Download Part A: True or False and Explain

ECON 301 Mock Final
Instructions. You have three hours for the exam. To obtain credit you must give arguments to support your
answer. Graphs are required where relevant. No electronic devices of any kind are permitted.
Part A: True or False and Explain
1. If the utility function u(x1 ; x2 ) = minfx1 ; x2 g represents a consumer’s preferences, then another utility function,
v(x1 ; x2 ) = minfx1 ; x2 g + 2; also represents her preferences
True.
2. A consumer considers apple pie and ice cream perfect complements. Then, his preferences are homothetic
True.
3. If marginal products are declining this means that the underlying technology exhibits decreasing returns to
scale.
False.
4. Carrie and Don both have Cobb-Douglas preferences over corn (good 1) and dairy (good 2). Carrie’s utility
function is xc1 x12 c and Don’s xd1 x12 d : If Carrie spends a higher proportion of her income on corn than Don
does, then c > d:
True.
5. At any output level the short run average total cost curve and the short run average variable cost curve must
slope in the same direction.
6. For a Gi¤en good, the negative income e¤ect dominates the positive substitution e¤ect as the price of the good
decreases.
True.
7. If preferences were given by u(x1 ; x2 ) = x21 x2 ; then both goods must be normal goods.
True. This is a Cobb-Douglas utility function. The demand functions for the two goods are
x1 =
2m
3 p1
x2 =
1m
3 p2
and
respectively. Both have income elasticities equal to one. Hence, both goods are normal goods.
8. A tangency condition is always both su¢ cient and necessary condition for utility maximization
False. In general the tangency condition is only a necessary condition for optimality, but not su¢ cient.
Optimality should satisfy two conditions (1) the tangency condition and (2) the budget line.
9. More than one utility function can represent a person’s preferences and two consumers with the same utility
function can have di¤erent preferences
True. Any monotonic transformation of a utility function is also a function that can represent the same
preference
10. In the case of quasi-linear utility functions, the indi¤erence curves are horizontal translate of each other.
True. In the case of a utility function that is linear in good 1
11. If a utility function is homothetic then the Engel curve is a straight line from the origin.
True. If preferences are homothetic, it means that when income is scaled up or down by t > 0; the demanded
bundle, (x1 ; x2 ) scales up or down by the same amount, i.e., the new bundle is (tx1 ; tx2 )
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12. Income e¤ect is zero if the utility function is quasi-linear
True. Because in the case of a quasi-linear preferences, the entire change in demand is due to the substitution
e¤ ect (because a shift in income causes no change in demand for good 1, for instance, when preferences are
quasi-linear)
13. Substitution e¤ect is always positive if we have perfect complement goods
False. There is no substitution e¤ ect in the case of perfect complements
14. Income e¤ect = total e¤ect, if we have perfect substitute goods
False. The total e¤ ect is due to substitution e¤ ect in case of perfect substitution, i.e., there is no income e¤ ect
15. If a consumer’s preferences are not transitive, then they can not be represented by a utility function
True. Actually, for a preference relation to exist, the …rst three axioms of consumer preference should be
satis…ed. In particular, preferences should be complete, transitive, and re‡exive.
16. For a gi¤en good, the negative income e¤ect dominates the positive substitution e¤ect as the price of the good
decreases.
True. Slutsky equation
17. Mary always consumes one slice of cheese cake with a cup of co¤ee. If co¤ee costs more than before, Mary’s
consumption of cheese cake must fall
True. There is no substitution e¤ ect in such a case. The total e¤ ect on the consumption of the good is due to
income e¤ ect
18. Angela considers Coke and Pepsi as perfect substitutes. If the price of Coke goes up, Angela will be strictly
worse o¤
False. If the price of Coke goes up, Angela will spend her income on Pepsi.
19. Indi¤erence curves will never intersects
True. Explain
20. Convexity of indi¤erence curves ensures that the consumers will never specialize in one good
True. Averages are preferred to extremes
21. A proportional increase in the prices and income will leave the consumer on the same budget constraint
True.
22. Indi¤erence curves exhibit diminishing marginal rate of substitution
False. Only convex indi¤ erence curves exhibit diminishing MRS
23. For homothetic preferences, the income o¤er curve will always be a ray, i.e., a straight line through the origin
True.
24. The demand curve of a Gi¤en good is, like any other good, downward sloping
False. Slutsky Equation
25. If a consumer has perfect complements preferences, then neither of the two goods she consumes can be a luxury
good
True. De…ne luxury good
26. A consumer always chooses the consumption bundle that maximizes her marginal utility given her budget
constraint
False. max utility given the budget constraint
27. The SE and IE reinforce each others, i.e., have the same sign, i¤ the good is normal.
True. Slutsky Equation
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28. Two …rms employ the same factors of production to produce the same product. Their technologies both exhibit
constant returns to scale. Thus, if the factors that …rm 1 uses are exactly twice the amount of those …rm 2
uses, …rm 1 must produce twice the output that …rm 2 produces
False. Although both …rms have technologies that exhibit CRS, we do not know whether they are using the
same technology. The statement would be true if they are
29. If the marginal cost curve lies above the average cost curve, then the average cost curve must be sloping upward
True. Graph
30. The short-run cost function is always greater than the long-run cost function
False. The short-run cost function and the long-run cost function can coincide
31. Increasing returns to scale is incompatible with the law of diminishing marginal product.
True. Returns to scale is a long-run phenomenon, while diminishing marginal product is a short-run one.
Therefore, both of them can not hold for a …rm at the same time
32. If the average product of an input is falling, then the average product must exceed the marginal product at
that input level
True. Graph
33. If the production function exhibits increasing returns to scale, average cost must be decreasing
True. Graph
34. Knowing a …rm’s production function is su¢ cient information for determining the …rm’s e¢ cient combination
of inputs?
False. The cost function should be known too. The …rm’s problem is
min C = w1 x1 + w2 x2
subject to
f (x1 ; x2 ) = y
35. If marginal cost is rising, then average cost must also be rising
False. MC attains its minimum before the AC
36. When Marginal cost curve attains its minimum, the marginal product is at its maximum
True. Graph
37. A pro…t maximizing …rm will always minimize cost
True.
38. When MC is increasing, AVC must be increasing too
False. See the graph
Part B: Problems on Consumer Choice
1. Consider a consumer who buys only two goods: food and alcohol. Her utility function is u(x1 ; x2 ) = x41 x2 ;
where x1 is food and x2 is alcohol. The prices of the two goods are p1 = $2 and p2 = $4: The consumer has
income m = $60:
(a) Does the utility function v(x1 ; x2 ) = 4 ln x1 + ln x2 also represent the consumer’s preferences?Why or Why
not?
(b) Derive the consumer’s demand function. What is the optimal consumption bundle?
(c) Suppose the government provides a $1 subsidy on food. Write down the new budget constraint. What is
the consumer’s optimal consumption bundle after the policy change?
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(d) Decompose the consumer’s consumption of food into the income e¤ect and the substitution e¤ect.
(e) Illustrate your results from (b) to (d) on a diagram?
2. Consider Gar…eld’s utility function u(x1 ; x2 ) = x2 + 12 ln x1 ; where x1 is lasagna and x2 is ‘everything else.’
Suppose his allowance from Jon is m = 30: The prices are p1 = 3 and p2 = 1: Illustrate your answers on the
same graph
(a) Derive his optimal consumption bundle
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(b) If the price of lasagna has tumbled due to the popularity of the Atkins diet. Now, it only costs p1 = 2 to
buy a unit of lasagna. What is his new consumption bundle now?
(c) Decompose the change in Gar…eld’s consumption of lasagna into substitution and income e¤ects
(d) Does Gar…eld have homothetic preferences for the two goods? Why or why not?
3. A consumer has a utility function u(x1 ; x2 ) = x1 x2 and a budget constraint p1 x1 + p2 x2 = m where p1 =
1; p2 = 1 and m = 1
(a) What are the consumer’s optimal choices? What is her utility level?
(b) Now, suppose the price of good 1 doubled. For any level of income, calculate her optimal choices and
utility levels.
(c) How much additional income does this consumer have to be given to compensate her for the price decrease?
4. Andy likes only Peanut butter, X1 , and Toast, X2 , and he always eat each toast with two ounces of peanut
butter. Suppose that Andy’s income is $300, the price of Peanut butter is $2 and the price of Toast is $6.
(a) Find the consumer’s utility function
(b) If preferences is give as
U = minf X1 ; X2 g
and the consumer budget line is
m = P1 X1 + P2 X2 ;
show that the demand functions for good 1 and good 2 are respectively
X1 =
m
P1 + P2
X2 =
m
P1 + P2
and
(c) Given your answer in (b), …nd the optimal bundle. Illustrate your answer graphically
(d) Suppose that the price of Peanut butter increased to $3, show the e¤ect on the consumer’s optimal choice
(e) If the increase in price in part (d) above is decomposed into income and substitution e¤ect, without further
calculation, …nd the substitution e¤ect and the income e¤ect in such a case.
5. Bill care only about how much beer he drinks. Therefore he considers two 12-ounce cans of beer to be exactly
as good as one 24-ounce bottle of beer. Suppose he consumes only these two kinds of beer. The price of a
12-ounce beer is $1 and that of 24-ounce is $3. Bill has $30
(a) Find Bill’s optimal choice. Illustrate your answer graphically
(b) Fond the demand functions for both goods
(c) If both prices were equal, what would be Bill’s choice in such a case
6. Consider a consumer who consumes only two goods: x1 and x2 : Her utility function is given as
U (x1 ; x2 ) = x31 x2
The prices of both goods are p1 = $3 and p2 = $2: The consumer has income m = $48
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(a) Does the utility function
v(x1 ; x2 ) = 3 ln x1 + ln x2
also represent the consumer’s preferences? Why or why not?
(b) Derive the consumer’s demand functions. What is the optimal choice. Illustrate your answer graphically
(c) Suppose the government provides $2 subsidy per unit on food. Write down the new budget constraint.
What is the consumer’s optimal consumption bundle after the policy change?
(d) Decompose the change in consumer’s consumption of food into the substitution e¤ect and the income
e¤ect. Illustrate your answer graphically
7. A consumer consumes two goods: milk, x1 , and "all other goods", x2 : The consumer’s utility function is
p
U (x1 ; x2 ) = 3 x1 + x2
Suppose her income m = $36. The prices are p1 = $0:5 and p2 = $1: The consumer consumes positive amounts
of both goods
(a) What kind of preferences do the consumer have. Illustrate graphically
(b) Write down the demand functions for good 1 and good 2 respectively
(c) What is the consumer’s optimal consumption bundle?
(d) Suppose his income increases to $24, what is his optimal consumption bundle?
(e) Does the consumer has homothetic preferences for the two goods? Why or why not?
8. Suppose the market demand curve is given by
Q = 30
P
and the market supply curve is given by
Q = 2P
The government imposes a per unit tax T = $3 on the consumers. Find the following
(a) The market equilibrium before imposing the tax
(b) The price elasticity of demand and supply at equilibrium
(c) The market equilibrium after imposing tax
(d) The tax burden and the net welfare e¤ect due to such policy.
9. Suppose the market demand for a good consists of two consumers, 1 and 2, where their individual demands
are given as
D1 (p) = 200 4p
and
D2 (p) = 100
p
(a) Derive the market demand
(b) Suppose supply is given by
p = 40;
…nd the amount purchased by each consumer.
(c) If the supply is instead given as
q = 20;
…nd the equilibrium market price, illustrate your answer with a graph.
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Part C: Problems on The Theory of The Firm
1. Consider a …rm that employs capital k and labor l in production. Its production function is f (k; l) = 2k 1=3 l1=2 :
The price of output is p = 1:
(a) Does this technology exhibit constant, increasing, or decreasing returns to scale?
(b) Find the marginal product of each factor and the technical rate of substitution. Determine if the technology
has diminishing marginal product in either factor.
(c) Suppose in the short run, capital is …xed at k = 27: What extra information do you need to determine
the exact amount of labour to be employed in order to maximize short-run pro…ts? Provide a suitable
number yourself, and …nd how much labor is employed. Illustrate your answer with a graph
2. Suppose it takes one unit of labor and one unit of sewing machine to make one sports shirt per hour. Let x1
denote the units of labor and x2 denote the number of sewing machines.
(a) Write down the production function which represents the technology of producing shirts. Draw some
isoquants representing this production technology
(b) Does this production function exhibit increasing, decreasing, or constant returns to scale?
(c) Let the factor prices be w1 and w2 : Derive the cost of production as a function of the number of sport
shirts.
(d) Derive and draw the marginal and average cost curves for the cost function you derived in (c) assuming
w1 = 1 and w2 = 5
3. A …rm has a production function
f (x1 ; x2 ) =
p
x1 +
p
x2
Let p be the price of the …rm’s output and let w1 and w2 be input prices. Suppose in order for a …rm to produce
a positive amount of output, a …xed cost, F; has to be spent.
(a) Compute the …rm’s cost function. That is, how much it costs the …rm to produce y units of output.
(b) suppose in the short run, the amount of input 1 is …xed at x1 : How much does it cost the …rm to produce
in the short run to maximize its pro…ts?
4. Suppose the production function of a typical producer is given as
Y = 2K 1=3 L2=3 ;
where K is capital and L is labor. If prices of capital and labor are PK
price of output P = 12: Answer the following
r = 1 and PL
w = 16, and the
(a) Does this technology exhibits constant, increasing, or decreasing returns to scale in production
(b) Find the marginal product of each factor and the technical rate of substitution. Show that this technology
exhibits diminishing marginal product for both factors.
(c) Now consider the short run scenario. Suppose that capital is …xed at K = 8; …nd the …rm’s optimal
production plan, i.e., …nd the …rm’s optimal input used, output produced, and pro…t realized. Illustrate
your answer with a graph
5. Suppose a …rm’s marginal cost function is given as
M C = 2y 2
2y + 30
and its …xed cost is zero. Answer the following questions
(a) Derive all cost functions
(b) Determine the level of output where the average cost functions reaches a minimum
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(c) Does the marginal cost is always upward sloping? (Hint: …nd the level of output at which marginal cost
starts to slope upward, i.e., at which marginal cost is minimized)
(d) Based on your …ndings in (c) above, if the …rm is producing y > 1=2;then which of the following is true
i. MC is de…nitely rising
ii. MC is de…nitely decreasing
iii. MC is unchanged
iv. Need additional information in order to assess the MC behavior
6. Suppose a …rm has a total cost function
c(y) =
2 3
y
3
y 2 + 30y
(a) Derive the mathematical expressions for the average and marginal cost functions
(b) Determine the level of output where the average cost function reaches a minimum
(c) Evaluate the marginal cost and average cost at the output level determined in part b
(d) Does the marginal cost slope upward at every point?
7. Consider a …rm with the following production function
f (l; k) = l1=3 k 1=2
where l is labor and k is capital. The factor prices are w = $1 and r = $48 for labor and capital respectively.
(a) Find the marginal product of each factor and the technical rate of substitution
(b) What is the …rm’s optimal long run choice of factors if it is to produce a target level of output at f (l; k) =
32?
(c) What is the minimum cost of producing 32 units of output in the long run?
8. Consider the following production function
Y = 2L + 4K
where K is capital and L is labor. If the prices of capital and labor are r = 14 and w = 2; and the price of
output P = 10. Answer the following questions
(a) Does this technology exhibits constant, increasing, or decreasing returns to scale in production
(b) Find the marginal product of each factor and the technical rate of substitution. Show that this technology
exhibits diminishing marginal product for both factors
(c) Consider the long-run scenario, for a given output Y = 20, …nd the optimal amount of capital and labor
that the …rm would employ and the pro…t realized. illustrate your answer graphically
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