1. Chapter 4 Individual Demands
Intermediate Microeconomics:
A Tool-Building Approach
Routledge, UK
© 2016 Samiran Banerjee

2. Preference maximization
A consumer’s demand for goods is found by maximizing her preferences on budget sets.
• Keep income, m, fixed
• Keep prices of goods, p1 and p2, fixed
• Keep preferences (i.e., utility function) fixed
• Maximizing preferences on this budget means to find the bundle (or bundles) that yield the highest utility
Given by utility function

3. At B, the budget line is tangent to the indifference curve
Interior solution
• Pick any commodity bundle A on the budget line
• Draw the indifference curve that passes through A
• Find a commodity bundle B on the budget whose utility is highest
• B is an interior bundle because x1 > 0 and x2 > 0 – –
At B, the budget line is tangent to the indifference curve p1 p2 =
MRS

4. Corner solution
• Sometimes the commodity bundle on the budget whose utility is highest is a corner solution
• It depends on the shape of the indifference curves
• A is a corner bundle because x1 > 0 and x2 = 0
• B is a corner bundle because x1 = 0 and x2 > 0 – – p1 p2 ≥
MRS† p1 p2 ≤
MRS*
(* In general; is < here)
(† In general; is = here)

5. Demands for u = 2x1 + x2 Linear Case 1: p1/p2 > MRS
• Utility-maximizing bundle is at A
• Demand function for x1, h1 = 0
• Demand function for x2, h2 = m/p2
Linear
Demand functions generally depend on all or some combination of income m, and prices, p1 and p2
Corner solution

6. Two corner, and infinitely many interior solutions!
Demands for u = 2x1 + x2
Case 2: p1/p2 = MRS
• Utility-maximizing bundle is any bundle on the budget
• Demand for x1 and x2 is any bundle (h1, h2) which satisfies p1h1 + p2h2 = m, e.g., C = (2, 6)
• There is no unique demand bundle!
Two corner, and infinitely many interior solutions!

7. Demands for u = 2x1 + x2 Case 3: p1/p2 < MRS
• Utility-maximizing bundle is at E
• Demand for x1, h1 = m/p1
• Demand for x2, h2 = 0
Corner solution

8. Demand functions for Leontief preferences depend on all: m, p1, and p2
Demands for u = min{x1, x2}
• The utility-maximizing bundle is where the diagonal meets the budget line, at A
• Solve x2 = x1 and p1x1 + p2x2 = m together
• h1 = m/(p1 + p2) and h2 = m/(p1 + p2)
Leontief
Demand functions for Leontief preferences depend on all: m, p1, and p2

9. Algorithm for interior solutions
At the interior solution B, the indifference curve is tangent to the budget line Step 1: Set MRS = MU1/MU2 equal to p1/p2 Step 2: Obtain an expression for either x1 or x2 Step 3: Substitute the expression in the budget equation, p1x1 + p2x2 = m to obtain one demand Step 4: Substitute from Step 3 into the expression in Step 2 to obtain the other demand

10. Demands for u = x1x2 Cobb- Douglas
Step 1: Set MRS = x2/x1 = p1/p2 Step 2: Obtain the expression, x2 = p1x1/p2 Step 3: Substitute x2 = p1x1/p2 into the budget equation, p1x1 + p2x2 = m. Solve for x1: h1 = m/(2p1) Step 4: Substitute h1 = m/(2p1) for x1 into Step 2. Solve for x2: h2 = m/(2p2)
The demand for this Cobb-Douglas utility lies on the midpoint of the budget line
The demand for x1 does not depend on p2, just as the demand for x2 does not depend on p1

11. Demands for u = 2√x1 + x2 Quasilinear
• Solve for the interior solution, A Step 1: Set MRS = 1/√x1 = p1/p2 Step 2: Obtain the demand, h1 = (p2/p1)2 Step 3: Substitute h2 for x2 into the budget equation Solve for x1: h2 = (mp1 – p22)/(p1p2)
Quasilinear
Note that h1 is always positive, but h2 is positive only when mp1 > p22

12. Condition for a corner solution
Demands for u = 2√x1 + x2
• Solve for the corner solution • From the interior solution, h2 is zero or negative for mp1 ≤ p22 • Since negative consumption is not possible, h2 = 0 whenever mp1 ≤ p22 • Then h1 = m/p1
Condition for a corner solution
The corner solution arises in cases when the budget is flat and income is low

13. Spending according to the demand functions uses up all income
Demand property #1
• A consumer’s demand satisfies budget exhaustion: p1h1 + p2h2 = m Ex. 1. Linear (case 1): h1= 0 & h2 = m/p2 p1(0) + p2(m/p2) = m Ex. 2. Leontief: h1 = m/(p1 + p2) & h2 = m/(p1 + p2) Ex. 3. Cobb-Douglas: h1 = m/(2p1) & h2 = m/(2p2)
Spending according to the demand functions uses up all income ✔

14. MATH DIGRESSION
Homogeneous function
• Suppose yo = f(x1, x2, x3) for some variable levels (x1, x2, x3) • Scale each variable by the factor t > 1, so yn = (tx1, tx2, tx3) • If yn = tryo, then f is homogeneous of degree r • Ex.1: yo = x1x2 yn = (tx1)(tx2) = t2x1x2 = t2yo • Ex.2: yo = 3x1 + 5x2 yn = 3(tx1) + 5(tx2) = t(3x1 + 5x2) = t1yo • Ex.3: yo = 2x1 /x2 yn = 2(tx1)/(tx2) = 2x1 /x2 = t0yo
The function y = x1x2 is homogeneous of degree 2
The function y = 3x1 + 5x2 is homogeneous of degree 1
The function y = 2x1/x2 is homogeneous of degree 0

15. Demand property #2
• A consumer’s demand function is homogeneous of degree zero in prices and income:  ho = f(p1, p2, m)  hn = f(tp1, tp2, tm)  hn = t0ho, or hn = ho Ex. 1 Cobb-Douglas: h1o = m/(2p1) & h2o = m/(2p2) h1n = tm/(2tp1) & h2n = tm/(2tp2) So h1o = h1n & h2n = h2o Ex. 2 Quasilinear: h1o = (p2/p1)2 & h2o = (mp1 – p22)/(p1p2)
Scaling all prices and income by the same factor leaves the budget set unchanged, so the utility-maximizing bundle is unchanged