Question

Mr. Krinsky is a wealthy lawyer who knows how to live well. No matter what his disposable income is, he will always devote at least $$\$ 30,000$$ to consumption each year. In addition, once his disposable income rises above $$\$ 40,000$$ he saves half of the portion above $$\$ 40,000$$ and spends the. rest on various luxuries. a) Draw Mr. Krinsky's consumption function. b) Over what range does dissaving occur, and over what range does saving occur? c) Describe the marginal propensity to consume over the course of the graph.

Short answer

The consumption function for Mr. Krinsky is a piecewise function, where: \[ C(Y)= \begin{cases} \$ 30,000, & \text{if } Y \le \$ 40,000 \\ \$ 30,000 + 0.5 * (Y - \$ 40,000), & \text{if } Y > \$ 40,000 \end{cases} \] Dissaving occurs when Y is in the range (0, $40,000], while saving occurs when Y is in the range ($40,000, +∞). The marginal propensity to consume (MPC) is 0 for Y ∈ (0, $40,000] and 0.5 for Y ∈ ($40,000, +∞).

Step-by-step solution

Understand Mr. Krinsky's consumption habits

Mr. Krinsky has a few rules when it comes to spending his disposable income: 1. He will always spend at least $30,000 on consumption each year. 2. If his disposable income is above \(40,000, he will save half of the amount above \)40,000 and spend the rest on various luxuries.

Define the consumption function

We'll now define the consumption function (C) using a piecewise function based on the information given: 1. If disposable income (Y) ≤ \(40,000: C(Y) = \)30,000, as he spends all his disposable income on consumption. 2. If disposable income (Y) > \(40,000: C(Y) = \)30,000 + 0.5 * (Y - \(40,000), as he saves half of the income above \)40,000. Thus, the consumption function C(Y) can be represented as: \[ C(Y)= \begin{cases} \$ 30,000, & \text{if } Y \le \$ 40,000 \\ \$ 30,000 + 0.5 * (Y - \$ 40,000), & \text{if } Y > \$ 40,000 \end{cases} \] #b) Over what range does dissaving occur, and over what range does saving occur?#

Identify dissaving and saving ranges

Dissaving occurs when consumption is greater than disposable income, which means that Mr. Krinsky is spending more than he earns. Saving happens when consumption is less than disposable income, implying that Mr. Krinsky spends less than he earns. Dissaving range: If Y ≤ \(40,000, Mr. Krinsky spends all of his income on consumption, so dissaving occurs when Y is in the range of (0, \)40,000]. Saving range: If Y > \(40,000, Mr. Krinsky starts saving half of the portion of his income above \)40,000. Therefore, saving occurs when Y is in the range of ($40,000, +∞). #c) Describe the marginal propensity to consume over the course of the graph.#

Describe the marginal propensity to consume (MPC)

The marginal propensity to consume (MPC) is the slope of the consumption function, reflecting the change in consumption resulting from a change in disposable income: 1. When Y ≤ \(40,000, C(Y) = \)30,000, which means that consumption remains constant at $30,000, so the MPC is 0. 2. When Y > \(40,000, C(Y) = \)30,000 + 0.5 * (Y - $40,000). In this case, the MPC = 0.5, as Mr. Krinsky spends half of the additional income on consumption. Therefore, the marginal propensity to consume varies over the course of the graph, being 0 (horizontal line) for Y ∈ (0, \(40,000], and 0.5 (upward sloping line) for Y ∈ (\)40,000, +∞).

Key concepts

Marginal Propensity to Consume

The Marginal Propensity to Consume (MPC) is a key economic concept that measures the proportion of an additional unit of disposable income that a consumer will spend on consumption, rather than saving it. This is relevant because it helps us understand consumer behavior and how changes in income levels can impact consumption patterns.

For example, if an individual has an MPC of 0.8, it means that for every additional dollar of income they earn, they will spend 80 cents and save the remaining 20 cents. Understanding MPC is essential for analyzing how households respond to changes in income and subsequently how these responses can affect economic growth.

In the given exercise, the consumption function demonstrates that initially, Mr. Krinsky does not change his consumption with changes in disposable income up to \(40,000, indicating an MPC of 0. However, once his income surpasses \)40,000, he increases consumption by 50 cents for every dollar earned above this threshold, showing an MPC of 0.5.

Dissaving

Dissaving occurs when a person's consumption expenditure exceeds their income. It represents a negative saving rate and typically implies that an individual is depleting their savings or incurring debt to finance current consumption.

In the context of the exercise, Mr. Krinsky's consumption function shows that dissaving happens when his disposable income is less than or equal to \(40,000. During this range, his consumption remains constant at \)30,000, regardless of income level. It means that even if he earns less than \(30,000, he will still spend \)30,000 on consumption necessitating the use of savings or borrowing.

Saving Range

The saving range is the portion of disposable income that is not spent on consumption but is instead set aside as savings. It's an important component of an individual or household’s financial health. Savings can serve as a buffer against economic uncertainty and as resources for future investments.

In Mr. Krinsky's case, the saving range is realized when his disposable income exceeds \(40,000. Beyond this income level, he starts to save half of every additional dollar. The saving range is therefore any income above \)40,000, as he's spending less than his total income, leading to a positive savings rate.

Piecewise Function

A piecewise function is a function that is defined by different expressions based on different intervals of the independent variable. It's like a mathematical tool that allows for the description of situations where a rule changes based on the input value.

In the exercise, Mr. Krinsky’s consumption function is modeled as a piecewise function with two parts. For disposable income (Y) less than or equal to \(40,000, the consumption is a constant \)30,000. For income levels above \(40,000, consumption increases at a rate of 50% of any income above this threshold. This piecewise nature of the function reflects the real-world scenario where Mr. Krinsky’s spending habits change at the pivotal income level of \)40,000.