Imperial MATH60142 - Mathematics of Business and Economics
| Front | Constrained utility-maximising bundle \(\underline{x}^* \in B\) |
| Back | \(\underline{x}^*\) will be independent under a strictly increasing transformation of utility function\(\underline{x}^*\) will, in general, be dependant both on prices \(\underline{p}\) and on the budget \(m\)\(\underline{x}^*\) is homogeneous of degree 0 jointly in prices and budget |
| Front | Profit maximisation for a noncompetitive firm FOC and SOC |
| Back | We seek \[ \max_{y > 0} \Big\{ p(y)y - c_s^*(\underline{w}, y)\Big\} \] where the FOC and SOC are \[ \begin{split} \text{FOC}: \ & \frac{\partial}{\partial y} \Bigg( p(y)y - c_s^* (\underline{w}, y) \Bigg) = 0 \Longrightarrow \frac{\partial p (y)}{\partial y} y + p(y) = \text{SMC}(y) \\ \text{SOC}: \ & \frac{\partial^2}{\partial y^2} \Bigg( p(y)y - c_s^* (\underline{w}, y) \Bigg) \leq 0 \Longrightarrow \frac{\partial^2 c_s^*(\underline{w}, \underline{x}_F, y)}{\partial y^2} \geq \frac{\partial^2 p(y)}{\partial y^2} y + 2 \frac{\partial p(y)}{\partial y} \end{split} \]The FOC can be rearranged to \[ p(y) \left( 1 + \frac{1}{\varepsilon_D(y)} \right) = \text{SMC}(y)\]where \( \varepsilon_D(y) := \varepsilon_D(p(y))\) |
| Front | Elasticity of Supply |
| Back | \[ \varepsilon_S = \frac{\partial S(p)}{\partial p} \cdot \frac{p}{S(p)} \] |