This figure presents a table that summarizes the trade-offs Karim faces between consumption and free time. It likely displays and compares the values for his Marginal Rate of Substitution (MRS) and the Marginal Rate of Transformation (MRT) at various potential choices, illustrating the data behind his decision-making process.

Figure 3.8 - Summary of Karim's Trade-Offs

This diagram visualizes Karim's optimal choice at a €30 wage. The horizontal axis represents daily hours of free time, ranging from 8 to 24, while the vertical axis shows daily consumption in euros, ranging from 0 to 600. The feasible frontier is depicted as a straight line connecting the points (8, 480) and (24, 0). Karim's optimal choice is shown at point E (17, 210), where a convex indifference curve is tangent to this feasible frontier, representing the highest utility he can achieve.

Figure 3.7a - Diagram of Karim's Optimal Choice at a €30 Wage

To determine the precise quantities of each good in an optimal bundle, a system of simultaneous equations must be solved. This system is composed of two key conditions: first, the optimality condition where the Marginal Rate of Substitution equals the Marginal Rate of Transformation (MRS = MRT), and second, the constraint that the chosen bundle must lie on the feasible frontier (e.g., $y+z=200$). Solving these two equations together allows for the calculation of the exact values for each variable (e.g., y and z) in the utility-maximizing outcome.

Solving for the Optimal Choice Using a System of Simultaneous Equations

A common critique of the economic model of choice is that it may not be a realistic description of how people actually make decisions. The model, which involves concepts like indifference curves and equating the MRS to the MRT, serves as a theoretical framework but likely does not reflect the conscious thought processes individuals use in their everyday choices.

The Realism of the Economic Model of Choice

With a €30 hourly wage, Karim's optimal choice without the influence of others' consumption is at Point E. This utility-maximizing bundle provides 17 hours of free time and €210 in consumption. It is located at the tangency point of his initial indifference curve (IC1) and the budget constraint. In comparative analysis with Veblen-influenced preferences, this original indifference curve is noted to be steeper than the new Veblen curves, reflecting a different marginal rate of substitution.

Karim's Optimal Choice at Point E (Assuming No Veblen Effect)

In a household's work-leisure choice model, the optimal allocation of resources is achieved at the point where the household's subjective valuation of the trade-off between consumption and non-working time (the Marginal Rate of Substitution, or MRS) is equal to the objective market trade-off, which is represented by the wage rate.

The Household's Optimality Condition (MRS = Wage)

An allocation is defined as Pareto efficient if the Marginal Rate of Substitution (MRS) equals the Marginal Rate of Transformation (MRT). This equality indicates that the subjective trade-offs of the agents involved are perfectly aligned with the objective, feasible trade-offs, ensuring that it is impossible to reallocate resources to make someone better off without making at least one other person worse off. This principle holds as a general rule across different economic models.

The Pareto Efficiency Condition (MRS = MRT)

The optimal choice for an individual is found where their subjective trade-off (MRS) equals the objective trade-off (MRT). This well-known optimality rule, expressed as the equation `MRS = MRT`, represents the first-order condition for a constrained optimization problem. This condition can be derived directly from calculus, where the MRS and MRT are represented as the derivatives of the utility and feasible frontier functions, respectively (e.g., $v'(t)$ and $g'(t)$). Equating them ensures that the highest possible utility is achieved given the constraints.

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Economic models of choice assume that individuals select the best possible option from their feasible set of combinations of goods. To model this, an individual's preferences are described using indifference curves, while their constraints are represented by a feasible set or budget constraint. The optimal, utility-maximizing choice is found at the tangency point between the feasible frontier (the boundary of the feasible set) and the highest attainable indifference curve. At this point, the subjective trade-off rate (Marginal Rate of Substitution, MRS) is equal to the objective trade-off rate (Marginal Rate of Transformation, MRT).

Optimal Choice as a Balance Between Two Trade-Offs

A fundamental principle that applies to any utility function is that the Marginal Rate of Substitution (MRS) is equivalent to the ratio of the marginal utilities of the two goods in question. For two goods, this relationship is expressed by a formula where the MRS equals the marginal utility of the good on the horizontal axis divided by the marginal utility of the good on the vertical axis. This principle holds true universally across different types of utility functions.

Marginal Rate of Substitution as the Ratio of Marginal Utilities

In the economic model of work-leisure choice with a constant wage, the Marginal Rate of Transformation (MRT) represents the objective market trade-off between consumption and free time. It is defined as the absolute value of the slope of the feasible frontier (the budget constraint). This value is constant and equal to the wage rate ($w$).

Marginal Rate of Transformation (MRT) as the Wage Rate (w)

In the context of solving a constrained choice problem with calculus, the Marginal Rate of Substitution (MRS) can be represented by the derivative of the utility component associated with a specific good. For instance, if utility from free time $t$ is given by a function $v(t)$, the MRS can be defined as $MRS = v'(t)$. This formulation is used alongside the derivative for the MRT to establish the first-order condition for optimization, $MRS = MRT$.

MRS as a Derivative of a Utility Component Function

For a constrained optimization problem defined with general functions, such as a utility component $v$ and a feasible frontier $g$, the first-order condition provides a universally applicable rule for identifying potential optima. This general formulation serves as the foundational principle from which more specific applications, like the MRS = MRT rule, are derived.

General Form of the First-Order Condition

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The Optimality Condition (MRS = MRT)

This example illustrates a standard constrained choice problem. An individual named Zoë has a fixed budget of £240 to spend on two goods: cinema tickets, which cost £10 each, and nights out, which cost an average of £16 each. Her objective is to maximize her satisfaction by acquiring as many of both goods as she can, but her purchasing power is limited by her budget constraint.

Zoë's Consumer Choice Problem with a Fixed Budget

This example introduces Alexei, a student, who faces a constrained choice problem. His decision revolves around how to allocate his time between studying and free time. The final grade he receives is dependent on the number of hours he studies, creating a trade-off between achieving a higher grade and enjoying more leisure time.

Alexei's Choice Between Study Hours and Final Grade

Mathematical solutions to constrained choice problems, such as calculus-based approaches using the substitution method or the MRS = MRT condition, offer more precision than graphical analysis. These general methods are robust and can be applied to solve optimization problems for various types of utility functions, not limited to just quasi-linear ones. For a more detailed mathematical explanation of these techniques, Sections 8.1 to 8.3 of the textbook 'Mathematics for Economists: An Introductory Textbook' by Pemberton and Rau are a recommended resource.

Mathematical Methods for Solving Constrained Choice Problems

An alternative to algebraic solutions, the graphical method solves a constrained choice problem by visually identifying the optimal bundle. The process involves finding the specific point of tangency where the budget constraint touches the highest possible indifference curve. This tangency point represents the best combination of goods for the individual, as it is the feasible bundle that provides the maximum utility.

Determining the Optimal Choice via the Graphical (Tangency) Method

The Marginal Rate of Substitution (MRS) for Karim is derived from his utility function, $u=(t-6)^2(c-45)$, by applying the principle that the MRS equals the ratio of the marginal utilities of free time and consumption. This calculus-based method, which involves finding the partial derivatives for each good and then their ratio, yields a specific formula for his MRS. Notably, the resulting formula is identical to the one obtained through alternative methods, such as directly calculating the slope from the indifference curve equation, confirming the consistency of the two approaches.

Karim's Marginal Rate of Substitution (MRS)

To calculate the Marginal Rate of Substitution (MRS) from a given utility function, a two-step method is used. First, the marginal utility for each good is determined by finding the partial derivative of the utility function with respect to that good. Second, these marginal utilities are substituted into the formula that defines the MRS as the ratio of the marginal utilities.

Method for Calculating the MRS from a Utility Function

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, $t$.

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