Consumption一 budgets The Numeraire A good is often referred to as a numeraire. what does this mean? Prices are relative things. They are exchange rates between goods, revealing how much of good I is required to buy some of good 2 and so on. If there were only I good, the price would be meaningless In the model so far there are three variables, Pi, P2 and m. One of these is redundant. For example, setting p2=1 Pir1+p212=m= x1+x2= This final equation can be written as pa1+I2=y where p= pi/p2 and y= m/p2. This formulation contains the information All prices are in terms of the price of the numeraire. Notice -p is the slope of the budget line. It is the price of good I in terms of good 2. y is the value of income in terms of the numeraire. Consumption- Budgets the consumer didnt start with an income m, but rather had an endowment of good I and of good 2. How The consumer starts with wn of good I and w of good 2. But a unit of good 1 is worth pi and of good 2 is worth P2. The total worth of the consumers endowment is Pw1 +P2w2. The endowment is written as w=(w1, u2 The total amount spent cannot exceed the value of the endowment. Hence P1x1+p2x2≤p141+p22 Budget Line, slope =-p/pz x1=(1a1+P22)/p The slope clearly remains the same. The endowment lies on the budget line. This is simply because the endowment is always just affordable- by definition. Price changes now alter the budget set in a more complicated way.Consumption — Budgets 5 The Numeraire • A good is often referred to as a numeraire. What does this mean? • Prices are relative things. They are exchange rates between goods, revealing how much of good 1 is required to buy some of good 2 and so on. If there were only 1 good, the price would be meaningless. • In the model so far there are three variables, p1, p2 and m. One of these is redundant. For example, setting p2 = 1 does not alter anything. Mathematically: p1x1 + p2x2 = m =⇒ p1 p2 x1 + x2 = m p2 • This final equation can be written as px1 + x2 = y where p = p1/p2 and y = m/p2. This formulation contains the same information as before but is simpler. It is equivalent to setting p2 = 1. Good 2 is the numeraire. • All prices are in terms of the price of the numeraire. Notice −p is the slope of the budget line. It is the price of good 1 in terms of good 2. y is the value of income in terms of the numeraire. Consumption — Budgets 6 Endowments • Suppose the consumer didn’t start with an income m, but rather had an endowment of good 1 and of good 2. How would this alter the budget set? • The consumer starts with ω1 of good 1 and ω2 of good 2. But a unit of good 1 is worth p1 and of good 2 is worth p2. The total worth of the consumers endowment is p1ω1 + p2ω2. The endowment is written as ω = (ω1, ω2). • The total amount spent cannot exceed the value of the endowment. Hence: p1x1 + p2x2 ≤ p1ω1 + p2ω2 . . ................................................................................................................................................................................................................................................................................ ............. ............. ..... . . . . . . . x2 0 x1 x1 = (p1ω1 + p2ω2)/p1 x2 = (p1ω1 + p2ω2)/p2 Budget Line, slope = −p1/p2 ω1 ω2 . ..................... • The slope clearly remains the same. The endowment lies on the budget line. This is simply because the endowment is always just affordable — by definition. Price changes now alter the budget set in a more complicated way