MGEB02: Price Theory: A Mathematical Approach
Winter 2025
Instructor: A. Mazaheri
Midterm
Instructions: This is a closed book test. You are allowed a Non-Programmable calculator.
You have 2 Hours.
Good Luck!
Last
Name:
First
Name:
ID
FOR MARKERS ONLY:
Q1
Q2
Q3
Q4
Q5
Total
43
10
15
15
17
100
Marks
Earned
Maximum
Marks
Possible
Answer all following 5 questions:
Question-1 [43 Points] Answer the following Short Questions:
a) [6 Points] Graph the indifference map (at least two indifference curves) for the
following individuals and write down the utility function when asked. (Make sure to label
your graph)
i) Adam is not picky at all he is willing to exchange 2 donuts for 1 coffee no matter what.
Write down his utility function and graph her indifference map.
Coffee (Y)
Donut (X)
ii) Gabriella likes tea but she knows her limit. She only drinks three tea cups a day and
throws away anything thereafter if offered.
Everything else
tea
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b) [5 Points] A consumer is currently consuming at point A. Suppose the price of spinach
has declined and that the consumer views spinach as a Giffen good. Graph your
hypothetical final bundle and show the substitution and income effects.
Y
A
spinach
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c) [5 Points] Use graph to show that if you believe in transitivity then crossing
indifferences curves will violate the non-satiation condition.
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d) [7 Points] Assume a utility function that is given by U(x,y) = 5x1/4 y3/4. Furthermore,
assume a budget of $150 When the prices where Px = 2 and Py = 1 the consumer
purchased x= 18.75 and y = 112.5, while when the prices changed to Px = 1 and Py = 1,
the consumer purchased x = 37.5 and y = 112.5. With the help of the following graph
decompose the total effect of the price change into the substitution and income effects.
Graphically:
150
112.5
18.75
37.5
150
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e) [5 Points] A consumer has $20. She spends it on two goods, Y and Muffins. A store
offers a new special, if a consumer purchases five Muffins at a regular price, she will be
offered two extra Muffins for free and a 40% discount for any extra unit purchased. Show
graphically how this affects her budget line. (assume prices of both goods are initially $1)
y
Muffin
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f) [5 Points] Carter is very picky; he always uses two Hamburgers (X) with one
Cheese(Y). Any extra Hamburger or Cheese outside that perfect combination will be
discarded. Assuming he spends all her money on X and Y. Write down her utility
function. Find her demand functions as well as Engel curves for X and Y.
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g) [10 Points] There are 100 consumers in the economy. Half of them live in city “AA”
and demand Cupcake according to the individual inverse demand curve P = 6− 3Q. The
other half live in city “BB” and demand Ice Cupcake according to the individual inverse
demand curve P = 9−6Q. Suppose that the market-clearing price for Cupcake is $3.
Write down the market demand for Cupcake in this economy and then graph it, make
sure label your graph? Is the market demand function linear? If not, where is the kink?
If the price increases from $3 to $10, how does the consumer surplus change?
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Question-2 [10 Points] The equilibrium price for a bottle of pop is $2.00, and the
number of units consumed each day is 200,000. The short-run price elasticity of demand
is -1.5, and the short-run elasticity of supply is 3.0.
a) [6 Points] Estimate supply and demand curves at the current price and quantity.
b) [4 Points] Suppose that the demand increases by 10%, what would be the new
equilibrium price and quantity? Show your solution using a graph.
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Question-3 [15 Points] Suppose your preferences for Gasoline (X) and a composite good
(Y) is set in accordance to U(X,Y) = X0.5 +2Y0.5. You have an annual income of
$100,000 and that the price of the composite good is $1.
a) [5 Points] If the government introduces a rationing system such that you can only consume
15,000 liters a year at $1 a liter. What would be your optimal consumption bundle?
b) [6 Points] If the government removes the rationing system and the free market price of
gasoline jumps to $2. What would be your new optimal consumption bundle? Are you better off
with or without rationing?
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c) [4 Points] Illustrate your solution in a clearly labeled graph.
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0.25
Question-4 [15 Points] Susan has the following utility function: U(x, y) = 2y + 4x
. Let px and
py be the corresponding prices and I her income.
a) [9 Points] Setup the Lagrangian function and find the first order conditions (FOCs). Use these
FOCs to find the expression for the marginal rate of substitution (MRS). Use the MRS to graph
the indifference map. Then find the demand functions?
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b) [6 Points] Suppose Susan lives in a city where px = 1, py = 4 and her job offers her I = 2. Find
the optimal consumption levels for x and y? Graph your solution.
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Question-4 [15 Points] Liam utility function is U = I1/2. He has a job that offers $400 a week
but there is 60% chance that he loses his job and ends up on social assistance that only pays
$100 a week.
a) (6 Points) Find expected payoff for the job. Find the expected utility and the utility of
expected. Find the certainty equivalent. Depict the utility of this consumer on the following
diagram and identify the calculated numbers.
U
I
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b) (6 Points) What is the maximum Liam is willing to pay to fully insure himself. Will he
purchase a 50% coverage for $100 in premium?
c) [5 Points] So far we have assumed that the probability of losing his job was 0.6. Now
suppose that is not necessarily the case. What would be the maximum probability such that
he is indifferent between insuring and not insuring herself if he is offered the partial
insurance as described in part (b)?
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