Price rigidity exists in oligopoly as their interdependence on each other stops them from changing prices without losing their competitive edge.
In the diagram above, P is the current market price a large firm in the market charges and Q is the quantity they supply.
The firms in an oligopolistic market are price searchers, so they can either set the price to be charged or the quantity to be supplied.
The elastic part of the demand curve before R (which is the kink), is elastic because the theory assumes that if the a firm increases the price it charges, competitors will not follow, as they will try to retain a competitive edge. Therefore, raising the price would result in a fall in revenue.
The inelastic part of the curve after R (the kink) is inelastic as the theory assumes that if a firm reduces the price it charges, competitors will follow as to remain competitive. Therefore, lowering the price would lead to a fall in revenue.
Since oligopolistic firms are profit maximisers, they will produce where MC=MR. Since, in this scenario, the MR curve between the points A and B is indeterminate, the theory states that firms in an oligopolistic market will keep supplying price P and quantity Q as long as the MC curve falls between the points A and B. Even if the curve shifts from MC to MC1 or even MC2, the firms will still charge P and supply Q as the MC curve would still fall within the region of indeterminacy.
This theory explains why price rigidity exists in oligopoly- the firms wish to remain profitable and their interdependence on each other stops them from changing prices.