7 - Constrained optimization

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Calculus of Multivariable Functions

Section 7 Use of Partial Derivatives in Economics; Constrained Optimization

Although there are examples of unconstrained optimizations in economics, for example finding the optimal profit, maximum revenue, minimum cost, etc., constrained optimization is one of the fundamental tools in economics and in real life. Consumers maximize their utility subject to many constraints, and one significant constraint is their budget constraint. Even Bill Gates cannot consume everything in the world and everything he wants. Can Mark Zuckerberg buy everything? Similarly, while maximizing profit or minimizing costs, the producers face several economic constraints in real life, for examples, resource constraints, production constraints, etc.

The commonly used mathematical technique of constrained optimizations involves the use of Lagrange multiplier and Lagrange function to solve these problems followed by checking the second order conditions using the Bordered Hessian. When the objective function is a function of two variables, and there is only one equality constraint, the constrained optimization problem can also be solved using the geometric approach discussed earlier given that the optimum point is an interior optimum. It should be mentioned again that we will not address the second-order sufficient conditions in this chapter.

Example 1: Maximize utility $u = f(x,y) = xy$ subject to the constraint $g(x,y) = x + 4y = 240$. Here the price of per unit $x$ is $1$, the price of $y$ is $4$ and the budget available to buy $x$ and $y$ is $240$. Solve the problem using the geometric approach.

Here the optimization problem is:
Objective function: maximize $u(x,y) = xy$
Subject to the constraint: $g(x,y) = x + 4y = 240$.

Step 1: $-\frac{f_{x}}{f_{y}} = -\frac{y}{x}$ (Slope of the indifference curve)

Step 2: $-\frac{g_{x}}{g_{y}} = -\frac{1}{4}$ (Slope of the budget line)

Step 3: $-\frac{f_{x}}{f_{y}} = -\frac{g_{x}}{g_{y}}$ (Utility maximization requires the slope of the indifference curve to be equal to the slope of the budget line.)
$$-\frac{y}{x} = -\frac{1}{4}$$ $$x = 4y$$ Step 4: From step 3, use the relation between $x$ and $y$ in the constraint function to get the critical values.
$$x + 4y = 240$$ $$4y + 4y = 240$$ $$8y = 240$$ $$y = 30$$ Using $y = 30$ in the relation $x = 4y$, we get $x = 4 \times 30 = 120$
Utility may be maximized at $(120, 30)$.

Example 2

Suppose a consumer consumes two goods, $x$ and $y$ and has utility function $u(x,y) = xy$. He has a budget of $$400$. The price of $x$ is $P_{x} = 10$ and the price of $y$ is $P_{y} = 20$. Find his optimal consumption bundle using the Lagrange method.

Here the optimization problem is:

Objective function: maximize $u(x,y) = xy$
Subject to the constraint: $g(x,y) = 10x + 20y = 400$.
This is a problem of constrained optimization.

Form the Lagrange function: $$L(x,y,\mu ) \equiv \color{red}{f(x,y)} - \mu (\color{purple}{g(x,y) - k})$$ $$L(x,y,\mu ) \equiv xy - \mu (10x + 20y - 400)$$ Set each first order partial derivative equal to zero: $$\frac{\partial L}{\partial x} = y - 10\mu = 0 \qquad\qquad\qquad \text{(1)}$$ $$\frac{\partial L}{\partial y} = x - 20\mu = 0 \qquad\qquad\qquad \text{(2)}$$ $$\frac{\partial L}{\partial \mu} = -(10x + 20y - 400) = 0 \quad \text{(1)}$$ From equations (1) and (2) we find: $$x = 2y$$ Use $x = 2y$ in equation (3) to get: $$10x + 20y = 400$$ $$40y = 400$$ $$\bf{y = 10}$$ $$\bf{x = 2y = 20}$$ See the graph below.

Example 3: The effects of a change in price

Suppose a consumer consumes two goods, $x$ and $y$ and has the utility function $U(x,y) = xy$. He has a budget of $$400$. The price of $x$ is $P_{x} = 10$ and the price of $y$ is $P_{y} = 20$.

Find his optimal consumption bundle.
What happens when the price of $x$ falls to $P_{x} = 5$, other factors remaining constant?

When $P_{x} = 10$, the optimal bundle $(x,y)$ is $(20,10)$ (See Example 2)
When the price of $x$ falls to $P_{x} = 5$,
New budget equation is: $5x + 20y = 400$
Slope of the indifference curve $= -\frac{y}{x}$
Slope of the budget line $= -\frac{5}{20} = -\frac{1}{4}$

Optimality requires that the slope of the indifference curve equals to the slope of the budget line: $$-\frac{y}{x} = -\frac{1}{4}$$ $$x = 4y$$ Use the relation $x = 4y$ in the new budget equation: $$5x + 20y = 400$$ $$5(4y) + 20y = 400 \;\;\;$$ $$20y + 20y = 400$$ $$\qquad\quad\;\; y = 10$$ $$x = 4y = 4(10) = 40$$ When the price of $x$ falls while all other factors remain constant, in this typical example, the consumption of $x$ increases.

Example 4: The effects of a change in income

Suppose a consumer consumes two goods, $x$ and $y$ and has utility function $U(x,y) = xy$. He has a budget of $$400$. The price of $x$ is $P_{x} = $10$ and the price of $y$ is $P_{y} = $20$.

Find his optimal consumption bundle.
What happens when the when the income rises to $B = 800$, other factors remaining constant?

When $P_{x} = $10$, $P_{y} = $20$ and $B = 400$, the optimal bundle is $(20,10)$. (See example 2)
When the income increases to $800$ while other factors remain constant,

New budget equation is: $10x + 20y = 800$

Slope of the indifference curve $= -\frac{y}{x}$

Slope of the budget line $= -\frac{10}{20} = -\frac{1}{2}$

Optimality requires that the slope of the indifference curve equals the slope of the budget line: $$-\frac{y}{x} = -\frac{1}{2} \;\;$$ $$x = 2y$$ Use the relation, $x = 2y$ in the new budget equation: $$10x + 20y = 800$$ $$10(2y) + 20y = 800 \quad$$ $$\; 20y + 20y = 800$$ $$\qquad\quad\;\; y = 20$$ $$\qquad\qquad\qquad\qquad\qquad\;\; x = 2y = 2(20) = 40 \quad$$ When consumer’s income increases, while other factors remain constant, for typical goods discussed in this example, consumption of both goods increases.