Question

Suppose that John's \(\mathrm{MPC}\) is constant at \(4 / 5\). If he had no income at all, he would have to borrow $$\$ 2,000$$ to meet all his expenses. Graph John's consumption function and write it out algebraically. Using the formula for John's consumption function, find his break-even point.

Short answer

John's consumption function is \(C = 2000 + \frac{4}{5}Y\). By graphing this function with Income (Y) on the horizontal axis and Consumption (C) on the vertical axis, we can visualize John's spending behavior. To find the break-even point, set \(C = Y\), which yields the equation: $$ Y = 2000 + \frac{4}{5}Y $$ Solving for \(Y\) gives us a break-even income level of \(Y = 10,000\). At this income level, John's consumption will also be $10,000, and he will not need to borrow any more money to meet his expenses.

Step-by-step solution

Determine Autonomous Consumption

Since John would have to borrow \(2,000 if he had no income, we can assume that his autonomous consumption, \)a\(, is equal to \)2,000.

Write the Consumption Function

John's consumption function can be written as: $$C = 2000 + \frac{4}{5}Y$$

Graph the Consumption Function

To graph John's consumption function, create a coordinate plane with Income (Y) on the horizontal axis and Consumption (C) on the vertical axis. 1. Plot the point (0, 2000) on the graph, which represents John's autonomous consumption. 2. Use the slope (MPC) of \(4/5\) to identify another point on the graph. For example, if John has an income of \(5,000, his consumption would be: \)C = 2000 + \frac{4}{5}(5000) = 6000$. 3. Connect the points on the graph with a straight line to represent John's consumption function.

Find the Break-even Point

The break-even point occurs when John's consumption equals his income. We can set \(C = Y\) and solve for \(Y\): $$ Y = 2000 + \frac{4}{5}Y $$ Subtract \(\frac{4}{5}Y\) on both sides to isolate \(Y\) term: $$ Y - \frac{4}{5}Y = 2000 $$ Combine like terms and solve for \(Y\): $$ \frac{1}{5}Y = 2000 $$ Multiply both sides by 5 to get the break-even income level: $$Y = 10,000 $$ John's break-even point occurs when his income is \(10,000. When John earns this amount, his consumption will also be \)10,000, and he will not need to borrow any more money to meet his expenses.

Key concepts

Marginal Propensity to Consume (MPC)

Let's dive into understanding the marginal propensity to consume, or MPC for short. MPC is an economic term that measures the proportion of additional income that a person or household is likely to spend on goods and services instead of saving it. In simpler terms, if you get an extra dollar, MPC tells us how much of that dollar you would typically spend. In John's case, we know his MPC is \(\frac{4}{5}\) or 0.8. This means that John spends 80% of any additional income he earns while saving the remaining 20%.

If John's income increases by \(1,000\), he will spend \(800\) on consumption and save \(200\). This relationship is crucial for understanding consumer behavior and economic policies because it influences how factors such as income changes affect overall spending in an economy.

• Higher MPC indicates a tendency to spend more of each additional dollar.
• Lower MPC suggests more saving and less spending of extra income.

Autonomous Consumption

Autonomous consumption is the amount of consumption that occurs regardless of income levels. It's the spending necessary to satisfy needs when income falls to zero. For John, this is the \(2,000\) he needs to borrow to cover his expenses when he has no earned income at all. Autonomous consumption signifies the baseline spending needed for essentials such as food, shelter, and clothing.

Even if John's earnings are \(0\), the need to consume basic necessities remains, which is why autonomous consumption comes into play. In the context of a consumption function, autonomous consumption is denoted by the constant in the equation: \(C = 2000 + \frac{4}{5}Y\). Here, "2000" serves as the autonomous consumption, establishing the starting point of the consumption line on a graph.

• Illustrates the consumption behavior even when income is absent.
• Represents non-discretionary spending.

Break-even Point

The break-even point represents the level of income where total consumption equals total income without needing to borrow or lend money. For John, determining this point is quite straightforward. At the break-even point, he spends exactly what he earns, balancing his budget.

Mathematically, it occurs when the consumption function equals the income, i.e., setting \(C = Y\). By solving the equation \(Y = 2000 + \frac{4}{5}Y\), we found that John's break-even income is \(10,000\). At this level of income, John's earnings are perfectly aligned with his expenditures, which means he meets his needs without incurring debt.

• It's a vital measure for financial stability — income covers expenses.
• Helps in budgeting and ensuring self-sufficiency.

Understanding the break-even point aids in financial planning and insights into economic health.