Application of Constrained Optimization in economics

1. CONSTRAINED OPTIMIZATION
The problem of optimization of some quantity subject to certain restrictions or constraints is
a common feature of economics, industry, defence, etc. Common economic situations are -
 The Maximization of Utility subject to the Budget Constraint of fixed income
 The Minimization of Cost subject to a fixed Output
 The Output Maximization subject to Cost constraint
If the constraints are in the form of equations, methods of calculus can be useful but if the
constraints are inequalities instead of equations we use the method of mathematical
programming (LPP).
Substitution Method
Utility Maximization Subject to Budget constraint
Consider a utility function of a consumer
Budget constraint, 20x + 10y = 200
Rewrite the above equation as
Then the original utility function becomes
𝑈 𝑈 = 𝑈x0.3
+ (20 − 2𝑈)0.3
Then find
𝑈𝑈
𝑈𝑈
and makes it zero to find value of x and do
𝑈d2
𝑈U
𝑈𝑈dx2 at that point(s) of x value(s)
If this second order
𝑈d2
𝑈U
𝑈𝑈dx2 is positive at that value of x, U will be minimized and if negative then U
will be maximized at that value of x.
Lagrange Method
If Objective function i.e. Profit, sales Revenue or Cost functions are nonlinear in nature then
Substitution method will be cumbersome & this method is applied. This method is fruitful to use
cases of Objective function like Linear or Nonlinear.
Given a function, Z = f(x, y)
Subject to a constraint equation g(x, y) = k, where k is a constant,
A new function F (Called Lagrangian Function) can be formed by
1) Setting the constraint equal to zero
2) Multiplying the constraint by language multiplier λ
3) Adding the product to the original function.
That is, F = f(x, y) + λ [k- g (x, y)]
The first order conditions are found by taking partial derivatives of F with respect to all three
independent variables and setting them equal to zero. That is, Fx = 0, Fy = 0, Fλ = 0
The second order conditions can be expressed in terms of bordered Hessian denoted as |H| and
can be written as

2. Example 2:
Maximise the utility function U = 2xy subject to a budget constraint equal to 3x + 4y = 90
using Lagrange method.

4. Constrained Cost Minimization
Cost minimization involves how a firm has to produce a given level of output with
minimum cost.
Consider a firm that uses labour (L) and capital (K) to produce output (Q). Let W is the price
of labour, that is, wage rate and r is the price of capital and the cost (C) incurred to produce
a level of output is given by
C = wL + rK
The objective of the firm is to minimize cost for producing a given level of output.
Let the production function is given by following.
Q = f (L, K)
In general there is several labour – capital combinations to produce a given level of
output. Which combination of factors a firm should choose which will minimize its total
cost of production.
Thus, the problem of constrained minimization is
Minimize C = wL +rK
Subject to produce a given level of output, say Q₁ that satisfies the following
production function Q1 = f (L, K)

5. Constrained Profit Maximization
Maximizations of profit subject to the constraint of fixed amount of Production can also
used to identify the optimum solution for a function.
Assumes that TR = PQ
TC = wL +rK
Profit Π = TR – TC
Π = PQ – (wL + rK)
Thus the objective function of the firm is to maximize the profit function subject to given
production target.

6. Constrained Optimization problem here is:
Maximize Π = PQ – (wL +rK)
Subject to the constraint Q = f (K, L)