immediately exist and won't be able to ever enter again.
Regarding your second question inelastic demand is in principle consistent with even perfect competition. You can only make some inferences about market power once you assume what the competition is (for example monopoly market power is inversely proportional to the elasticity of demand). However, it would be beyond scope of SE answer to list effect of inelastic demand on competition across all market structures. You can have look at  Paul Belleflamme &amp;  Peitz Industrial Organization: Markets and Strategies for overview of this.