2 marks, −0.667MicroeconomicsBudget constraint and consumer equilibriumNon-standard budgets: bundles, rationing and quantity limitsHardabout 3 min

GATE 2021 Question 24: Budget constraint and consumer equilibrium

Trisha's consumption preference on biryani ( $x$ ) and pudding ( $y$ ) is given by the utility function $U(x,y)=x+4y$ . The price per unit of biryani is ₹2 and the price per unit of pudding is ₹3. Trisha's total income is ₹120. However, she faces an extra quantity constraint as she is not allowed to consume biryani more than 60 units and pudding more than 30 units. The optimum quantity of biryani and pudding consumed by Trisha is

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Answer and worked solution

Answer: (B)

With linear utility $U=x+4y$ , $MU_x=1$ and $MU_y=4$ (constants).

Compare marginal utility per rupee:

$\frac{MU_x}{P_x} = \frac12 = 0.5, \qquad \frac{MU_y}{P_y} = \frac43 \approx 1.33$

Since $\dfrac{MU_y}{P_y} > \dfrac{MU_x}{P_x}$ , pudding gives more utility per rupee spent — with perfect-substitutes preferences like this, Trisha wants to spend as much as possible on pudding first.

Without the quantity cap, she would spend her entire ₹120 income on pudding alone: $120/3=40$ units of $y$ . But the quantity constraint caps pudding at 30 units — so this cap binds before the budget does.

Applying the binding cap: buy the maximum allowed 30 units of pudding, costing $3\times30=90$ . Remaining income: $120-90=30$ . Spend the rest on biryani: $30/2=15$ units of $x$ — well under its own cap of 60, so the biryani cap is not binding.

Therefore the optimum bundle is $(x^*,y^*)=(15,30)$ , matching option (B).

The trap: (D)

(x^*,y^*)=(60,0)

Students who correctly identify that pudding is preferred per-rupee might jump straight to the 'corner solution' answer of spending everything on the OTHER good's cap ((D), all biryani) by confusing which good the linear utility function actually favors, or might forget to check the quantity cap at all and compute the unconstrained optimum (which would be all pudding, 40 units — not even listed as an option, since it violates the cap), missing that the specific binding constraint here is the quantity cap on pudding, not the plain budget line.

Plain-text version

Trisha's consumption preference on biryani (x) and pudding (y) is given by the utility function U(x,y)=x+4y. The price per unit of biryani is ₹2 and the price per unit of pudding is ₹3. Trisha's total income is ₹120. However, she faces an extra quantity constraint as she is not allowed to consume biryani more than 60 units and pudding more than 30 units. The optimum quantity of biryani and pudding consumed by Trisha is (A) (x*,y*)=(30,20) (B) (x*,y*)=(15,30) (C) (x*,y*)=(30,15) (D) (x*,y*)=(60,0)

Budget constraint and consumer equilibrium has come up 99 times in 7 exams since 2004.

What this question tests

This question is on Non-standard budgets: bundles, rationing and quantity limits, part of Budget constraint and consumer equilibrium in Microeconomics.

Budget sets with bundled offers, tie-ins, quantity caps, block prices or liquidity constraints.

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