Optimization (Basics) and Integration (Core Tools for Economics)

1) Optimization: what economists do with it

Economics frequently involves choosing the best option (maximize/minimize):

• Consumers maximize utility subject to budget constraint.
• Firms maximize profit or minimize cost.
• Governments may minimize inflation loss, etc.

Optimization needs:

• derivatives (marginal conditions),
• second-order checks,
• constraints (often handled by Lagrange method).

2) Unconstrained optimization (single variable)

Let $y=f(x)$. To maximize/minimize:

Step 1: First-order condition (FOC)

Step 2: Second-order condition (SOC)

• Maximum if $f''(x)<0$ at critical point.
• Minimum if $f''(x)>0$.

Example (profit maximization idea)

Profit $\pi(Q)=R(Q)-C(Q)$. FOC:

SOC for max: $d^2\pi/dQ^2<0$.

3) Multivariable unconstrained optimization

For $z=f(x,y)$: FOCs:

SOC uses the Hessian matrix (concept):

• For CUET PG, you mainly need the idea that second derivatives determine max/min.