Question

7. We know that well-defined preferences over two goods have the properties that () indifference curves are negatively sloped, and (ii) that indifference curves are convex (so that chords between two points on the indifference curve lie in the set ((c, y) such that (c,y)(coyo) when (co. o) is a point on the indifference curve). Suppose that good y is "clean air" and good c is consumption of all other goods. This problem gets you to determine what these two properties mean for the shape of the indifference curve if we were to plot consumption against pollution, b, where the relationship between clean air and pollution is that y = y"-b, where y. is a measure of how clean the air is when pollution b is zero, and where each unit of pollution reduces the cleanlieness of air by one unit. (a) Draw a graph, which should take about 2/3 of a page in height and width, with a vertical and a horizontal line crossing at the center of the graph, forming a giant sign. The center of the cross is the origin for each of the four quadrants. The upper vertical axis measures y, the left-extending horizontal axis measures b, and the remaining two axes each measure c. In the lower-right (c-quadrant, draw a line through the origin that has slope -1, which allows you to trace consumption values from the (c-y) quadrant to the (c-b) quadrant. (b) Draw an indifference curve in the upper right (c-y) quadrant which exhibits the two properties. Use this graph to show that starting from some point (coyo) and increasing c in equal increments Δ > 0 to c1 = co + Δ, then to c2 = cı + Δ and finally to cs = ce+ Δ, and show that this implies that the amounts of clean air one is willing to give up for more consumption goods is diminishing. Indicate the corresponding clean air levels as yo-yı, y2, and y3. (c) Now draw a line representing the relationship between y and b in the upper-left (-b) quadrant. Remember that since this quadrant has b increasing to the left that this curve will have slope +1 (not -1) and intercepts y" on both the y and b axis. Assume that y" > yo

Answer 1

(a), (b) and (c) are instructions to construct a diagramatic representation of the relationship between y, b and c.

following is the diagram

upward axis is y - clean air

downward and rightward axis represent c - consumption of all other goods

leftward axis is b - pollution

(a) the lower right quadrant has the -1 slope line (same variable, 'c' on both the axes, rightward and downward)

(b) the top right quadrant has the Indifference curve, IC between clean air, y, and consumption of all other goods, c. for different levels of c i.e. c_0, c₁, c₂ and c₃ , there are corresponding levels of clean air, y, i.e y₀, y₁, y₂ and y₃ respectively. as given in the question, increments from c₀ to c₁, c₁ to c₂ and c₂ to c₃ are equal. remember c₀< c₁ < c₂ < c₃

c₁ - c₀ = c₂ - c₁ = c₃ - c₂ but from the diagram it is clear that corresponding changes in y are as follows

y₀ - y₁ > y₁ - y₂ > y₂ - y₃

this implies for every equal increase in c, y decreases at a decreasing rate. that means the amount of y willing to be forgone for every increase in c is diminishing for every increase in c.

(c) this is represented int he top left quadrant as instructed.

the intercept on the upward axis is y* and on the leftward axis is b' where y* = b'

0s 1C