BEEM101 Microeconomics Final exam
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BEEM101 Microeconomics
Final exam (mock paper)
To achieve full points for a problem it is not sufficient to provide the correct final answer. I must be able to retrace every step that took you to your answer.
1. Problem - Indifference curves (16 points)
Let X = R be the set of all bundles of two goods. Denote by x1 and x2 the amounts of goods 1 and 2 contained in bundle x ∈ X . For each of the preference relations on X represented by the following utility functions, sketch an indifference curve map.
Make sure to include axis labels and minimal grid information such that the most important char- acteristics of the indifference curve map (e.g. slope, vertical/horizontal intercept, direction of im- provement, the location of important points) can be concluded from your pictures.
a) u(x1 ,x2 ) = x2 − x1
b) u(x1 ,x2 ) = x1 + 2x2
c) u(x1 ,x2 ) = min{x1 , 2x2 }
d) u(x1 ,x2 ) = x1(0) ,5 x2(0) .5
a) Good 1 is a bad, good 2 is a good. The indifference curves are straight lines with a slope of +1 . The direction of improvement is to the upper left.
b) Perfect substitutes . The indifference curves are straight lines with a slope of − . The direction
c) Perfect complements . The indifference curves are L-shaped. The location of the kinks is at bundles for which the proportion between goods 1 and 2 is 2:1 . The direction of improvement is to the upper right.
d) Cobb Douglas . The indifference curves are convex. They approach the axes in the limit but never cross them. The slope of indifference curves is negative and decreasing. The direction of improvement is to the upper right.
2. Problem - Common assumptions about consumer preferences (10 points)
Let X = R be the set of all bundles of two goods. Denote by x1 and x2 the amounts of goods 1 and 2 contained in bundle x ∈ X . Show that the preference relation on X represented by the utility function u(x1 ,x2 ) = x1 + x2 satisfies strong monotonicity. Furthermore, show that it is not strictly convex.
❼ Strong monotonicity: We have to show that for any two bundles x,y ∈ X : x1 ≥ y1 , x2 ≥ y2 and x y ⇒ x ≻ y .
For any two such bundles it holds that x1 + x2 > y1 +y2 ⇔ u(x1 ,x2 ) > u(y1 ,y2 ) which implies x ≻ y .
❼ Strict convexity does not hold: We have to show that there exist two bundles x,y ∈ X with x ≽ y such that for some λ ∈ (0, 1), y ≽ λx + (1 − λ)y . Consider x = (2, 0) and y = (0, 2) . We have u(x) = u(y) = 2 such that x ∼ y . Now take the mixture of x and y with λ = 0.5 . We have λx + (1 − λ)y = (1, 1) such that u(λx + (1 − λ)y) = 2 and therefore y ∼ λx + (1 − λ)y .
3. Problem - Common assumptions about consumer preferences (5 points)
Briefly discuss why continuity is a crucial assumption needed to conduct economic analysis of consumer preferences and choice.
Continuity assures that the preference relation on the set of all bundles of two goods is representable by a (continuous) utility function. Thus, if continuity is fulfilled on top of the basic assumptions about preference relations (completeness, reflexivity and transitivity) we can conduct economic anal- ysis of consumer preferences as if the consumer uses a utility function. As a result, we can use utility maximization to study the consumer’s choices .
4. Problem - Marginal rate of substitution (16 points)
Let X = R be the set of all bundles of two goods. Denote by x1 and x2 the amounts of goods 1 and 2 contained in bundle x ∈ X . Consider the preference relation on X represented by the utility function u(x1 ,x2 ) = x1(0) .2
a) Calculate the marginal rate of substitution for good 1 in units of good 2.
b) Show that the marginal rate of substitution for good 1 in units of good 2 is decreasing.
c) In your own words, briefly explain the intuition behind a decreasing marginal rate of substitution.
d) To which of the general families of preference relations that we discussed in class does the one given in the question belong to?
e) Does the alternative utility function v(x1 ,x2 ) = x1(0) .4 x2(1) .6 represent the same preference relation? Please explain your answer.
a) MRS = 0.25
b) = −0.25 < 0
c) The more I already have of good 1, the less I am willing to give up of good 2 in order to get one additional unit of good 1 .
d) Cobb Douglas
e) Yes, it does . The alternative utility function is a transformation of the original utility function with the increasing function f(u) = u2 (u(x1 ,x2 ) ≥ 0 for all x ∈ X) . Such transformations of
the utility function are admissible because they do not affect the preference ordering of bundles .
5. Problem - The consumer’s choice problem (15 points)
Suppose the consumer is endowed with wealth w = 10 and the prices are p1 = 1 for good 1 and p2 = 2 for good 2.
a) Provide the mathematical definitions of the consumer’s budget set and budget line and explain the difference between the two in your own words.
b) Draw the consumer’s budget set and budget line. Make sure to include proper labeling such that I can distinguish the two. Furthermore, include axis labels and minimal grid information such that the location of vertical and horizontal intercept as well as the slope of the budget line can be concluded from your picture.
c) Suppose the price of good 1 increases such that the new price is given by p = 2. Sketch the new budget line in your drawing from b). Make sure to label the new budget line such that I can distinguish it from the initial one. Again, include any additional grid information such that I can conclude vertical and horizontal intercept as well as the slope of the new budget line from your picture.
a) Budget set: B(p1 = 1,p2 = 2,w = 10) = {(x1 ,x2 ) ∈ R : x1 + 2x2 ≤ 10}, budget line: {(x1 ,x2 ) ∈ R : x1 + 2x2 = 10} While the budget set is the set of all bundles the consumer can afford, the budget line is the set of all bundles that fully exhaust the consumer’s wealth.
b) The budget line is a line with vertical intercept of 5, horizontal intercept of 10 and thus a slope of − . The budget set is the triangular surface enclosed by the budget line and the two axes .
c) The new budget line is a line with vertical intercept of 5, horizontal intercept of 5 and thus a slope of −1 .
6. Problem - Consumer demand (17 points)
Let X = R be the set of all bundles of two goods. Denote by x1 and x2 the amounts of goods 1 and 2 contained in bundle x ∈ X . Consider a consumer with budget set B(p1 ,p2 ,w) whose preference relation on X is represented by the utility function u(x1 ,x2 ) = min{x1 , 2x2 }.
a) Write down the consumer’s problem.
b) Derive the consumer’s demand function.
c) Are the consumer’s preferences differentiable? Provide an intuitive argument, you do not need to give a formal proof here.
d) At the consumer problem’s solution x∗ , do we have = ? Use the answer you provided for c) to explain.
e) Show whether good 1 is regular or Giffen for the consumer.
a) max(x1 ,x2 )∈X min{x1 , 2x2 } subject to p1 x1 + p2 x2 ≤ w
b) x∗ (p1 ,p2 ,w) = ( , )
c) No, they are not differentiable . The indifference curves are not smooth, in particular each indifference curve has a kink located at the bundle in which the proportion between the two goods is 2:1 .
d) No . Since the consumer’s preferences are not differentiable, the MRS cannot be calculated using partial derivatives of u . In particular, that means we cannot calculate the slope of indifference curves in the usual way. The solution to the consumer’s problem is located at the kink of the corresponding indifference curve . At the kink, the slope of the indifference curve is actually not defined. Therefore, we cannot find the solution to the consumer’s problem by equating slope of indifference curve and slope of budget line .
e) = − < 0
7. Problem - WARP (6 points)
Let X = R be the set of all bundles of coffee and sugar. Suppose you observe a consumer choosing a bundle consisting of 5 cups of coffee and 5 spoons of sugar when the price per cup of coffee is pc = 1 and the price per spoon of sugar is ps = 0.1. Also, you observe the same consumer choosing a bundle consisting of 2 cups of coffee and 3 spoons of sugar when the price per cup of coffee is p = 2 and the price per spoon of sugar is p = 0.05. Can you conclude a violation of the weak axiom of revealed preference (WARP) from the consumer’s choices? Explain your answer.
No . The first bundle is revealed preferred to the second bundle since the second bundle was affordable when the first bundle was chosen (5pc + 5ps = 5.5 > 2pc + 3ps = 2.3) . However, the second bundle is not revealed preferred to the first bundle since the first bundle was not (necessarily) affordable when the second bundle was chosen (5p + 5p = 10.25 > 2p + 3p = 4.15) .
8. Problem - Producer behavior (15 points)
Consider a producer who produces output y using only one input a. The producer’s production function is given by f(a) = ^a. Let p > 0 denote the price for one unit of output and w > 0 denote the price for one unit of input. Assume that the producer takes all prices as given.
a) Suppose the producer wants to maximize profits. Derive her input demand a ∗ (w,p).
b) What is the producer’s optimal output y ∗ (w,p)?
c) Derive the producer’s cost function C(y).
d) Derive the producer’s marginal cost function MC(y).
e) Briefly explain the intuition of why at the producer’s optimal output y ∗ it must hold that p = MC(y∗ ).
a) a ∗ (w,p) = ()2
b) y ∗ (w,p) =
c) C(y) = pf−1(y) = wy2
d) MC(y) = C\ (y) = 2wy
e) Because the cost function is convex, the marginal cost function is increasing. That means, the more output the producer already produced, the higher the marginal cost of producing one additional unit. At the same time, the price the producer receives per unit of output remains constant. It is profitable for the producer to produce additional units as long as the price she gets for the additional unit exceeds the marginal cost she has to incur for producing that additional unit. As soon as the marginal cost of producing the next unit exceeds the price she gets for the additional unit, that additional unit decreases her overall profit.
2023-01-11