Source: Intermediate Microeconomics, Problem Set 2, Texas A&M University
Tags: Cobb-Douglas utility, marginal utility, MU, optimal choice, utility maximisation, tangency condition, MRS equals price ratio, consumer optimality, newspapers and arcade games, ECON 323
A Cobb-Douglas utility function (U = X^a · Y^b) produces well-behaved preferences with a clean optimality rule: spend a fixed fraction of income on each good, determined by the exponents. At the optimum, the ratio of marginal utilities equals the ratio of prices, which is the tangency condition between the indifference curve and the budget line.
Cobb-Douglas utility function
A utility function of the form U = X^a · Y^b, where a and b are positive constants (often summing to 1). It produces smooth, convex indifference curves and guarantees an interior solution.
Marginal utility (MU)
The additional utility gained from consuming one more unit of a good, holding the other good constant. For U = X^a · Y^b: MU_X = a · X^(a−1) · Y^b and MU_Y = b · X^a · Y^(b−1).
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to trade one good for another while staying on the same indifference curve. MRS = MU_X / MU_Y. For Cobb-Douglas: MRS = (a/b) · (Y/X).
Tangency condition (optimality condition)
At the utility-maximising bundle, the MRS equals the price ratio: MU_X / MU_Y = P_X / P_Y. Equivalently, MU_X / P_X = MU_Y / P_Y (equal marginal utility per pound/dollar spent).
Spending share rule (Cobb-Douglas shortcut)
For U = X^a · Y^b, the consumer spends fraction a/(a+b) of income on X and fraction b/(a+b) on Y. This is unique to Cobb-Douglas and makes solving much faster.
For Cobb-Douglas utility U = N^(1/4) · G^(3/4), the exponents are a = 1/4 and b = 3/4, and a + b = 1.
The spending shares are:
Spend on N: (1/4) / 1 = 1/4 of income
Spend on G: (3/4) / 1 = 3/4 of income
With income M = $10:
Spending on N = (1/4)(10) = $2.50
Spending on G = (3/4)(10) = $7.50
At P_N = $0.50: quantity of N = 2.50 / 0.50 = 5
At P_G = $0.25: quantity of G = 7.50 / 0.25 = 30
The optimal bundle is N = 5, G = 30 (option a in the problem).
Always verify: P_N · N + P_G · G = (0.50)(5) + (0.25)(30) = 2.50 + 7.50 = 10.00. This equals income, so the bundle is on the budget line. Good.
For U = N^(1/4) · G^(3/4):
MU_N = (1/4) · N^(−3/4) · G^(3/4)
MU_G = (3/4) · N^(1/4) · G^(−1/4)
The ratio MU_N / MU_G simplifies:
MU_N / MU_G = [(1/4) · N^(−3/4) · G^(3/4)] / [(3/4) · N^(1/4) · G^(−1/4)]
= (1/4) / (3/4) · G / N
= (1/3) · (G / N)
At the optimum (N = 5, G = 30):
MU_N / MU_G = (1/3)(30/5) = (1/3)(6) = 2
This should equal the price ratio P_N / P_G = 0.50 / 0.25 = 2. It does, confirming the tangency condition holds.
The consumer maximises utility when:
MU_N / MU_G = P_N / P_G
Or equivalently:
MU_N / P_N = MU_G / P_G
This says: at the optimum, the last dollar spent on newspapers yields exactly the same marginal utility as the last dollar spent on arcade games. If it didn't, the consumer could do better by reallocating spending toward the good with the higher marginal utility per dollar.
Cobb-Douglas utility:
U = X^a · Y^b
Marginal utilities:
MU_X = a · X^(a−1) · Y^b
MU_Y = b · X^a · Y^(b−1)
MRS for Cobb-Douglas:
MRS = (a / b) · (Y / X)
Optimality (tangency) condition:
MU_X / MU_Y = P_X / P_Y
or equivalently: MU_X / P_X = MU_Y / P_Y
Cobb-Douglas spending shares:
Expenditure on X = [a / (a + b)] · M
Expenditure on Y = [b / (a + b)] · M
⚠️ The spending share shortcut only works for Cobb-Douglas utility. Do not apply it to other functional forms (quasi-linear, perfect complements, etc.).
⚠️ When computing MU_N / MU_G, most of the terms cancel. Write it out algebraically before plugging in numbers to avoid arithmetic errors.
⚠️ On a multiple-choice question, check each candidate bundle against the budget constraint first. Any bundle that doesn't satisfy P_N · N + P_G · G = M can be eliminated immediately. In the problem above, option (a) is the only one consistent with the spending shares, but checking the budget constraint is a fast first filter.
⚠️ The two forms of the optimality condition (MRS = price ratio, and equal MU-per-dollar) are equivalent. Know both. Exam questions may ask you to "state the condition" in one form or the other.
Q: For U = N^(1/4) · G^(3/4) with P_N = $0.50, P_G = $0.25, and M = $10, what fraction of income is spent on each good?
A: One-quarter of income ($2.50) on newspapers, three-quarters ($7.50) on arcade games. This comes directly from the exponents summing to 1.
Q: What does MU_N / MU_G = 2 mean in plain language?
A: At the optimum, the consumer's marginal utility from one more newspaper is twice the marginal utility from one more arcade game. This makes sense because newspapers are twice as expensive as arcade games ($0.50 vs $0.25), so the consumer needs twice the marginal utility from a newspaper to justify the higher price.
Q: Why must MU_X / P_X = MU_Y / P_Y at the optimum?
A: If the marginal utility per dollar were higher for one good, the consumer could increase total utility by shifting a dollar of spending from the lower-return good to the higher-return good. At the optimum, no such reallocation can improve utility, so the ratios must be equal.
Q: A student claims the optimal bundle is N = 10, G = 20. Without computing utility, how can you tell this is wrong?
A: Check the spending share. The exponent on N is 1/4, so spending on N should be 1/4 of income = $2.50. At P_N = $0.50, that means N = 5, not 10. Alternatively, spending on N at (10, 20) is $5.00 and on G is $5.00, a 50/50 split, which contradicts the 1/4 to 3/4 ratio implied by the exponents.
Cobb-Douglas preferences, utility maximisation, consumer optimum, marginal utility ratio, price ratio, tangency condition, equimarginal principle, equal bang-per-buck, interior solution, spending shares, budget allocation, MRS equals price ratio