Source: Intermediate Microeconomics, Problem Set 2, Texas A&M University
Tags: marginal rate of substitution, MRS, marginal rate of transformation, MRT, income effect, substitution effect, Slutsky decomposition, Hicks decomposition, normal good, inferior good, Giffen good, budget constraint, price increase, ECON 323
The MRS measures how much of one good a consumer is willing to give up for another; the MRT measures the rate at which the market allows them to trade. At the optimum these are equal. When a price changes, the total effect on quantity demanded can be decomposed into a substitution effect (movement along the indifference curve) and an income effect (shift to a new indifference curve), which reveals whether a good is normal, inferior, or Giffen.
Marginal rate of substitution (|MRS| of C for F)
The absolute value of the slope of the indifference curve at a given bundle. It tells you how many units of C the consumer would willingly give up to get one more unit of F, staying at the same utility. At the optimum, |MRS| = P_F / P_C.
Marginal rate of transformation (MRT of C for F)
The rate at which the market lets you convert good C into good F. It equals the price ratio P_F / P_C. Graphically, it is the absolute value of the slope of the budget line.
Substitution effect
The change in quantity demanded due purely to the change in relative prices, holding utility constant. Always moves opposite to the price change (if F gets more expensive, the substitution effect reduces F). Shown graphically by sliding along the original indifference curve to a point tangent to a hypothetical budget line with the new price ratio.
Income effect
The change in quantity demanded due to the change in purchasing power caused by the price change. The direction depends on whether the good is normal or inferior.
Normal good
A good for which the income effect reinforces the substitution effect. When income (or purchasing power) falls, you buy less of it. Both effects push consumption in the same direction after a price increase.
Inferior good
A good for which the income effect works against the substitution effect. When purchasing power falls, you buy more of it (perhaps because you can no longer afford the preferred alternative).
Giffen good
An extreme case of an inferior good where the income effect is so large it overwhelms the substitution effect, causing quantity demanded to rise when the price rises. Very rare in practice.
Sarah has $400, with P_C = $40 and P_F = $20.
Budget constraint equation:
40C + 20F = 400
Divide through by 20 for a cleaner form:
2C + F = 20, or equivalently, C = 10 − (1/2)F
Intercepts (C on the vertical axis, F on the horizontal axis):
Vertical intercept (F = 0): C = 400 / 40 = 10
Horizontal intercept (C = 0): F = 400 / 20 = 20
The optimal bundle is C = 4, F = 12. Check: 40(4) + 20(12) = 160 + 240 = 400. Confirmed.
At the optimum, the indifference curve is tangent to the budget line. So:
|MRS| = P_F / P_C = 20 / 40 = 1/2
This means Sarah is willing to give up 1/2 unit of C to get one more unit of F, and the market requires exactly that trade-off. No better deal is available.
Note on direction: because C is on the vertical axis and F is on the horizontal axis, the slope of the budget line is −P_F / P_C = −1/2. The |MRS| is the absolute value of this slope.
The MRT is the absolute value of the budget line's slope:
MRT = P_F / P_C = 20 / 40 = 1/2
At the optimum, |MRS| = MRT. This is exactly the tangency condition, restated.
New budget constraint:
40C + 40F = 400, which simplifies to C + F = 10
New intercepts:
Vertical intercept (F = 0): C = 400 / 40 = 10 (unchanged, because C's price didn't change)
Horizontal intercept (C = 0): F = 400 / 40 = 10 (was 20, now 10)
The budget line pivots inward around the vertical intercept. The feasible set shrinks. The new optimal bundle is C = 6, F = 4.
The total change in F: from 12 down to 4, a decrease of 8 units.
To decompose this graphically (Hicks method):
Step 1: Draw the compensated budget line. This is a hypothetical budget line with the new price ratio (slope = −P_F / P_C = −40/40 = −1) but shifted outward until it is tangent to the original indifference curve. The point of tangency is the "decomposition bundle."
Step 2: Identify the substitution effect. The substitution effect is the movement from the original optimum (C = 4, F = 12) along the original indifference curve to the decomposition bundle. Because F became relatively more expensive, the substitution effect reduces F and increases C.
Step 3: Identify the income effect. The income effect is the movement from the decomposition bundle to the new optimum (C = 6, F = 4). This is a parallel shift of the budget line (same slope, lower purchasing power).
The problem states you do not need exact numeric values for the decomposition bundle, only the graphical intuition.
Look at the income effect alone. The price of F rose, which reduced Sarah's real purchasing power (like a drop in real income). In response:
Sarah's consumption of F fell from 12 to 4 overall
The substitution effect definitely reduces F (it always works against the price increase)
The income effect also reduces F (from the decomposition bundle to F = 4)
Since both effects reduce F when the price rises, F is a normal good. When purchasing power falls, Sarah consumes less F, not more.
For a price increase in good F:
Normal good: Income effect reduces F (reinforces substitution effect). Total effect is a decrease in F.
Inferior good (non-Giffen): Income effect increases F (works against substitution effect), but the substitution effect is larger. Total effect is still a decrease in F.
Giffen good: Income effect increases F and overwhelms the substitution effect. Total effect is an increase in F when price rises.
Budget constraint (general):
P_C · C + P_F · F = M
Slope of budget line (C on vertical, F on horizontal):
Slope = −P_F / P_C
At the optimum:
|MRS| = MRT = P_F / P_C
Total effect = Substitution effect + Income effect
Substitution effect: along the original indifference curve, from old bundle to compensated bundle
Income effect: from compensated bundle to new optimum (parallel shift)
⚠️ When the question says "|MRS| of C for F," it is asking: how much C would you give up per unit of F? This equals P_F / P_C, not P_C / P_F. The "of X for Y" phrasing means "how much X you sacrifice per unit of Y gained."
⚠️ The MRT always equals the price ratio, regardless of the optimum. It describes the market trade-off, not the consumer's preferences. The MRS describes the consumer's willingness to trade.
⚠️ For the income/substitution decomposition, the compensated budget line has the new slope but is tangent to the old indifference curve. A common error is making it tangent to the new indifference curve instead.
⚠️ To determine if a good is normal or inferior, look only at the direction of the income effect, not the total effect. The substitution effect always goes against the price change, so it doesn't help you classify the good.
⚠️ The bonus question asks you to construct a hypothetical scenario where F is inferior. To do this, pick a new optimum after the price rise where the consumption of F is higher than the decomposition bundle (income effect increases F) but still lower than the original 12 (substitution effect dominates). For a Giffen good, the new F would be higher than 12.
Q: Sarah has $400 with P_C = $40 and P_F = $20. Write her budget constraint and find the intercepts with C on the vertical axis.
A: 40C + 20F = 400. Vertical intercept: C = 10. Horizontal intercept: F = 20.
Q: At Sarah's optimum (C = 4, F = 12), what is |MRS| of C for F, and what does the number mean?
A: |MRS| = P_F / P_C = 20/40 = 0.5. It means Sarah is willing to give up 0.5 units of C to obtain one additional unit of F, and the market requires exactly that exchange rate.
Q: If P_F rises from $20 to $40, which intercept of the budget line changes?
A: Only the horizontal intercept (F-axis) changes, falling from 20 to 10. The vertical intercept stays at 10 because P_C is unchanged.
Q: The substitution effect of a price increase in F always reduces F. True or false?
A: True. The substitution effect always moves consumption away from the good that became relatively more expensive. This holds regardless of whether the good is normal, inferior, or Giffen.
Q: How do you tell whether a good is normal or inferior from a graph showing income and substitution effects?
A: Look at the income effect arrow for that good. If the income effect moves consumption in the same direction as the substitution effect, the good is normal. If it moves in the opposite direction, the good is inferior. If the income effect is large enough to reverse the total direction, the good is Giffen.
MRS, marginal rate of substitution, MRT, marginal rate of transformation, tangency condition, Hicks decomposition, Slutsky decomposition, compensated budget line, compensating variation, substitution effect, income effect, normal good, inferior good, Giffen good, price change, real income, purchasing power, consumer theory, indifference curve analysis