Source: Practice MCQs for Exam 1, Texas A&M University
Tags: budget constraint, budget line slope, utility maximisation, MRS equals price ratio, Cobb-Douglas optimisation, perfect complements optimisation, perfect substitutes corner solution, income change parallel shift, price doubling budget line
The budget line shows what a consumer can afford. Its slope is the negative ratio of prices. Utility is maximised where the indifference curve is tangent to the budget line, which gives the condition MRS = P_x/P_y. For perfect complements (min functions), you use the constraint x = y. For Cobb-Douglas utility, there is a clean formula based on income shares. Special cases like constant MRS (perfect substitutes) produce corner solutions.
Budget constraint / budget line
The set of all bundles the consumer can just afford. For two goods X and Y: P_x · X + P_y · Y = I, where I is income.
Slope of the budget line
Equal to -(P_x / P_y) when X is on the horizontal axis and Y is on the vertical axis. It represents the rate at which the market allows you to trade X for Y.
Utility maximisation condition (interior solution)
MRS_xy = P_x / P_y. The consumer's willingness to trade (MRS) equals the market's rate of trade (price ratio). This is the tangency condition.
Corner solution
The consumer spends all income on one good only. This occurs with perfect substitutes when the MRS does not equal the price ratio at any interior point.
For two goods, apples (X-axis) and oranges (Y-axis), with prices P_a and P_o:
Slope = -(P_a / P_o)
Example: P_apple = $2, P_orange = $1.20.
Slope = -(2 / 1.20) = -(5/3)
The slope tells you how many oranges the consumer must give up to buy one more apple at market prices.
If income increases and prices stay the same, the budget line shifts outward in a parallel fashion. The slope does not change because the slope depends only on the price ratio, which has not changed.
If both P_A and P_B double, the new slope is -(2P_A / 2P_B) = -(P_A / P_B). The slope does not change. However, the budget line shifts inward (the consumer can afford less). In real terms, this is equivalent to income falling by half.
If income and all prices increase by the same percentage, the budget line does not move at all and the utility-maximising bundle stays the same.
For u(x, y) = x^a · y^b, the optimal quantities are:
x* = (a / (a + b)) · (I / P_x)
y* = (b / (a + b)) · (I / P_y)
The consumer spends fraction a/(a+b) of income on x and fraction b/(a+b) on y.
Worked example: u(x, y) = x⁴y⁵, P_x = 2, P_y = 1, I = 180.
x* = (4/9) × (180/2) = (4/9) × 90 = 40
y* = (5/9) × (180/1) = (5/9) × 180 = 100
For u(x, y) = min{x, y}, the consumer always buys x = y. Substitute into the budget constraint:
P_x · x + P_y · x = I
x(P_x + P_y) = I
x = I / (P_x + P_y)
Worked example: P_x = 1, P_y = 2, I = 60.
x = 60 / (1 + 2) = 60 / 3 = 20
For a constant MRS, compare MRS to the price ratio:
If MRS > P_x/P_y, buy only X (X gives more utility per dollar).
If MRS < P_x/P_y, buy only Y.
If MRS = P_x/P_y, any combination on the budget line is optimal.
Worked example: Sunny's MRS of coffee for tea = 4/3 (she gives up 3 teas for 4 coffees, so MRS = 4/3). If tea and coffee have the same price, P_c/P_t = 1.
Since MRS (4/3) > price ratio (1), coffee gives more bang per pound. Sunny buys only coffee.
At each point along a price consumption curve, utility is maximised and all income is spent. The price of one good varies while income and the other price are held fixed. Each point is a full optimum.
Budget constraint:
P_x · X + P_y · Y = I
Slope of budget line (X horizontal, Y vertical):
Slope = -(P_x / P_y)
Cobb-Douglas optimum:
x* = [a / (a + b)] · (I / P_x)
y* = [b / (a + b)] · (I / P_y)
Perfect complements optimum (min{x, y}):
x* = y* = I / (P_x + P_y)
⚠️ The slope of the budget line is -(P_x / P_y), not -(P_y / P_x). Check which good is on which axis.
⚠️ When both prices double, the slope does not change. When income rises with no price change, the line shifts out but the slope is unchanged.
⚠️ If income and all prices rise by the same percentage, the optimal bundle stays the same. Purchasing power has not changed.
⚠️ For Cobb-Douglas, the income shares a/(a+b) and b/(a+b) are the key. Memorise the formula or derive it from the tangency condition.
⚠️ For perfect complements, set x = y (or whatever the fixed-proportion ratio is) and plug into the budget constraint.
⚠️ For perfect substitutes, compare MRS to the price ratio to determine which corner solution applies.
⚠️ Along the price consumption curve, both conditions hold: utility is maximised, and all income is spent.
Q: Lindie has income $10, P_apple = $2, P_orange = $1.20. What is the slope of her budget constraint (apples on horizontal axis)?
A: -5/3. Slope = -(P_apple / P_orange) = -(2 / 1.2) = -5/3.
Q: Hudson's utility is u(x, y) = min{x, y}. P_x = $1, P_y = $2, I = $60. How much x does he buy?
A: 20. Set x = y. Then 1(x) + 2(x) = 60, so 3x = 60, x = 20.
Q: Huxley's utility is u(x, y) = x⁴y⁵. P_x = $2, P_y = $1, I = $180. How much x does he buy?
A: 40. x* = (4/9)(180/2) = (4/9)(90) = 40.
Q: Both prices of goods A and B double. What happens to the slope of the budget line?
A: The slope does not change. -(2P_A / 2P_B) = -(P_A / P_B), which is the same as before.
Q: Michael's income increases, prices unchanged. How does his budget line change?
A: It makes a parallel shift outward. The slope stays the same because prices have not changed.
Q: Sunny has a constant MRS of 4/3 (coffee for tea). Tea and coffee cost the same. What does she buy?
A: Only coffee. Her MRS (4/3) exceeds the price ratio (1), so coffee provides more utility per dollar spent.
Q: What is true at each point along a price consumption curve?
A: Utility is maximised and all income is spent.
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