When comparing Angela's optimization problem as a tenant to that of an independent farmer, the core components of her decision-making remain constant, provided she has quasi-linear preferences. A fixed rent payment only lowers her consumption and does not alter her Marginal Rate of Substitution (MRS), which depends solely on free time. Similarly, the Marginal Rate of Transformation (MRT), or the slope of the feasible frontier, is also unchanged because the frontier merely shifts downward in parallel. Since both the MRS and MRT are identical in both scenarios, the optimal condition (MRS = MRT) is met at the same point, leading to an identical choice of free time and work hours.

Angela's Optimization Problem as a Tenant vs. an Independent Farmer

A key consequence of Angela's quasi-linear preferences is that her optimal choice of free time and grain production is independent of any fixed rent payment ($c_0$). The reason for this is that paying a fixed rent, which lowers her consumption, does not affect her Marginal Rate of Substitution (MRS) between consumption and free time. Due to her quasi-linear utility, the MRS depends only on her amount of free time, not her level of consumption. Since the Marginal Rate of Transformation (MRT) also remains unchanged, her optimal choice of hours—where MRS equals MRT—is identical to her choice as an independent farmer. Her production level is therefore the same, although her final consumption is reduced by the rent.

Independence of Optimal Work Hours and Production from Rent Due to Quasi-Linear Preferences

With quasi-linear preferences, it becomes possible to quantify the difference in utility between two different allocations using the units of the 'linear' good (in this case, grain). Because the Marginal Rate of Substitution is independent of grain, the vertical distance between any two indifference curves remains constant. This unique property allows for the preference for a bundle on a higher curve over a bundle on a lower one to be expressed as a concrete quantity of that good. For instance, stating that one allocation is preferred over another by an amount equivalent to 17 bushels of grain is a valid utility comparison under this assumption.

Measuring Utility Differences with Quasi-Linear Preferences

Figure 5.3b (also referred to as Figure E5.3) is a diagram that visually illustrates the quasi-linearity of Angela's preferences for free time ($t$) and grain consumption ($c$). As stated by the figure's caption, Angela's Marginal Rate of Substitution (MRS) is determined by her amount of free time but is independent of the amount of grain she possesses. This property is shown by the parallel tangents to her indifference curves at any given level of free time, such as at $t=18$ hours. The identical slope of these tangents confirms that her MRS remains constant at 18 hours of free time, irrespective of her grain consumption.

Figure 5.3b/E5.3 - Illustration of Quasi-Linear Preferences

A key implication of the quasi-linear utility function, $u(x, m) = v(x) + m$, is that the Marginal Rate of Substitution (MRS) between the linear good (income, $m$) and the non-linear good ($x$) depends exclusively on the quantity of $x$. This occurs because the marginal utility of income is 1, causing the MRS to simplify to the marginal utility of good $x$, which is $v'(x)$. Graphically, this means that for any given quantity of $x$, the slope of all indifference curves is identical.

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Quasi-linear preferences represent a specific preference structure that serves as a valuable analytical simplification, despite being a restrictive assumption. Their primary utility in economic modeling is that they permit the measurement of benefits and costs (utility) in monetary terms. This structure is defined by an individual's desire for one good being independent of the quantity of another good they possess. Graphically, this results in indifference curves that are vertical translations of each other, meaning their slopes are identical for any given amount of the good on the horizontal axis. Consequently, the Marginal Rate of Substitution (MRS) is determined solely by the quantity of the good on the horizontal axis.

Quasi-linear Preferences

The Marginal Rate of Substitution (MRS) for a quasi-linear utility function, $u(t, c) = v(t) + c$, is given by the formula $v'(t)$. This result can be derived in two primary ways. The first method uses the ratio of marginal utilities: since the marginal utility of free time is $\frac{\partial u}{\partial t} = v'(t)$ and the marginal utility of consumption is $\frac{\partial u}{\partial c} = 1$, the MRS is $\frac{v'(t)}{1} = v'(t)$. Alternatively, one can start from the indifference curve equation, $v(t) + c = u_0$. By rearranging to express consumption as a function of time, $c = u_0 - v(t)$, and then differentiating with respect to $t$, the slope of the indifference curve is found to be $\frac{dc}{dt} = -v'(t)$. The MRS, which is the absolute value of this slope, is therefore also $v'(t)$. Both methods confirm that for quasi-linear preferences, the MRS depends only on the amount of free time, $t$.

Derivation of the MRS for a Quasi-Linear Utility Function

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CORE Econ - The Economy 2.0: Microeconomics

This example illustrates a specific type of preference where an increase in non-labor income does not lead to a change in the amount of free time chosen. For a student with these preferences, depicted in Figure 3.12, the Marginal Rate of Substitution (MRS) is identical at every level of free time across different indifference curves. Consequently, this student would maintain their original amount of leisure and allocate the entire \$1,000 gift towards increased consumption.

Zero Income Effect on Free Time (Figure 3.12)

The general mathematical expression for a quasi-linear utility function is $u(x, m) = v(x) + m$. In this form, $x$ represents a particular good (or a 'bad' if it negatively impacts utility), and $m$ represents an individual's income that is spent on other goods. The function is termed 'quasi-linear' because it is linear with respect to income ($m$) but typically non-linear concerning good $x$. This functional form has the important property that the marginal utility of good $x$ is independent of income. For the preferences to be considered well-behaved, $v(x)$ must be an increasing function, so its first derivative must be positive ($v'(x) > 0$), and it is generally assumed to be concave.

General Form of a Quasi-Linear Utility Function

Indifference curves derived from a quasi-linear utility function are convex if they exhibit a diminishing Marginal Rate of Substitution (MRS), which means they become flatter as one moves to the right along the curve. Since the MRS for such preferences is given by the formula $v'(t)$, the condition for convexity is that $v'(t)$ must be a decreasing function of $t$. In other words, as the quantity of good $t$ increases, the value of the first derivative of its utility component, $v'(t)$, must fall.

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