How to Calculate the Marginal Rate of Substitution
The marginal rate of substitution equals MUx ÷ MUy, the units of good y a consumer will give up for one more unit of good x.
The MRS formula
Calculator
Enter both marginal utilities and both prices to get the MRS and compare it with the price ratio.
MUx, the extra utility from one more X. A sandwich worth 30 utils in the example.
MUy, the extra utility from one more Y. A juice worth 10 utils.
Sandwiches at $6 in the worked example.
Juice at $3 in the worked example.
The consumer will give up 3 units of good Y to get one more unit of good X, which is the absolute slope of the indifference curve.
- Price ratio (Px ÷ Py)
- 2
- Gap between MRS and the price ratio
- 1
- What to do next
- Buy more X
The market makes one more unit of X cost 2 units of Y, which is the absolute slope of the budget line.
The consumer values X 1 units of Y above what the market charges.
At the optimum MRS equals Px ÷ Py, where the indifference curve is tangent to the budget line.
How to calculate MRS, step by step
- 1Find the marginal utility of each good. MUx is the extra utility from one more unit of x, and MUy is the extra utility from one more unit of y.
- 2Divide MUx by MUy. MRS = MUx ÷ MUy, the units of y the consumer will trade away for one additional unit of x.
- 3Compare it to the price ratio. Set MRS against Px ÷ Py; when MRS is larger, x is worth more to the consumer than it costs, so buy more x.
- 4Adjust until they match. At the optimum MRS = Px ÷ Py, the point where the indifference curve is tangent to the budget line.
Worked example: MRS
A sandwich has MU = 30 utils and a juice has MU = 10 utils, so MRS = 30 ÷ 10 = 3, meaning the consumer will give up 3 juices for 1 sandwich. If sandwiches cost $6 and juice costs $3, the price ratio is 6 ÷ 3 = 2, so a sandwich only costs 2 juices. Since 3 is greater than 2, buying more sandwiches raises utility until the falling MU of sandwiches pulls MRS down to 2.
MRS questions
Why does MRS fall as you move along an indifference curve?
As the consumer gains more x and gives up y, MUx falls while MUy rises, so the ratio MUx ÷ MUy shrinks. That diminishing MRS is what makes indifference curves bow toward the origin.
What does MRS equal at the consumer's optimum?
At the optimum MRS equals the price ratio Px ÷ Py. That is the same condition as MUx ÷ Px = MUy ÷ Py, just rearranged.
Is MRS the same as the slope of the indifference curve?
MRS is the absolute value of that slope. The slope itself is negative, because gaining one good requires giving up some of the other.
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