In the context of the student's choice diagram (Figure 3.10), where the horizontal axis represents days of free time and the vertical axis shows consumption, the Marginal Rate of Transformation (MRT) is defined by the slope of the budget constraint. This constraint is represented by the straight, downward-sloping line connecting the point of maximum consumption (0 days of free time, \$6,300) to the point of maximum free time (70 days, \$0 consumption). The constant slope of this line is -\$90, signifying that the opportunity cost of an additional day of free time is \$90. Consequently, the MRT is \$90.

Marginal Rate of Transformation (MRT) for the Student's Budget Constraint (Figure 3.10)

The Marginal Rate of Transformation (MRT) for a linear feasible frontier, such as one defined by the equation $y+z=200$, is calculated by finding the absolute value of its slope. By rearranging the equation into the slope-intercept form $y = 200 - z$, the slope is identified as -1. Consequently, the MRT is $|-1| = 1$. This value indicates that the two goods can be transformed or exchanged at a one-to-one rate.

Calculating MRT for a Linear Feasible Frontier (y + z = 200)

The Marginal Rate of Transformation (MRT) can be calculated precisely using calculus. If the feasible frontier is expressed as a function $c = g(t)$, where $c$ is consumption and $t$ is free time, the MRT is the absolute value of the slope of this function. Mathematically, it is given by the derivative of the function $g(t)$ with respect to $t$: $MRT = |g'(t)|$. This method is essential for analyzing economic models, especially those with non-linear frontiers where the trade-off rate varies.

MRT as the Derivative of the Feasible Frontier Function g(t)

In Angela's model, the Marginal Rate of Transformation (MRT) is a measure of the trade-off between her free time and grain production. At any given point on her feasible frontier, the MRT specifies the exact amount of grain she would have to forgo to gain one additional hour of free time. This value is determined by the absolute value of the slope of her feasible frontier at that point.

MRT for Angela's Trade-off between Free Time and Grain

Angela's optimal choice, which maximizes her utility, is at point A, corresponding to 16 hours of free time and 46 bushels of grain. At this point, the two critical trade-offs she faces are perfectly balanced. The trade-off she is subjectively willing to make between grain and free time, her Marginal Rate of Substitution (MRS), becomes equal to the objective trade-off she is constrained to make by her production technology, the Marginal Rate of Transformation (MRT). This condition (MRS = MRT) signifies the point of tangency between her feasible frontier and the highest possible indifference curve, as depicted in Figure 5.7.

Angela's Optimal Choice (Point A) where MRS = MRT

A key similarity between the Marginal Rate of Transformation (MRT) and the Marginal Rate of Substitution (MRS) is that both are conventionally expressed as positive numbers. This is an important clarification because they are derived from the slopes of the feasible frontier and indifference curves respectively, which are negative. The MRT, for instance, is precisely defined as the absolute value of the slope of the feasible frontier.

MRT and MRS as Positive Values

The Marginal Rate of Transformation (MRT) is a fundamental economic concept that maintains its core meaning across different applications. For instance, the MRT in an intertemporal choice model, which involves trading consumption between the future and the present, is conceptually identical to the MRT in production models, such as trading produced goods for free time. In all these contexts, it measures the quantity of one good that must be sacrificed to obtain one additional unit of another along the feasible frontier.

Conceptual Equivalence of MRT across Economic Models

In an intertemporal choice model, the Marginal Rate of Transformation (MRT) measures the trade-off of converting future consumption into present consumption. The value of the MRT is one plus the interest rate, `1 + r`. This formula signifies that for each unit of a good consumed now, `1 + r` units must be sacrificed from future consumption. The term `1 + r` also represents the absolute value of the slope of the intertemporal feasible frontier.

Marginal Rate of Transformation (MRT) in Intertemporal Choice

The Marginal Rate of Transformation (MRT) quantifies the trade-off along a feasible frontier, for instance, the trade-off between free time and grain. It is strictly defined as the absolute value of the slope of the feasible frontier. While the slope of the frontier is inherently negative (as more of one good requires sacrificing another), the MRT is expressed as a positive value that represents the rate of this exchange.

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The feasible frontier is the boundary of the feasible set, representing the maximum feasible quantity of one good for a given quantity of another. It is comprised of all the points that are on the edge of what is possible. In many economic models, such as one involving a budget, the budget constraint line serves as the feasible frontier.

Feasible Frontier

The geometric properties of the feasible frontier, specifically its downward slope and concavity, can be rigorously verified using calculus. The frontier is confirmed to be downward-sloping if its first derivative with respect to free time is negative. Furthermore, its curvature is determined by differentiating a second time. A negative second derivative with respect to free time proves that the frontier is strictly concave, a feature that typically reflects diminishing marginal returns.

Verification of Feasible Frontier Properties using Differentiation

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

Marginal Rate of Transformation (MRT)

In contrast to simple models with straight-line feasible frontiers (like a budget constraint from a constant wage), many economic models feature non-linear frontiers. In these more realistic scenarios, the boundary of the feasible set is a curve, which signifies a changing rate of trade-off between goods. This often occurs due to factors like declining marginal productivity, where an individual's hourly income is not fixed but decreases as they work more hours.

Non-Linear Feasible Frontiers

When a feasible frontier is a straight line, its slope is constant. Consequently, the Marginal Rate of Transformation (MRT)—the absolute value of the slope—is the same at every point along the frontier. This signifies that the trade-off an individual faces remains unchanged regardless of their position on the frontier. Karim's model in Unit 3 provides a clear example, as his straight-line feasible frontier results in a constant MRT.

MRT for a Straight-Line Feasible Frontier (Budget Constraint)

This diagram, identified as Figure 4.11 and reproduced as E4.1, illustrates how an altruistic Zoë chooses to distribute her £200 lottery winnings. It graphs 'money for Zoë' (horizontal axis) against 'money for Yvonne' (vertical axis). The feasible frontier, representing all possible splits, is a line from (200, 0) to (0, 200). Zoë's preferences are depicted by downward-sloping, convex indifference curves. Her optimal choice, which maximizes her utility, is at point A (140, 60), where the feasible frontier is tangent to the highest attainable indifference curve. This means she keeps £140 and gives £60 to Yvonne.

Figure 4.11 (reproduced as E4.1) - Zoë's Optimal Altruistic Choice

This diagram illustrates Julia's consumption choices, plotting 'consumption now' against 'consumption later' on axes scaled from 0 to 110 dollars. It displays a straight, downward-sloping feasible frontier from (0, 100) to (90, 0). Also shown is a convex, downward-sloping indifference curve that crosses the frontier at two points, C (12, 89) and E (58, 28), which represent feasible but suboptimal choices. A key feature of this indifference curve is that its slope is steeper at point C compared to point E. The diagram also includes a higher indifference curve that is tangent to the frontier at F (30, 60), indicating Julia's optimal consumption bundle.

Julia's Optimal and Suboptimal Choices on the Feasible Frontier

This figure demonstrates how the interest rate, which represents the price of borrowing, defines the boundary of the feasible set for consumption choices. It uses the specific case of Julia to illustrate how different interest rates (e.g., 10% and 78%) alter the feasible frontier, thereby showing how a higher interest rate contracts the set of available borrowing options.

Figure 9.3: Borrowing, the Interest Rate, and the Feasible Set

This diagram presents Julia's feasible frontier on a two-dimensional graph. The horizontal axis, labeled 'consumption now,' spans from 0 to 110 dollars, while the vertical axis, 'consumption later,' ranges from 0 to 180 dollars. Julia's initial endowment is located at the point (0, 100). Her feasible frontier is illustrated by a straight line that slopes downwards, connecting her endowment point with the point (83, 0) on the horizontal axis.

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