Consumption- Preferences Well-Behaved Preferences Two further assumptions are made in order to obtain well-behaved preferences. monotonicity: If ri y and r2 y2 or if ri yI and rz y2 then r>y. In other words, more is better. 2. Convexity: If r w y then Ar+(1-A)y>r. In other words, averages are better than extremes. It is easiest to see what these assumptions entail by looking at the diagrams below. The second diagram is ruled out by the convexity assumption. It is these two assumptions which give indifference curves their nice regular shape that will become so familiar during the course. Monotonicity makes the curves downward sloping and convexity makes thembowed Consumption-Preferenees The Marginal Rate of Substitution The marginal rate of substitution or MRS is the rate at which the Ist willing to give up a small of good 1 in order to gain a small amount of good 2. It is the private rate of erchange In the diagrams this is represented by the slope of the indifference Slope=MRs=△x2/△x Arn and AIz are both taken to be very small numbers- hence the word marginal More mathematically derivatives, or difference ratios involving A(capital delta) The second diagram shows how the MRS changes along the indifference curve. Notice it is decreasing(in absolute value)as Il increases. This is called a diminishing marginal rate of substitution. It is a direct consequence of the convexity assumption. The consumer is willing to give up of good 1 in exchange for good 2 as the amount ofConsumption — Preferences 7 Well-Behaved Preferences • Two further assumptions are made in order to obtain well-behaved preferences. 1. Monotonicity: If x1 ≥ y1 and x2 > y2 or if x1 > y1 and x2 ≥ y2 then x  y. In other words, more is better. 2. Convexity: If x ∼ y then λx + (1 − λ)y º x. In other words, averages are better than extremes. • It is easiest to see what these assumptions entail by looking at the diagrams below. ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. x2 x1 Well-Behaved x2 x1 Non-Convexity 0 0 . ................................................................................................................... .................................................................................................. . ......... ........ .... • The second diagram is ruled out by the convexity assumption. It is these two assumptions which give indifference curves their nice regular shape that will become so familiar during the course. Monotonicity makes the curves downward sloping and convexity makes them “bowed”. Consumption — Preferences 8 The Marginal Rate of Substitution • The marginal rate of substitution or MRS is the rate at which the consumer is just willing to give up a small amount of good 1 in order to gain a small amount of good 2. It is the private rate of exchange. • In the diagrams this is represented by the slope of the indifference curve. ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. x2 x1 The MRS x2 x1 Diminishing MRS 0 0 . . ......................................... .................................................................................................................................................................................................................................................................................................... ......................................... . .............................................. ∆x2 -∆x1 Slope = MRS = ∆x2/∆x1 ............ ............ ............ ............ ............ ............. ..................... . ............................................................................................................... . ............................................................................................................... • ∆x1 and ∆x2 are both taken to be very small numbers — hence the word marginal. More mathematically ∆x2/∆x1 ≈ dx2/dx1, the derivative. The concept of marginal is central to economics and is always associated with derivatives, or difference ratios involving ∆ (capital delta). • The second diagram shows how the MRS changes along the indifference curve. Notice it is decreasing (in absolute value) as x1 increases. This is called a diminishing marginal rate of substitution. It is a direct consequence of the convexity assumption. The consumer is willing to give up more of good 1 in exchange for good 2 as the amount of good 1 increases