Consider an economy that produces two goods, x and y. The production possibilities frontier (PPF) for this economy is linear (i.e. a straight line) and the current production point is x=75 and y=100. The marginal rate of transformation (MRTxy) is constant and equal to 4 for the entire PPF. The economy also has an economy-wide indifference curve that is bowed in to the origin. You know that the marginal rate of substitution (MRSxy) is equal to
y

x
Error parsing MathML: error on line 1 at column 70: Opening and ending tag mismatch: br line 1 and mi
What is the optimal production value of x?

Question

Consider an economy that produces two goods, x and y. The production possibilities frontier (PPF) for this economy is linear (i.e. a straight line) and the current production point is x=75 and y=100. The marginal rate of transformation (MRTxy) is constant and equal to 4 for the entire PPF. The economy also has an economy-wide indifference curve that is bowed in to the origin. You know that the marginal rate of substitution (MRSxy) is equal to 
y

x
Error parsing MathML: error on line 1 at column 70: Opening and ending tag mismatch: br line 1 and mi
 What is the optimal production value of x?

BlackTom AI · Accepted Answer

We start by restating the setup and identifying the key relationships.
The economy produces two goods, x and y, with a linear PPF. The current production point is x = 75 and y = 100. The MRTxy is constant at 4 along the PPF, which means the PPF slope represents the amount of y that must be given up to gain one more unit of x, and since the PPF is drawn with x on the horizontal axis, dy/dx = -MRTxy = -4. A linear PPF with slope -4 has equation y = b − 4x, where b is the y-intercept.
Next, we determine b from the current production point: 100 = b − 4(75) ⇒ 100 = b − 300 ⇒ b = 400. Thus the PPF is y = 400 − 4x.
We also have an economy-wide indifference curve that is bowed in to the origin, which is the typical convex-to-origin shape associated with utility functions like U = x y, for which MRSxy = MUx/MUy = (∂U/∂x)/(∂U/∂y) = y/x. The optimal (efficient) production point on the PPF occurs where the slope of the PPF equals the slope of the indifference curve, i.e., MRTxy = MRSxy.
Now set MRTxy equal to MRSxy: 4 = MRSxy = y/x. This implies y = 4x.
Substitute this into the PPF equation to find the tangency (optimal) x: 4x = 400 − 4x ⇒ 8x = 400 ⇒ x = 50.
Thus the optimal production value of x is 50. If you want the corresponding y, you can plug back into the PPF: y = 400 − 4(50) = 400 − 200 = 200, giving the tangency point (x, y) = (50, 200).
In summary, the key steps are: identify the PPF equation from the current point and MRT, impose the tangency condition MRT = MRS with MRS = y/x for a bowed-in indifference curve, solve y = 4x along with y = 400 − 4x, and obtain x = 50.
Note: if the indifference curve had a different functional form, MRSxy would have a different expression, but the requirement MRTxy = MRSxy would still pin down the optimal x; the given bowed-in shape commonly implies MRSxy = y/x, as used here.

类似问题

Specialization and international trade will

How can an increase in economic growth be represented on a Production Possibilities Frontier (PPF)?

Which of the following is NOT an assumption of production possibilities frontier (PPF)?

Use the following production possibilities frontiers to answer the next question. Curve (a) is the current frontier for the economy. Focusing on curve (a), point N suggests that the economy currently produces

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单