How to equation of a budget line?

The budget line, also known as the budget constraint, exhibits all the combinations of two commodities that a customer can manage to afford at the provided market prices and within the particular earning degree.

The budget line is a graphical delineation of all possible combinations of the two commodities that can be bought with provided income and cost so that the price of each of these combinations is equivalent to the monetary earnings of the customer.

It is important to keep in mind that the slope of the budget line is equivalent to the ratio of the cost of two commodities. The slope of the budget constraint possesses distinctive importance.

In other words, the slope of the budget line can be described as a straight line that bends downwards and includes all the potential combinations of the two commodities which a customer can purchase at market value by assigning his/her entire salary. The concept of the budget line is different from the Indifference curve, though both are necessary for consumer equilibrium.

The two basic elements of a budget line are as follows:

Read link: Deriving a Demand Curve from Indifference Curves and Budget Constraints

To understand the concept of a budget line in a detailed manner, it is important to understand the mentioned equation. The equation of the budget line equation can be represented as follows:

M = Px × Qx + Py × Qy

Where,

Px is the cost of product X.

Qx is the quantity of product X.

Py is the cost of product Y.

Qy is the quantity of product Y.

M is the consumer’s income.

Additional Reading: What is the Government Budget?

Radha has ₹50 to buy a biscuit. She has a few options to allocate her income so that she receives maximum utility from a limited salary.

To get an appropriate budget line, the budget schedule given can be outlined on a graph.

The budget set indicates that the combinations of the two commodities are placed within the affordability margin of a consumer.

Also, read: What is a Budget Set?

Some of the properties of the budget line are as follows:

Negative slope: If the line is downward, it shows a reverse correlation between the two products.

Straight line: It indicates a continuous market rate of exchange in individual combinations.

Real income line: It denotes the income and the spending size of a customer.

Tangent to indifference curve: It is the point when the indifference curve meets the budget line. This point is known as the consumer’s equilibrium.

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The budget line is mostly based on the assumption and not reality. However, to get clear and precise results and summary, the economist considers the following points in terms of a budget line:

Two commodities: The economist assumes that the customers spend their income to purchase only two products.

Income of the customers: The income of the customer is limited, and it is designated to buy only two products.

Market price: The cost of each commodity is known to the customer.

Expense is similar to income: It is assumed that the customer spends and consumes the whole income.

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A budget line includes a consumer’s earnings and the rate of a commodity. These are the two important factors that shift the budget line.

Shift due to change in price: The amount of the product either increases or decreases from time to time. For instance, if the price and income of product A remains constant and the price of product B decreases, then the buying potential of product B automatically increases. Similarly, if the price of B increases and the other factors remain steady, the demand for product B automatically decreases.

Shift due to change in income: Change in income makes a huge difference that leads to a change in the budget line. High income means high purchasing possibility and low income means low purchasing potential, making the budget line to shift.

The preference-indifference pattern of the consumer based on the axioms of preferences. The indifference map of the consumer shows that the points lying on any particular IC give the consumer the same level of utility. It also shows that the points lying on a higher IC give the consumer a higher level of utility than the points on a lower IC.

Therefore, the points on the former curve are strictly preferred to the points on the latter curve. The consumer would always want to climb up on a higher IC because then he would be able to obtain a higher level of utility. But the real world consumer is constrained by his income or budget.

His income may not permit him to climb on the ICs beyond a certain height. The consumer’s income and the prices of the goods together give us the budget constraint of the consumer, and his budget is as important as his preference-indifference pattern in the determination of his behaviour in the market.

While discussing the budget of the consumer, for the present, assume the following:

(i) The prices of the two goods (X and Y) have been determined in the market, and they are given and constant for the consumer.

(ii) The consumer has a fixed amount of money income to spend on the two goods.

(iii) Both the goods are MIBs.

(iv) The consumer wants to maximise utility.

(v) He spends all his money on the goods because of (iii) and (iv).

(vi) He can purchase any combination of the goods if his budget permits, i.e., there are no (absolutely) maximum limits on the purchase of the goods.

The above assumptions would lead to the properties of the budget equation and the budget line of the consumer.

Suppose that the consumer’s money income is denoted by M and the prices of the goods are given to be px and py.

Then the equation M = px. x + py . y ……(6.14)

is known as his budget equation.

As is evident from (6.14), the budget equation is linear in

M̅, x and y, px and py being given and constant. Now, assume that the consumer’s money income is fixed at M, then from his budget equation (6.14), is known as his budget line.

The equation of the budget line would be:

M̅ = px.x + py.y …..(6.15)

Since M̅, px and py are constants, (6.15) is a linear equation in x and y, i.e., it is the equation of a straight line in a two dimensional commodity space. The budget line gives us the combinations of x and y that the consumer can purchase with his fixed money income M.

It may be noted that equation (6.14) represents a family of parallel budget lines. Because, at each value of M, a separate budget line from this equation and the slope of each such budget line, is −px/py = constant (px and py remain constant).

The equation of the budget line is:

M̅ = px.x + py.y [eq. (6.15)]

The features of this line are:

(i) It is a straight line in the commodity space, since it is a linear equation in x and y.

(ii) Its slope is negative being equal to − px/py (... px, py > 0).

Therefore, the numerical slope of the budget line is px / py which is equal to the ratio of the prices of X and Y. Since the numerical slope of the line represents the price ratio, or, the relative price of good X in terms of good Y, this line is also called the price line.

(iii) The intercept form of the budget line is:

This gives the x-intercept of the budget line is M̅ / px which is equal to the quantity of good X that the consumer would be able to buy if he spends all his money income (M) on X. Similarly, the y-intercept of the budget line is M / py which is equal to the quantity of good Y that the consumer would be able to buy with all his money income.

(iv) Since M̅, px and py are given, can immediately know the intercepts M̅/px and M̅/ py, and is also know the meeting points of the budget line with the x-axis and y- axis. These points would be (M̅/px, 0) and (0, M̅/py) respectively.

By joining these two points at once have the budget line of the consumer. It may be noted here that, at the point (M̅/px, 0), the consumer would buy only good X, and the point (0, M̅/py), he would by only good Y. At all other points on the line he would buy a combination of the goods.

The concept of the budget line can be illustrated with the help of a simple example. Suppose M̅ = Rs 500, px =  Rs 10 and py = Rs 5. Then, putting these values in (6.15), the budget line of the consumer as 500 = 10x + 5y (6.17)

The x-intercept of the budget line (6.17) = M̅/px = 500/10 = 50 is obtained, i.e., the consumer would be able to buy 50 units of good X with all his income; and the y-intercept of the line = M̅/py = 500/5 = 100, i.e., the consumer would be able to buy 100 units of good Y with all his income. Therefore, (50, 0) and (0, 100) are the two combinations that lie on the line which has been shown in Fig. 6.6. Some other combinations along with these two have been shown in the Table 6.1.

Table 6.1 Some of the Combinations of Goods that Lie on the Budget Line in Fig. 6.6