Suppose that individual A’s utility function is UA=4T+C and that individual B’s utility function is UB=2T+C, where T is pounds of tea and C is pounds of coffee. Suppose there are fixed amounts of 20 pounds of coffee and 20 pounds of tea. (a) Compute the marginal rate of substitution for each individual. (1 point) (b) Draw the Edgeworth Box with tea in the horizontal axis and coffee in the vertical axis. Put individual A in the bottom left corner and individual B in the top right corner. Draw individuals A and B indifference curves and identify the locus of all Pareto efficient allocations, including a
brief explanation. (2 points) (c) Suppose that the initial allocation is 10 pounds of coffee and 10 pounds of tea for each individual. Identify numerically and graphically the subset of Pareto efficient allocations that are feasible through trade given the initial allocation. (2 points) Please upload your handwritten answer below.

Question

Suppose that individual A’s utility function is UA=4T+C and that individual B’s utility function is UB=2T+C, where T is pounds of tea and C is pounds of coffee. Suppose there are fixed amounts of 20 pounds of coffee and 20 pounds of tea. (a) Compute the marginal rate of substitution for each individual. (1 point) (b) Draw the Edgeworth Box with tea in the horizontal axis and coffee in the vertical axis. Put individual A in the bottom left corner and individual B in the top right corner. Draw individuals A and B indifference curves and identify the locus of all Pareto efficient allocations, including a
brief explanation. (2 points) (c) Suppose that the initial allocation is 10 pounds of coffee and 10 pounds of tea for each individual. Identify numerically and graphically the subset of Pareto efficient allocations that are feasible through trade given the initial allocation. (2 points) Please upload your handwritten answer below.

BlackTom AI · Accepted Answer

We start by clearly restating what is given and what is being asked, even though there are no multiple-choice options. The problem provides two individuals with linear utility functions and a fixed total of two goods.

- Individual A has UA = 4T + C, where T is pounds of tea and C is pounds of coffee.
- Individual B has UB = 2T + C, with the same goods.
- The total endowments are T_total = 20 and C_total = 20, so any allocation is defined by A’s bundle (T_A, C_A) with B receiving (T_B, C_B) = (20 − T_A, 20 − C_A).

Part (a): Marginal rate of substitution (MRS) for each individual
- For A, compute marginal utilities: MU_T^A = ∂UA/∂T = 4 and MU_C^A = ∂UA/∂C = 1. The MRS of tea for coffee for A is MRS_TC^A = MU_T^A / MU_C^A = 4 / 1 = 4. This means A is willing to give up 4 units of coffee for an extra unit of tea, holding utility constant.
- For B, compute marginal utilities: MU_T^B = ∂UB/∂T = 2 and MU_C^B = ∂UB/∂C = 1. The MRS of tea for coffee for B is MRS_TC^B = MU_T^B / MU_C^B = 2 / 1 = 2. This means B is willing to give up 2 units of coffee for an extra unit of tea, holding utility constant.

Part (b): Draw the Edgeworth Box and the contract curve (in words, since we are not drawing here)
- The Edgeworth Box for two goods with 20 units of tea on the horizontal axis (T from 0 to 20) and 20 units of coffee on the vertical axis (C from 0 to 20) has the origin for A at the bottom-left corner (0,0) and B’s origin at the top-right corner (20,20). Feasible allocations for A are the points (T_A, C_A) within the square [0,20] × [0,20], with B’s bundle automatically determined as (20 − T_A, 20 − C_A).
- Indifference curves: Because both U functions are linear, each agent’s indifference curves are straight lines with constant slope in the (T, C) plane. For A, IC_A has slope −MU_T^A / MU_C^A = −4 (a steep downward-sloping line). For B, IC_B has slope −MU_T^B / MU_C^B = −2 (a less steep downward-sloping line). In plain terms, A values tea relatively more than coffee (kofier substitution) than B does.
- The Pareto efficient allocations (the contract curve) in an Edgeworth Box with linear (non-strictly convex) preferences are found where no unilateral small reallocation can improve one person’s utility without harming the other. Since A’s MRS is constantly 4 and B’s MRS is constantly 2, their MRS are not equal except in degenerate cases. With linear preferences, the contract curve tends to lie along the boundary where one agent is at a corner of their feasible consumption in order to prevent mutual improvements: i.e., at allocations where one good is exhausted for one agent or where one agent consumes the maximum or minimum possible of a good given the other’s consumption, effectively placing the allocation on the box edges. Concretely, the relevant Pareto-efficient locus consists of the portions of the edges of the Edgeworth Box where any small reallocation would require moving along a feasible path that worsens at least one agent’s utility because the fixed ratios make simultaneous improvements impossible across interior points.

Part (c): Starting from the initial allocation (T_A, C_A) = (10, 10) for A and (T_B, C_B) = (10, 10) for B, identify Pareto-efficient allocations reachable by trade
- Feasible reallocations must keep the total endowments fixed: T_A + T_B = 20 and C_A + C_B = 20.
- From the initial symmetric point (10,10) for A, notice that A’s MRS is 4 and B’s MRS is 2. Because their MRS differ, there exist trade directions that can improve one agent without harming the other only up to the boundary conditions where one agent’s bundle hits a corner (0 or 20) on at least one good. In other words, you can move along the feasible line in the direction that benefits the agent whose marginal rate of substitution makes them better off more quickly per unit of the traded good, until you hit a boundary (T_A = 0 or 20 or C_A = 0 or 20).
- Numerically, consider moving a small amount δ of tea from A to B while compensating with coffee to respect feasibility. The change in A’s utility is ΔU^A ≈ 4(ΔT_A) + (ΔC_A). If we move δ units of tea from A to B, then ΔT_A = −δ, and to keep the total constant we must adjust C_A accordingly; the net sign of ΔU^A will depend on how we adjust coffee. Because the problem is linear, the set of Pareto-efficient allocations reachable from (10,10) by trading until a boundary is reached is the line segment along which either T_A or C_A hits 0 or 20. Specifically, you can push A toward consuming more tea and less coffee until one of these goods is exhausted for A, or push A toward consuming less tea and more coffee until the other boundary is reached. The precise efficient set reachable from (10,10) is therefore the portion of the boundary of the box that can be attained without forcing both agents to move away from their preferred trade-off in a way that would allow a welfare-improving (for one) move for both.
- Graphically, this corresponds to the boundary rays starting from (10,10) that head toward the corners of the Edgeworth Box where A or B hits 0 or 20 units of a good. For example, moving toward increasing A’s tea and decreasing A’s coffee (i.e., heading toward T_A approaching 20 and/or C_A approaching 0) until you hit a corner, or the opposite direction toward T_A approaching 0 and/or C_A approaching 20, until you hit a corner. The resulting set is the feasible Pareto-efficient subset reachable by trade from the initial endowment, consisting of those allocations on the box boundary between the two corner endpoints determined by the direction of feasible, welfare-improving trades given the differing MRS values.

Notes and common points to keep in mind as you proceed:
- MRS_A = 4 and MRS_B = 2 are constant across all bundles due to linear utilities; this shapes the contract curve as non-curved and lying on or near the box edges in this two-good, two-agent setting.
- Pareto efficiency in this context often implies corner solutions or boundary segments, since equalizing MRS perfectly across interior points is not possible with these two constant MRS values.
- When you actually sketch or compute, you’ll mark A’s and B’s straight-line indifference curves with slopes −4 and −2, respectively, and then identify where feasible tangencies cannot be sharpened by any small move without harming someone, which will align with the boundary portions described above.

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