Economics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4


 Georgina Hodges
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1 Economics 00a / HBS 4010 / HKS API111 FALL 010 Solutions to Practice Problems for Lectures 1 to Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with a quantity discount and an ordinary budget line is the change in price for purchases of x beyond Q* units. This change in price induces a change in slope of the budget line. Therefore, the budget line is piecewise linear with (1) slope = p / q = 1 if 0 < x < Q*. () slope = p / q = 1/4 if 0 > Q*. We now need to identify the yintercept and point of change of slope to fully describe the budget constraint. yintercept If the consumer purchases only good y, she can purchase W / q = 3 / 4 = 8 units. So, one boundary point for the budget constraint is point A = (0, 8). Change of slope The slope of the budget constraint changes at x = Q*. At x = Q*, the consumer spends 4Q* on good x and has wealth W 4Q* to spend on good y corresponding to a total of (W 4Q*) / q = 8 Q* units of y given that W = 3 and q = 4. That is, the point on the budget line where the price of x (and thus the slope of the budget line) changes is B = (Q*, 8 Q*). Piecewise Linear Budget Constraint The equation for the first (of two) linear segment of the budget constraint is y = yintercept + slope * x, which in this case yields the equation y = 8 x. We can solve for the second linear segment of the budget constraint by using the equation y = constant + slope * x. We know that the slope of this second segment of the budget constraint is 1/4 and the and that it starts at the point B = (Q*, 8Q*). This gives the equation 8Q* = constant ¼ * Q* OR constant = 8 (3/4) Q* Thus, the second component of the budget constraint is given by y = 8 (3/4) Q*  (1/4) x.
2 With this as background, we can now substitute the appropriate values of Q* to find the budget constraints in (a1), (a), (a3). (a1) Component 1: Straight line y = 8 x between A = (0, 8) and B = (5, 3). Component : Straight line y = (17 x) / 4 between B = (5, 3) and C = (17, 0). (a) Component 1: Straight line y = 8 x between A = (0, 8) and B = (16/3, 3). Component : Straight line y = 4 x/4 between B = (16/3, 8/3) and C = (16, 0). (a3) Component 1: Straight line y = 8 x between A = (0, 8) and B = (6, ). Component : Straight line y = (14  x) / 4 between B = (6, ) and C = (14, 0). The graph below illustrates the budget set with Q* = 6. The boundary of the budget set is in bold. The dotted line indicates the convex combination of the intercepts of the budget set (0, 8) and (14, 0), showing that the budget set is not convex. Problem 1.1: Budget Set if Q* = Good (6, ) Good 1 (b) The quantity discount affects the budget constraint if the point (Q*, 0) lies within the budget constraint i.e. if the consumer can afford to purchase enough units of good x to reach the quantity discount: p Q* < W, or Q* < w / p. (c) The budget set is not convex for any positive value of Q* that influences the budget constraint i.e. Q* < W / p.
3 It is straightforward to see this in a graph. The line between any point on the budget constraint with x < Q* and any other point on the budget constraint with x > Q* is entirely outside the budget set. The quantity discount expands the budget constraint, and thereby expands the set of points included in convex combinations of the budget constraints including many points that remain outside the budget set. To make the same point algebraically, suppose that Q* = 6 and consider the straight line between the xintercept (3 3Q*, 0) = (14, 0) and the yintercept (0, W / q) = (0, 8) of the budget set. (It is also possible to make this same argument without assuming a specific value of Q*, but then the notation may make the exposition rather unwieldy.) A convex combination of these two points takes the form [14 α, 8(1 α)]. For α = ½, this convex combination produces the bundle [7, 4], which has associated value 41 (10 units at original prices 4 per unit and one unit of x purchased at quantity discount), and is not affordable with budget 3. Intuitively, each convex combination of points on the boundary of the budget set would be in the budget set based on the average prices of those goods. At (14, 0), the average price of x is 3 / 14, while at (0, 8) the average price of y is 4. Thus, the bundle (7, 4) would be affordable at average price 3/14 for x and average price 4 for y. But with the quantity discount, the average price of x declines with the number of units purchased exactly the opposite of what would be required for convex combinations to remain in the budget set.
4 1.. Discounts on Initial Units and the Budget Constraint (a) As in Problem 1.1, The only distinction between the budget line with a quantity discount and an ordinary budget line is the change in price for purchases of x beyond Q* units. This change in price induces a change in slope of the budget line. Therefore, the budget line is piecewise linear with slope = p / q = 1/ if 0 < x < Q*. slope = p / q = 1 if 0 > Q*. We now need to identify the yintercept and point of change of slope to fully describe the budget constraint. yintercept If the consumer purchases only good y, she can purchase W / q = 4 / 4 = 6 units. So, one boundary point for the budget constraint is point A = (0, 6). Change of slope The slope of the budget constraint changes at x = Q*. At x = Q*, the consumer spends Q* on good x and has wealth W Q* to spend on good y corresponding to a total of (W Q*) / q = 8 Q* / units of y given that W = 4 and q = 4. That is, the point on the budget line where the price of x (and thus the slope of the budget line) changes is B = (Q*, 6 Q* / ). Piecewise Linear Budget Constraint The equation for the first (of two) linear segment of the budget constraint is y = yintercept + slope * x, which in this case yields the equation y = 6 x /. We can solve for the second linear segment of the budget constraint by using the equation y = constant + slope * x. Using slope = 1 and the values (Q*, 6 Q*/), we can solve for the constant and identify the constant on the second piece of the budget constraint: y = constant + slope * x: 6  Q* / = constant Q* 6 + Q* / = constant, Thus, the second component of the budget constraint is given by y = 6 + Q* / x. With this as background, we can now substitute the appropriate values of Q* to find the budget constraints in (a1), (a), (a3).
5 (a1) Component 1: Straight line y = 6 x / between A = (0, 6) and B = (, 5). Component : Straight line y = 7  x between B = (, 5) and C = (7, 0). (a) Component 1: Straight line y = 6 x / between A = (0, 6) and B = (4, 4). Component : Straight line y = 8 x between B = (4, 4) and C = (8, 0). (a3) Component 1: Straight line y = 6 x / between A = (0, 6) and B = (6, 3). Component : Straight line y = 9  x between B = (6, 3) and C = (9, 0). Problem 1: Budget Set with Q* = Good 4 (4, 4) Good 1 The graph below illustrates the budget set with Q* = 6. By contrast to the graph for Problem 1.1 above, it is apparent from inspection that this budget set is convex. (b) The budget set is convex for every positive value of Q*, It is straightforward to see this in a graph. The line between any point on the budget constraint with x < Q* and any other point on the budget constraint with x > Q* is always inside the budget set. By comparison to a pricing rule that maintains the same (lower initial) price for all units of good y, the price increase for later units of y causes the budget constraint to contract in a way that ensures that all convex combinations of points on the boundary of the budget set are still contained in the budget set.
6 1.3. Revealed Preference and Transitivity (Problem 1 on Problem Set 1) Solution will be included with Problem Set 1 Solutions 1.4. (Not) The Subsidy Principle (Problem on Problem Set 1) Solution will be included with Problem Set 1 Solutions 1.5 Revealed Preference and Compensated Demand (a) Suppose that the new bundle chosen by the consumer is (x*, y*). The Law of Compensated Demand says that (x*  5)(8 10) + (y*  5)(1 10) < 0 for the first case where prices change to p 1 = 8 and p = 1 and wealth stays at 100. This simplifies to the equation (y*  5) (x*  5) < 0, or x* > y*. The original bundle (5, 5) is still available under these conditions. So, the Law of Compensated Demand rules out all bundles with x* < 5. Similarly, The Law of Compensated Demand says that (x*  5)(16 10) + (y*  5)(4 10) < 0 for the second case where prices change to p 1 = 16 and p = 4 and wealth increases to 00. This simplifies to the equation 6x* + 14y* < 0. The budget constraint is given by 16 x* + 4 y* = 00 since the consumer spends all money on goods. Rearranging terms to isolate y*, we find y* = 5 / 3 x* / 3. Substituting back into the inequality, we find 6 x*  14 (/3) x* + 14 * 5 / < 0, which simplifies to x* > 5. Once again, the Law of Compensated Demand rules out all bundles with x* < 5. In fact, these two cases correspond to the budget line, since prices and wealth simply double from the first case to the second. (b) Let (x*, y*, z*) be the bundle chosen under the new conditions. By the Law of Compensated Demand, (x*  5)(8 10) + (y*  10) (10 10) + (z*  10) (1 10) < 0, which simplifies to x* > z*. But it is possible for demand for any individual good to increase or decrease. For example, the bundle (, 13.4, 0) satisfies this inequality with demand for good x falling despite its drop in price, while demand for good increases and demand for good z falls. Similarly, the bundle (11, 0, 31/6) satisfies this inequality with demand for goods x and z increasing while demand for good falls.
7 Note that (, 13.4, 0) and (11, 0, 31/6) were not available at the original prices, so WARP does not rule out the possibility of choosing either of them over (5, 5, 5), the bundle selected at the original prices. While we cannot conclude that demand for any one particular good must increase or fall, we can rule out some combinations of changes. If demand for good z rises, then demand for good x must rise even more, while if demand for good x falls, then demand for good 3 must fall even more. It seems natural that demand for good x would rise and demand for good z would fall, since the price of good x declined while the price of good z increased, but this need not be the case. OPTIONAL PROBLEM FROM MWG 4. MWG.D.4: You should be able to identify the nonconvexity of this budget set immediately: if you were to draw a line connecting the points (4,0) and (a, M), the points on this line would lie outside the budget set. What this means in practical terms is that the individual can choose to have 4 hours of leisure (no work) and no consumption, or a hours of leisure and M consumption, but if the individual were to choose an amount of leisure exactly ½ of the way between a and 4, they would get less than ½ M in consumption. [The same is true if you pick any other fraction.] This makes it unlikely that individuals will choose (leisure, consumption) bundles that place them on the nonconvex portion of a budget set. Try drawing some indifference curves and you will see that depending on how steep or flat these curves are, people will tend to locate either at (4,0) or (a,m). Moreover, two people with very similar indifference curves might end up choosing very different bundles because of this nonconvexity. To demonstrate this nonconvexity mathematically, the strategy is to find the slope of the line connecting (4,0) and (a,m), and then show that at the point on this line where leisure=16 (or alternatively, work=8 hours), you get a consumption level above the point on the budget set (16,b). Thus, the first helpful step is to find the values for a and b. The point (16,b) is attained with 8 hours of labor at a wage rate of s per hour, so b=8s. The value of a indicates how much leisure you will have left, if you work enough hours to consume M. You can consume M by working 8 hours at wage rate s and (16a) hours at wage rate s : M = 8s + (16 a) s' M 8s = 16 a s' M 8s a = 16 s' If we work 8 hours (leisure=16), where would this put us on the line between (4,0) and (a,m)? If we calculate the slope of the line with respect to work hours instead of with
8 respect to leisure hours, we get rise/run = (M0)/(4a). Working 8 hours would thus give consumption of: 8M M 8s 8 + s' And then we need only show that this quantity is larger than 8s. There are many ways to do this; the most elegant is to notice that if we changed s to s in the above expression: 8M 8M = = 8s M 8s M 8 + s s Since we know from the problem that s >s, 8M 8M 8M > = = 8s M 8s M 8s M s' s s This demonstrates that the line is outside the budget set. NOTE: A full mathematical proof is ideal, but not necessary. Instead, a descriptive answer showing a clear understanding of the mathematical meaning of convexity as well as its practical interpretation in this problem, along with an appropriate graphical illustration, would be sufficient..1 Transformations of Utility Functions (a) When x = y, u(x, y) = xy = x. The desired transformation function f solves the equation f(u(x,y)) = x when x = y OR f(x ) = x, which can be solved by inspection: f(z) = SQRT(z). Then v(x, y) = f(u(x, y)) = SQRT(xy) = x 1/ y 1/ Next compute the Marginal Rate of Substitution for u and for v. u / x = y u / y = x, so ( u / y) / ( u / x) = x / y. v / x = 1/ x 1/ y 1/ v / y = 1/ x 1/ y 1/
9 so ( v / y) / ( v / x) = x / y. (b) When x = y, u(x, y) = x + y = x + x. The desired transformation function solves the equation f(u(x, y)) = x when x = y OR f(x + x ) = x. This indicates that f(x) is the inverse function to g(x) = x + x To invert this function, solve the equation z = x + x z + 1 = x + x + 1 z + 1 = (x+1) SQRT (z + 1) = x + 1 SQRT(z + 1) 1 = x So if g(x) = x + x, then g 1 (x) = SQRT(x+1) 1 i.e. f(x) = SQRT(x+1) 1. Then, v(x, y) = f(u(x,y)) = SQRT(x + y 1) 1. Next compute the Marginal Rate of Substitution for u and for v. u / x = u / y = y, so ( u / y) / ( u / x) = y.. v / x = (x + y 1) 1/ v / y = y (x + y 1) 1/ so ( v / y) / ( v / x) = y.. Convex Preferences and Quasiconcave Utility Functions Consider the utility functions u 1 (x, y) = xy, u (x, y) = x + y, u 3 (x, y) = x + SQRT(y). (a) As shown in the graph, these three utility functions produce indifference curves with qualitatively distinct properties. For indifference curves corresponding to u 1, convex combinations of bundles on an indifference curve lie above the indifference curve. For indifference curves corresponding to u, convex combinations of bundles on an indifference curve lie exactly on the indifference curve. For indifference curves corresponding to u 3, convex combinations of bundles on an indifference curve lie below the indifference curve. That is, u 1 appears to be strictly quasiconcave, u appears to be weakly quasiconcave and u 3 does not appear to be quasiconcave.
10 Problem.1 Graph
11 (b) Taking partial derivatives: u 1 / x = y u 1 / y = x; u / x = 1 u / y = 1; u 3 / x = 1 u 3 / y = y. MRS 1 MRS MRS 1 = ( u 1 / x) / ( u 1 / y) = y / x. = ( u 1 / x) / ( u 1 / y) = 1. = ( u 1 / x) / ( u 1 / y) = 1 / y. Along an indifference curve as x increases and y decreases, MRS 1 is decreasing, MRS is constant, and MRS 3 is increasing. This indicates that u 1 is strictly quasiconcave, u is weakly quasiconcave, and u 3 is not quasiconcave..3 Skiing & Violins (a) Graph for Problem A (0, 15) B (10, 10) 4 C (30, 0)
12 (b) As shown in the graph, a convex combination of two points on the same indifference curve lies below the indifference curve, indicating that with these indifference curves, the consumer strictly prefers the weighted average of utilities of two different bundles to the convex combination of those bundles. Thus, the preferences represented by these utility functions are not convex. (c) The lowest indifference curve is tangent to the budget line at B = (10, 10). The second indifference curve intersects with the budget line at A = (0, 15) and at (0, 5). The third indifference curve intersects with the budget line only at the boundary point C = (30, 0). Comparing these points and indifference curves, the optimal bundle on the budget line is at C = (30, 0). All other bundles on the budget line lie below the last indifference curve and therefore C = (30, 0) is strictly preferred to all other options on the given budget line. (d) The nonconvexity of preferences induces a choice of bundle that specializes in consumption of a single good. In essence, once we knew that the preferences were nonconvex as shown (with indifference curves that are mirror images of those for convex preferences) then it was almost certain that the optimal bundle would be at one of the ends of the budget line at A = (0, 15) or C = (30, 0).
13 .4 Piecewise Linear Indifference Curves and Optimal Bundles (a) First equate each bundle to a different bundle with x = y and then use transitivity to compare the bundles. The consumer is indifferent between bundle (x 1, y 1 ) and bundle (k 1, k 1 ) where k 1 = (x 1 + y 1 ) / 3. The consumer is indifferent between bundle (x, y ) and bundle (k, k ) where k = ( x + y ) / 3. The consumer s preference between the bundles can be identified from the comparison between k 1 and k. If k 1 > k, i.e. (x 1 + y 1 ) / 3 > ( x + y ) / 3, then the consumer strictly prefers (x 1, y 1 ) to (x, y ). Similarly, if k > k 1, i.e. (x + y ) / 3 > ( x 1 + y 1 ) / 3, then the consumer strictly prefers (x, y ) to (x 1, y 1 ). Finally, if k 1 = k, i.e. (x 1 + y 1 ) / 3 > ( x + y ) / 3, then the consumer is indifferent between (x 1, y 1 ) and (x, y ). (b) On a graph, show this consumer s indifference curves through (4, 4) and (8, 8). Graph for Problem.4(b) (c) As shown in the graph below, the bundle (4, 4) lies on the budget line and is the optimal choice.
14 Graph for Problem.4(c) (d) As shown in the graph below, the bundle (4, 4) lies on the budget line and is the optimal choice. Graph for Problem.4(d) (e) This utility function builds in a preference for equal consumption of both goods, as indicated by the change in slope of each indifference curve at the point where x = y.
15 With equal prices, as in (c), the consumer will naturally choose a bundle with equal amounts of each good. However, if the price of one good is much larger than the other (as in (d), when p > q) then the degree of the consumer s preference for equal consumption is outweighed by the difference in prices and the consumer will specialize consumption to just one good..5 Perfect Complements Solution will be included with Problem Set 1 Solutions
16 3.1 CobbDouglas Utility (a) Start by writing the Lagrangian: L(x, y) = x α y 1α λ (px + qy  W) Taking partial derivatives gives the firstorder conditions L / x = α x α1 y 1α λp = α (x / y) α1 λp = 0; (1) L / y = (1α) x α y α λq = (1α) (x / y) α λq = 0; () L / λ = W px qy = 0. (3) Rearrange and solve for x and y however you like. For example, you could start by solving (1) and () for λ, then using those two equalities to solve for y as a function of x. λ = α (x / y) α1 / p λ = (1α) (x / y) α / q; Setting these equal, α (x / y) α1 / p = (1α) (x / y) α / q; OR α (y / x) / p = (1α) / q; OR y = (1α) px / αq. (4) Finally, we can solve for the optimal bundle by substituting (4) into (3) which is simply the budget constraint. First rewrite (3) as x = (W qy) / p (5) then substitute (5) into (4): y = (1 α) (W qy) / αq OR y = (1 α) W / αq  (1 α) y / α OR αy = (1 α) W / q  (1 α) y OR y = (1 α) W / q. (6) This is the formula for the choice of y in the optimal bundle. Substituting (6) into (5) yields the formula the choice of x in the optimal bundle. x = (W  (1 α) W) / p OR x = αw / p. (7)
17 (b) Use the same FOCs as above to solve for the Lagrange multiplier: λ = (1α) (x / y) α / q; From (6) and (7), x / y = αq / (1α) p, implying λ = (1α) (αq / (1α) p) α / q, which can be simplified slightly to λ = (α / p) α [(1α) / q) 1α. (c) Returning to the results from (a), the optimal choice of x is linear in W, α and 1 / p, and similarly the optimal choice of y is linear in W, 1α and 1 / q. One property of these results is that total expenditure on x is equal to px*(p, W) = αw, while total expenditure on y is equal to qx*(p, W) = (1α)W. That is, the parameter α determines the proportion of wealth spent on each good, then the prices p and q, translate that proportional expenditure into particular quantities of x and y. (d) Optimal Bundle With transformed utility function v(x, y) = α ln x + (1α) ln y, L(x, y) = α ln x + (1α) ln y λ (px + qy  W) Taking partial derivatives gives the firstorder conditions L / x = α / x λp = 0; (1 ) L / y = (1α) / y λq = 0; ( ) L / λ = W px qy = 0. (3 ) Solving (1) and () for λ, λ = α / px; λ = (1α) / qy; Setting these equal, α / px = (1α) / qy OR y = (1α) px / αq, which is identical to (4). Thus, the optimal bundle is unchanged by the transformation of the utility function.
18 Lagrange Multiplier Now solving λ = (1α) / qy for λ gives λ = (1α) / q[(1 α) W / q] OR λ = 1 / W. Using the interpretation of λ as the shadow price of the budget constraint, the number of utility units achieved per additional unit of money varies with the form of the utility function. Therefore, since the transformation of the utility function changed the definition of one unit of utility, the Lagrange multiplier must also change as a result of this transformation. 3.. Skiing & Violins revisited (a) This is a separable utility function with associated firstorder condition (MRS form) ( u / x) / p = ( u / y) / q OR 10 / p = y / q. With p = 10 and q = 0, the firstorder condition is given by 10 / 10 = y / 0 OR y = 10. (b) From.3, we know that the utility function is not quasiconcave so that preferences are not convex and in fact, the firstorder condition identifies a global minimum (i.e. minimum utility available on the budget constraint) rather than a maximum. (c) When the firstorder conditions identify a minimum rather than a maximum, then the optimal bundle must be at one of the endpoints of the budget line, in this case (30, 0) and (0, 15). Computing u(30, 0) = 300 and u(0, 15) = 5, we conclude that the optimal bundle is (30, 0), since this gives higher utility in the comparison of the endpoints.
19 3.3 Comparative Statics with One Good (Problem 3 on Problem Set ) Solution will be included with Problem Set Solutions 3.4 Comparative Statics and the Marginal Rate of Substitution (Problem 1 on Problem Set ) Solution will be included with Problem Set Solutions 3.5 Utility Maximization with Quantity Discount (Problem on Problem Set ) Solution will be included with Problem Set Solutions 3.6. Utility Maximization with Discounts on Initial Units (a) The budget constraint is piecewise linear, with slope 1/ between the point A = (0, 6) and point B = (, 5) (the point where the quantity discount applies). The budget constraint is kinked at point B, with slope changing to 1 between B = (, 5) and C = (7, 0). The natural method for solving this problem is to find the optimal bundle on each separate linear portion of the budget line, and then compare these two bundles (a1) Between A = (0, 6) and B = (, 5) On this part of the budget line, p =, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x /. The equation for the line between A and B is y = 6 x /. Substituting the firstorder condition into the budget line equation gives the solution x = 6, corresponding to a point beyond the end of the line between A and B. This indicates that the firstorder conditions for a maximum cannot be satisfied on the line between A and B and instead that the optimal bundle on this line must be at either A or B. Since A yields utility 0, the optimal bundle on this line must be at B. TECHNICAL COMMENT: The firstorder condition identifies the optimal bundle for a budget line from (0, 6) to (1, 0) the budget line that applies with no increase in the price of x for x > Q*. Under these conditions, the optimal bundle would be (6, 3), producing utility 18. When the firstorder condition is not satisfied on the line between A and B, the utility function must be strictly increasing (or strictly decreasing) on that line. In this case, the utility function starts at 0 at point A and is strictly increasing in x until
20 reaching point B where x =. Therefore, the maximum value of the utility function on the line between A and B occurs at the boundary, point B. (a). Between B = (, 5) and C = (7, 0), On this part of the budget line, p = 4, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x. The equation for the line between B and C is y = 7 x. Substituting the firstorder condition into the budget line equation gives the condition x = 7, which yields the solution x = 3.5, corresponding to the bundle ( ). This is the unique point on the line between B and C that satisfies the firstorder condition. (a3) Comparing the candidate solutions. We have two candidate bundles for the consumer s optimum. Bundle (, 5) yields utility 10 and is the optimal choice on the budget line between A and B. By comparison, bundle (3.5, 3.5) yields utility 49/4 and is the optimal choice on the budget line between B and C. Since 49/4 > 10, bundle (7/, 7/) is the global optimum. (b) The solution follows the same method as in (a), but with slightly different budget equations throughout the analysis. As before, the budget constraint is piecewise linear, with slope 1/ between the point A = (0, 6) and point B = (4, 4) (the point where the quantity discount applies) and separate slope 1 between B = (4, 4) and C = (8, 0). (b1) Between A = (0, 6) and B = (4, 4), On this part of the budget line, p =, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x /. As in (a1), the firstorder condition identifies a bundle beyond the end of the line between A and B. so the optimal bundle on this part of the budget line is at a boundary point, in this case at B = (4, 4). (b) Between B = (4, 4) and C = (8, 0), On this part of the budget line, p = 4, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x. The equation for the line between B and C is y = 8 x. Substituting the firstorder condition into the budget line equation gives x = 8, which yields the solution x = 4, corresponding to the bundle (4. 4). This is the unique point on the line between B and C that satisfies the firstorder condition.
21 (b3) Comparing the candidate solutions. We ve identified the same bundle (4, 4) as the optimal bundle on each separate component of the budget constraint. Therefore, the bundle, which lies exactly at the boundary between the two parts of the budget constraint, must be the global optimum. (c) The solution follows the same method as above, but with again slightly different budget equations throughout the analysis. As before, the budget constraint is piecewise linear, with slope 1/ between the point A = (0, 6) and point B = (6, 3) (the point where the quantity discount applies) and separate slope 1 between B = (6, 3) and C = (9, 0). (c1) Between A = (0, 6) and B = (6, 3) On this part of the budget line, p =, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x /. identifying a bundle (6, 3) that lies at the boundary of the line between A and B. (c) Between B = (6, 3) and C = (9, 0), On this part of the budget line, p = 4, q = 4 and the firstorder condition is given by ( u / x) / ( u / y)) = p / q OR y = x. The equation for the line between B and C is y = 9 x. Substituting the firstorder condition into the budget line equation gives x = 9, which corresponds to a bundle with x = 4.5 that does not lie on the line between B and C. The optimal bundle on this line must be at one of the boundaries and in this case is at point B = (6, 3). (c3) Comparing the candidate solutions. We ve identified the same bundle (6, 3) as the optimal bundle on each separate component of the budget constraint. Therefore, the bundle, which lies exactly at the boundary between the two parts of the budget constraint, must be the global optimum. (d) For purchases of good x beyond Q* units, the prices of the two goods are equal, so with the given utility function, the consumer would then want to set the quantities of these two goods equal. However, if Q* is relatively large by comparison to W, the consumer has reason to consume more of x than y (setting x = y if possible) in order to take advantage of the reduced price for x on initial purchases. In (a) and (b), when Q* is low, the consumer simply sets x = y and the value of Q* only affects the total number of units of each good that the consumer can choose.
22 In (c), where Q* is higher, the incentives change and the consumer sets x = y. (e) One immediate implication of the convexity of the budget set is that there is a unique optimum in every part of this problem (since the utility function is quasiconcave). In addition, since the utility function is continuous, any small change in Q* translates into a small change in utility value for each bundle implying that the optimal bundle for a given Q* must be close to the optimal bundle for Q* + ε. That is, the choice of optimal bundle is continuous in Q*. This property is reflected by the overlap of optimal bundles on each component of the budget constraint in (b) and (c).
23 4.1 Quasilinear Utility and Boundary Solutions (Problem 1 on Problem Set 3) Solution will be included with Problem Set 3 Solutions 4.. Quasilinear Utility and FirstOrder Conditions a) Suppose the consumer purchases positive quantities of all goods, then in the optimum, the following relation must hold: A 1 1 = = * * p1 p x px 3 3 p x = = p * 1 * 1, x 3 4Ap Ap3 * The consumer must enough wealth to purchase at least x and x * 3 quantities of good and 3, respectively. * * p1 p1 So, w> px + p3x3 = + 4Ap A 9 3 b) Substitute ( p1, p, p 3) = (3,4,5) into the last equation, we get w + 16A A c) A = 1, then the cutoff value for w is (i) Solve the problem without setting up a Lagrangian. 57 Suppose wealth w, then in the optimum, the consumer purchases 0 amount of good 16 1, and her consumption for good and good 3 satisfies 1 1, px p3x3 w p x = px + = 3 3 Solve the equations, we get x3 = p ( 1+ w 1), x = + w 1+ w p3 p p p Substitute ( p1, p, p 3) = (3,4,5), we get w 4 ( x1, x, x3) = (0,+ 4 + w, ( 4+ w )) If the wealth w >, the consumer spends 57 on good and 3 such that =, px + px 3 3 =, and spends all the remaining on good 1. p px x w w0 w ( x1, x, x3) = (,+ 4 + w0, ( 4+ w0 )), where w 0 =
24 (ii) Now set up a Lagrangian. 1/ L= x1+ x + ln x3 λ(3x1+ 4x + 5 x3 w) First order conditions give that 1 3λ 0, with equality if x 1 > x 3 x 4λ 0, with equality if x > 0. 5λ 0, with equality if x 3 > 0. p1x1+ px+ p3x3= w Solving this system will give the same result as above. 57 w 4 So, if w, ( x1, x, x3) = (0,+ 4 + w, ( 4+ w )) ; w w0 w if w >, ( x1, x, x3) = (,+ 4 + w0, ( 4+ w0 )), where w 0 = Homothetic Utility and Price Indices (Problem on Problem Set 3) Solution will be included with Problem Set 3 Solutions 4.4 Price Indices and Revealed Preference (a) Assume that wealth is the same in both periods and that Walras Law holds, so that the bundles consumed in each period have value W in that period: p 1 x* 1 + q 1 y* 1 = p x* + q y* = W Algebraically, the condition for the Laspeyres index to be less than 1 is CPI L = (p x* 1 + q y* 1 ) / (p 1 x* 1 + q 1 y* 1 ) < 1; OR (p x* 1 + q y* 1 ) / W < 1; OR p x* 1 + q y* 1 < W. Algebraically, the condition for the Paasche index to be less than 1 is CPI P = (p x* + q y* ) / (p 1 x* + q 1 y* ) < 1; OR W / (p 1 x* + q 1 y* ) < 1; OR p 1 x* + q 1 y* > W. Given equal levels of wealth in both periods, (1) the Laspeyres index is less than 1 if the bundle consumed in the first period is within the budget set in period ; () the Paasche
25 index is less than 1 if the bundle consumed in the second period is not in the budget set in period 1. To see this graphically, suppose that W = 150, p 1 = 5, q 1 = 5, p = 3, q = 6. This example reflects a case where the price of good y increases and the price of good x falls over time making it possible that a price index could be either greater than 1 (if it puts sufficient weight on y) or less than 1 (if it puts sufficient weight on x). With W = 150 and p 1 = 5, q 1 = 5, the budget line for period 1 is y 1 = 30 x 1. With W = 150 and p = 3, q = 6, the budget line for period is y = 5 x /. Figure 5.3(a) uses a solid line for the budget constraint in period 1 and a dashed line for the budget constraint in period. The budget lines intersect at point A = (10, 0). Let (x* 1, y* 1 ) represent the optimal bundle in period 1 and (x*, y* ) represent the optimal bundle in period. Since wealth is the same in both periods, we can identify the cost of bundles in period (relative to 150) by observing whether they are above or below the dashed budget line. Similarly, we can identify the cost of bundles in period 1 by observing whether or not they are above the solid budget line. From the discussion above, the Laspeyres index will be less than 1 if (x* 1, y* 1 ) is not in the budget set in period meaning that (x* 1, y* 1 ) is on the budget line in period 1 to the right of the intersection of the budget lines at point A = (10, 0). Similarly, the Paasche index will be less than 1 if (x*, y* ) is not in the budget set in period 1 meaning that (x*, y* ) is on the budget line in period 1 to the right of the intersection of the budget lines at point A = (10, 0).
26 Figure 5.3: Laspeyres and Paasche Indexes < A = (10, 0) C 5 B Figure 5.3 shows a possible combination of optimal bundles for this combination of first period and second period prices that produces both a Laspeyres index less than 1 and a Paasche index less than 1. Here, point B represents the choice in period 1 and point C represents the choice in period. Since point B is below the dotted budget line, it is within the budget set in period implying that the Laspeyres index is less than 1. Since point C is above the other budget line, it is not in the budget set in period 1, implying that the Paasche index is less than 1. Intuitively, bundles to the right of point A put greater weight on good x than y relative to this given change in prices so that computations based on these bundles emphasize good x (whose price declined in this example) and create price indexes less than 1. By similar reasoning, bundles to the left of point A create price indexes greater than 1. (b) If the Laspeyres index is less than 1, then (x* 1, y* 1 ) is inside the budget set in period. By local nonsatiation, the consumer can improve on the choice of (x* 1, y* 1 ) in period and therefore, the consumer must strictly prefer (x*, y* ) to (x* 1, y* 1 ). (c) If the Laspeyres index is exactly equal to 1 and wealth is equal in periods 1 and, then the optimal bundle for period 1 lies on the budget line for period (i.e. the bundle at point A must be chosen in period 1). If the optimal bundle in period is not the same as the optimal bundle in period 1, then by revealed preference, (x*, y* ) must be strictly preferred to point (x* 1, y* 1 ) and is not affordable in period 1. In Figure 5.3, this choice of
27 (x*, y* ) corresponds to a point like C a bundle to the right of the intersection of the budget lines since it is not affordable in period 1. (Note that this choice would also cause the Paasche index to be less than 1.) This result reflects an application of the Law of Compensated Demand, which says that consumption must move in the opposite direction of prices in response to a compensated price change. If the consumer chooses the bundle at point A in period 1, the price change is defined to be compensated (even though wealth is unchanged) because (x* 1, y* 1 ) is on the budget line in period. Further, the price change represents a decrease in the price of x and an increase in the price of y, so the Law of Compensated Demand indicates that the consumer will increase consumption of good x and reduce consumption of good y. (d) If the Laspeyres index is less than 1, then the optimal bundle in period 1 lies inside the budget set in period and must be to the right of point A on the solid budget line in Figure 5.4. Further, if the Paasche index is greater than 1, then the optimal bundle in period lies inside the budget set in period 1 and must be to the left of point A on the dashed budget line. This configuration is inconsistent with revealed preference because it indicates that (x* 1, y* 1 ) is inside the budget set in period and similarly (x*, y* ) lies inside the budget set in period 1. By revealed preference, (x*, y* ) must be strictly preferred to (x* 1, y* 1 ) and simultaneously (x* 1, y* 1 ) must be strictly preferred to (x*, y* ), producing a contradiction.
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