Question

7. We know that well-defined preferences over two goods have the properties that (i) 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) (co, yo)) when (co,yo) 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-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 (cb) 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 (co, yo) and increasing c in equal increments Δ > 0 to ci-co + Δ, then to C2 = cut Δ and finally to C3 = C2 + 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 (y-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 (d) Now, find the corresponding bo, bi, b2, and b3 points along the b axis and the corresponding co, c1, c2, and c3 points along the bottom (vertical) c-axis. Then use these ponts to trace out the shape of the indifference curve between b and c. (e) Finally, draw a new graph in the bottom third of your page, with b on the vertial axis and c on the horizontal axis showing the indifference curve shape you have derived. You might have to hold up your paper so that you can see your graph from behind to understand how to draw it! (Ain't topology fun?) Describe in words what diminishing MRS means when a good and a bad are graphed together

Answer 1

Given that, From the above question a, b & c are instructions to construct a diagrammatic representation of the relation ship b/w y, b & c. Below is the diagram. upward axis is y-clear all downward and right ward axis represent c - consumption of all other goods leftward axis is b - pollution. \r\n 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 indifferences Curve, 1c between clear air, y, and consumption of all other goods, c_1 for different levels of c. i.e c_0, c_1, c_2, c_3 there are corresponding levels of clear air, y, i.e, y_0, y_1, y_2, y_3 respectively, as given in the question, increment from c_0 to c_1, c_1 to c_2 & c_2 to c_3 are equal remember c_0 < c_1 < c_2 < c_3 c_1 - c_0 = c_2 - c_1 = c_3 - c_2 but from the diagram it is clear that corresponding changes in y are as follows.