**3.3 Indifference curves**

Suppose that an individual consumes \$100 worth of consumption goods and 2 hours of leisure a day (point A in **Figure 1**). Combining the \$100 worth of consumption and 2 hours of leisure give the individual 60 units of happiness. If the individual only consumes \$60 worth of consumption goods and 5 hours of leisure (point B in **Figure 1**), the individual receives the same level of happiness.

Suppose that the individual is still consuming \$100 worth of consumer goods. At the same time, he/she spends 5 hours on leisure instead of 2 hours (point C in **Figure 1**). Combining the \$100 worth of consumption and 5 hours of leisure give the individual 90 units of happiness. The indifference curves are the graphical representation of the utility function.<sup>1</sup>

#### **3.4 The budget constraint**

An individual's income consist of two parts: earned income and unearned income. **Earned income** is the money you make in exchange for the work you do. For most people, almost all the money they make is earned income. Any money earned in professional wages or fees—including tips—counts as earned income. Reimbursements from your employer for travel expenses, including meals, accommodations,

<sup>1</sup> Indifference curves have five important properties: 1. indifference curves are strictly downward sloping; 2. indifference curves are convex to the origin; 3. indifference curves never cross; 4. higher indifference curves mean higher levels of utility; 5. indifference curves are continuous, with no gaps.

and transportation, also count as earned income. Unearned income involves the money you make without having performed a professional service. **Unearned income** includes money-making sources that involve interest, dividends, and capital gains. Additional forms of unearned income include retirement account distributions, annuities, unemployment compensation, social security benefits, and gambling winnings. Other forms of income, such as money from an estate, trust, or partnership, may also be considered unearned income [2].

Let *K* denote an individual's unearned income. Let *L* be the number of hours the individual works and *w* be the hourly wage rate. *C* is the quantity of the consumption goods. The individual's budget constraint can be written as

$$\mathbf{C} = \mathbf{w}L + K \tag{2}$$

Eq. (2) means the expenditures on the consumption goods equals to the sum of earned income and unearned income.

Since the total time the individual can allocate a day is *T* and the individual can only allocate his/her time between two activities: *work (L) and leisure (I)*, we have *T* ¼ *L* þ *I*. Thus, *L* ¼ *T* � *I*. We can then rewrite the budget constraint as

$$C = w(T - I) + K \tag{3}$$

Rearrange Eq. (3)

$$\mathbf{C} = (wT + K) - wI \tag{4}$$

**Figure 2** shows the consumption-leisure budget line. If the individual chooses to spend all of his/her time on leisure, the individual's expenditure on the consumption goods is *K*, which is his/her unearned income (point D in **Figure 2**). If the individual chooses to spend all of his/her time on work, the individual's expenditure on the consumption goods is *wT* þ *K*, which is the sum of his/her earned income and unearned income (point E in **Figure 2**). Moreover, the slope of the consumptionleisure budget line is �*w*.

**Figure 2.** *Consumption-leisure budget line.*

### **3.5 Worker's optimal choice**

The budget line shows the combinations of the consumption goods and the hours of leisure that the individual can afford. Notice that the individual can choose any combination of the consumption goods and leisure hours on the budget line or in the area below the budget line. We want to determine which combination within the budget gives the individual the highest level of utility. **Figure 3** illustrates the solution to this problem.

Combinations that lie on indifference curves above the budget line, such as point E, are not in the budget set. Even though the individual prefers point E to point A or B, he/she cannot afford the combination of the consumption goods and the hours of leisure at point E.

Although the individual can afford any combination inside the budget line, such as point B, he/she can always find a better affordable combination such as point D. Thus, the individual will not choose any combination of the consumption goods and leisure hours in the area below the budget line.

Points A and C lie on the budget line as point D. However, the indifference curve that passes through point D is higher than the indifference curve that pass through points A and C. Therefore, the combination of the consumption goods and leisure hours at point D gives the individual higher level of utility (higher indifference curves mean higher levels of utility). In fact, the point where an indifference curve is tangent to the budget line is the affordable combination of the consumption goods and leisure hours that gives the individual the highest level of utility.

#### **3.6 Substitution and income effects**

**Figure 3** demonstrates the optimal combination of the consumption goods and leisure hours an individual will choose given the hourly wage rage at *w*. What happens if the wage rate increases? Would the individual choose more work and less leisure because he/she can earn more per hour? Or would the individual choose less

**Figure 3.** *Worker's optimal choice of the combination of the consumption goods and leisure hours.*

work and more leisure because he/she has a higher income to maintain the same standard of living by working less?


Both the substitution and income effects are present for all individuals. When the substitution effect dominates the income effect, a higher wage reduces leisure hours and, therefore, increases work hours. On the other hand, when the income effect dominates the substitution effect, higher wage increases the hours of leisure and, therefore, decreases work hours.

