A budget constraint is a representation of the quantities and prices of various goods that can be purchased within a specified budget. This story explores the concept of budget constraint with examples.

### Please Remember

For a firm/company, a small budget constraint can be solved by borrowing; however, in the long run, certain other factors need to be considered, like rent, salaries, etc.

In simple, plain words, ‘budget constraint’ can be defined as a situation wherein one is within a tight budget and all purchases have to constrained within that limit. The term is used differently in economics and related disciplines.

Budget constraint analytics prove useful in determining consumer choices. This concept can be better explained using graphs with corresponding values. The next paragraphs will explain the budget constraint equation, formula, and graphs, along with suitable examples.

## THE CONCEPT

Let’s begin with a simple concept of budget management in a household.

● Everyone of us designates a stipulated amount every month from our salary for household expenses, which include expenses for food, clothing, daily utilities, repair, bill payments, etc.

● Now, these expenses heavily depend on your lifestyle and your way of expenditure.

● For example, if you have a designated budget, you might buy only a particular brand of coffee beans instead of buying a more expensive brand. Or, you may pay a smaller installment on your house every month because you need some amount to save.

● If the market situation changes, i.e., to simplify, if the price of that particular can of coffee or a particular grocery item increases, you will immediately settle for a less expensive brand of the same product. Or, you may reduce your electricity and water usage so as to save on your bills.

● These situations arise because you need to maintain your budget and your expenditure. This is called budget constraint, in layman terms.

The concept of budget constraint in microeconomics though, is almost based on similar terms.

● If the consumer has a predetermined budget and he needs to buy goods, he has to consider the prices and quantities of those goods, and manage the purchases such that they fit exactly within his budget. The aim is to maximize your utility.

### Example

● Consider a very basic example. Assume that a customer has USD 20 in his hands, and he has to buy apples and oranges.

● Let the price of one apple be USD 2 and the price of one orange be USD 4.

● In this case, the customer has to adjust his purchases such that the total cost adds up to the money he has on hand.

● Therefore, you can formulate a budget constraint equation as follows:

Price of one apple P(A) = USD 2

Price of one orange P(O) = USD 4

Price of 4 apples (4A) = 4 X 2 = USD 8

Price of 3 oranges (3O) = 3 X 4 = USD 12**Therefore, budget constraint = 4A + 3O = 8 + 12 = 20**

Thus, budget constraint is obtained by grouping the purchases such that the total cost equals the cash in hand. Hence, we can deduce a simple budget constraint formula as follows:

P(G1) X Q(G1) + P(G2 + Q(G2) = I

**P(G1) =** Price of one good**P(G2) =** Price of the other good**Q(G1) =** Quantity of one good**Q(G2) =** Quantity of the other good**I =** total income

A basic budget constraint graph is as mentioned:

● In the graph, the X and Y axes denote the goods, i.e., apples and oranges, respectively.

● Let the Y axis indicate oranges, and the X axis indicate apples.

● The point on the Y axis indicates that the entire income has been spent on oranges. Therefore, the point (0, 5) has been plotted, indicating that the total purchase is 5 oranges.

● Similarly, on the X axis, the point (10, 0) has been plotted, indicating that the total purchase is 10 apples.

● Therefore, mathematically, the slope of budget constraint is (-5 / 10), i.e., (-1 / 2). Technically, this means that one orange must be reduced to be able to buy 2 more apples.

## IMPORTANT TERMS

### Intertemporal Budget Constraint

● It is a term that requires an individual to curtail his expenditure within the available budget, over a long period of time.

● For an individual, his intertemporal budget constraint is the total spending limit he defines, i.e., the amount he may earn over a period plus the costs of the assets he owns.

● Analyzing an intertemporal budget constraint helps deduce future income and future expenditure.

### Example

● Mathematically, assume that over 2 separate time periods, incomes i1 and i2 are earned. The consumptions in these periods are c1 and c2, and the corresponding prices are p1 and p2.

● If an individual does not save any money, his income is going to be equal to his consumption, i.e., i1 = c1 and i2 = c2.

● Thus, the incomes can be directly plotted on the graphs, making it a basic budget constraint diagram.

● Now, assume that the consumer has had no expenses in the first period, thus, making his savings equal to his entire income. Let the interest rate be r.

● The total savings from period 1 will be (1+r)i1, which means that the total income for the second period is not just i2, but i2 + (1+r)i1.

● This will also be the total expenditure if the consumer spends it all, i.e., c2 = i2 + (1+r)i1.

● Thus, the graph will have a point (0, i2 +(1+r)i1) on the Y axis, if all the income from the first period is saved.

● Now, contrary to the first assumption, let the total expenditure in the first period be equal to the income. In this case, the consumer has to borrow.

● Assuming that the borrowed amount is b1, the value of i2 will be (1+r)b1. Therefore, the maximum expenditure that can take place will be equal to i1 + b1, i.e., i1 + i2/(1+r).

● The graph will now have a value (i1 + i2/(1+r),0) on the X axis, when the consumer borrows maximum amount for the first period.

● This graph represents an intertemporal budget constraint, depicting the expenditures and incomes when the consumer chooses to save or borrow.

### Utility Function and Marginal Utility

● A utility function helps derive alternate solutions, depending on utility.

● Marginal utility helps determine the quantity of a product, a customer will buy.

### Example

● In this scenario, the utility function can be mathematically represented as the (XY)1/2

● Consider the same example of apples and oranges given previously. Now, the budget constraint equation is P(G1) X Q(G1) + P(G2 + Q(G2) = I.

● Assume that the prices of both the goods are same, say USD 1. In this case, in order to obtain a basic budget line, you have to figure out the marginal utility.

● If the price of both the goods is the same, the marginal utility will be equal for both of them, indicating that we need to purchase the same quantity of both the goods.

● If, the price of one of the goods increases, the utility function will change, so will the marginal utility.

● Let the price of one orange increases by 3, thus, making it USD 4. In this case, the marginal utility will differ, i.e., the price of one orange is 4 times the price of one apple; therefore, for every orange purchased, we must purchase 4 apples.

● If the income is divided equally and spent completely on each of the goods, we can purchase 10 apples and 2.5 oranges.

● Using these values in the utility function, in the first case (before price changes), the function is (10 X 10) ^ 0.5, which gives us the value 10.

● In the second case, we have (10 X 2.5) ^ 0.5, which gives us the value 5; half of 10.

● Thus, the utility function in this case indicates that we cannot maintain the same budget and increase the price of the oranges; if we do, we have very less marginal utility. Therefore, to increase the utility, we need to increase the budget. This is called utility maximizing choice.

### Indifference Curve

● Indifference curves are obtained when, for a given number of choices, the value of the utility function is the same.

● For a utility function U = (XY)1/2, the value of the constant will change.

● This equation can be optimized further in order to obtain the consumer’s demand curves.

● The graph of the indifference curves shows a combination of goods that provides the customer with equal utility. Every point of the curve represents the same utility, and the customer cannot prefer another consumption bundle while being on the same curve.

### Example

● Use the equation from the graph in the previous example, i.e.,** PxX + PyY =** I and let the utility function be **U =** (XY)1/2.

● From the equation, we can write: **Y =** (I – Px) / Py. Substitute this in the utility function.

● Therefore, **U =** (XY)1/2

[(X)1/2] [((I – Px) / Py)1/2]

● Taking the first derivative of this term (use the product rule f(x) = uv), we obtain

(1/2)(X)-1/2 [((I – Px) / Py)1/2] + (1/2)(X)1/2 [((I – Px) / Py)-1/2] (-(Px) / (Py))

● Considering the utility function as zero, we obtain

(1/2) [((I – Px) / Py)1/2] [(X)1/2] = (Px) / 2(Py) [(X)1/2] [(X)1/2] [((I – Px) / Py)1/2]

● Solving this equation, we get X = I/2(Px) and Y = I/(Py)

● Using these values, we can obtain different demand curves, by changing the value of I.

## BUDGET CONSTRAINT AND CHANGING PRICES

● We have already seen that budget constraints vary with changing prices.

● When the cost of one product increases compared to the other, the budget constraint rotates in a clockwise direction.

● The change in either of the goods causes the budget curve to move closer to the origin, introducing new intercepts.

### Effect of changing prices on budget constraint

**Budget constraint with changes**

**Budget constraint with x intercept**

**Budget constraint with y intercept**

Budget constraint is a very important term to analyze consumer behavior. Based on the budget constraint graph, the consumer should choose the indifference curve that is tangential to their constraint. That is what will ensure maximum utility.