Question

Suppose that John's MPC is constant at \(3 / 4\). If his breakeven point occurs at $$\$ 7,000$$, how much will John have to borrow when his income is $$\$ 3,000 ?$$

Short answer

John will have to borrow $1000 when his income is $3,000.

Step-by-step solution

Understand the MPC formula

The marginal propensity to consume (MPC) is the change in consumption for a change in income. Mathematically, the MPC formula is: \(MPC = \frac{dC}{dI}\) where C is consumption, I is income, and d represents the change in the variable. In this exercise, the MPC is constant at \(3/4\).

Determine the consumption function

Since MPC is constant in this case, we can integrate the MPC formula with respect to income, I, to find the consumption function: \(C = \int MPCdI\) Plug in the value of MPC: \(C = \int (3/4)dI\) Now integrate with respect to I: \(C = \frac{3}{4}I + k\) where k is the constant of integration.

Find the constant of integration using the breakeven point

At the breakeven point, consumption is equal to income. We know that the breakeven point occurs when income is $7,000. Therefore, we have: \(7000 = \frac{3}{4}(7000) + k\) Now solve for k: \(k = (1 - \frac{3}{4})(7000)\) \(k = \frac{1}{4}(7000)\) \(k = 1750\) Now we have the complete consumption function with the constant k: \(C = \frac{3}{4}I + 1750\)

Determine John's consumption at an income of $3,000

Now we use the consumption function to find John's consumption when his income is $3,000: \(C = \frac{3}{4}(3000) + 1750\) \(C = 2250 + 1750\) \(C = 4000\) John's consumption will be \(4,000 when his income is \)3,000.

Calculate how much John needs to borrow

Finally, we find the difference between John's consumption and his income to determine how much he needs to borrow: Amount to borrow = Consumption - Income Amount to borrow = \(4000 - \)3000 Amount to borrow = $1000 When his income is \(3,000, John will have to borrow \)1,000.

Key concepts

Consumption Function

The consumption function is an essential economic concept that describes the relationship between income and consumption. It determines how changes in income levels affect spending behavior. In the context of this exercise, using the Marginal Propensity to Consume (MPC), we ascertain the consumption function formula through integration.

With a constant MPC of \( \frac{3}{4} \), we integrate this over income \( I \) to arrive at a consumption function: \

• Start with the MPC formula: \(C = \int MPCdI\).
• Substitute the constant MPC: \(C = \int \frac{3}{4}dI\).
• The integration yields: \(C = \frac{3}{4}I + k\), where \(k\) is the constant of integration.

This function plays a pivotal role by allowing us to find the specific relationship governing the consumption behavior of individuals, in this case, John, as his income changes.

Breakeven Point

The breakeven point is a critical threshold in economics, where income is equivalent to expenditure. At this point, there is neither a deficit nor a surplus in the budget. Understanding the breakeven point is crucial for budgeting and financial planning.

In our exercise, John's breakeven income is given as \(7,000. This means that at an income of \)7,000, John's consumption is exactly $7,000 as well. Here’s how it influences the consumption calculation:

• Use the general consumption function: \(C = \frac{3}{4}I + k\).
• At the breakeven point, set \(I = 7000\), thus \(C = 7000\).
• Equate to find \(k\): \(7000 = \frac{3}{4}(7000) + k\).
• Solving gives \(k = 1750\).

The breakeven point is used to find the "constant of integration," which in turn allows the determination of exact consumption levels at different incomes.

Constant of Integration

The constant of integration \(k\) comes into play when resolving indefinite integrals, such as the integration of the MPC to form the consumption function. In economics, it helps to align the function to specific real-world conditions, like the breakeven point.

From the exercise: once we determine that \(C = \frac{3}{4}I + k\), the breakeven condition provides the needed bridge to solve for \(k\). By having \(C = I\) at $7,000:

• We set the equation \(7000 = \frac{3}{4}(7000) + k\).
• Simplifying gives \(k = 1750\).

This constant \(k = 1750\) completes the consumption function: \(C = \frac{3}{4}I + 1750\). It helps to accurately reflect John's actual spending needs at varying income levels and accounts for factors not captured solely by the MPC.