# Elasticity of Supply for a Monopolist

Is the supply of output for a monopolist less elastic than that of a perfectly competitive firm with the same production function?

On one hand, it seems impossible to calculate the elasticity of supply for a monopolist since it has no supply curve (i.e. no explicit relationship between quantity supplied and prices since it determines both quantity and therefore indirectly, prices by producing where MR = MC). As a result, it would be impossible to calculate the elasticity of supply for a monopolist and the above statement is false.

However, it would seem to make sense that the elasticity of supply is lower for a monopolist because if for example there is an increase in demand, leading to higher prices, the additional output produced by a competitive firm would be higher than a monopolist as the monopolist would tend to restrict output to keep prices higher. Thus, the monopolist's supply is less elastic and responsive to price changes.

Both sets of reasoning seem to be correct to me. What am I missing?

Contrary to what is said in some bachelor textbooks, the monopoly has a supply curve, but it cannot directly be seen on the MR=MC Figure. As the price is endogenous in the case of monopoly, the price elasticity of supply is not defined, unless you have some exogenous variable shifting the price. For instance if the inverse output demand function is $$p=P(y,z)$$ where $$z$$ denotes an exogenous variable (population size, aggregate unemployment rate, etc), then it is possible to compute the elasticity, even in the case of a monopoly. Then you can compare the responsiveness to shifts to $$z$$ in both the competitive and the monopoly cases. At the monopoly optimum, the condition $$P(y,z)+\frac{\partial P}{\partial y}(y,z)y=c'(y)$$ implicitly defines the supply function $$y^*(z)$$ and it can be shown by the implicite function theorem that $$\frac{\partial y^*}{\partial z}(z)=-\frac{\partial P/\partial z +y^*\partial^2P/\partial y\partial z}{2\partial P/\partial y+y^*\partial^2P/\partial y^2-c''},$$ where for simplicity I omitted the arguments of the different functions. From the second order condition for an optimum, we know that the denominator is negative. It turns out that for demand shifts $$z$$ which increase $$P$$ the monopoly output increases (provided the term $$y^*\partial^2P/\partial y\partial z$$ is well behaved, if $$P$$ is linear in $$z$$ for instance). In the competitive case, the comparable expression is $$\frac{\partial y^c}{\partial z}(z)=\frac{\partial P/\partial z }{c''},$$ which is positive as well. Now compare both expressions, and conclude…. As we are interested in elasticities, we should actually compare $$\frac{\partial y^*}{\partial z}\frac{z}{y^*}(z) \text{ and } \frac{\partial y^c}{\partial z}\frac{z}{y^c}(z),$$ and consider that $$y^*>y^c$$. If the inverse demand is linear, comparison is easy, but is involved in general as it depends on global properties of all functions involved in the expressions above.