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
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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
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 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.

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