Tutorial week 5
Lam, Wing Tung
ECON2210 Fall 2025
Monotonic transformation of utility function
Exercise. Consider the following utility functions
√
• U C (x1 , x2 ) = x1 x2 = [ x1 x2 ]2
• U D (x1 , x2 ) = ln x1 + ln x2 = 2 ln
√
x1 x2
The above utility functions are all monotonic transformation of the Cobb-Douglas
√
utility function U A (x1 , x2 ) = x1 x2 , so they represent the same preference
1. Show that the marginal utility M U1 are respectively given by
• M U1C (x1 , x2 ) = x2
• M U1D (x1 , x2 ) = x11
2. For each utility function, how does marginal utility M U1 change with x1 and x2 ?
3. Verify that all utility functions share the same marginal rate of substitution
M RS12 (x1 , x2 ) = xx12
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Budget constraint
Changes in prices and income
Exercise. Suppose during a stagflation, the prices of good 1 and 2 increase by the same
percentage while the household’s income level remains the same. Illustrate how the
budget line changes in a graph.
Exercise. Suppose the prices of good 1 and good 2 are $5 and $2. Susan’s preference
is well-behaved. She recently has a pay raise of $30. She then buys three more units
of good 1. Calculate the change in quantity of good 2 consumed by Susan.
Common economic policies
Exercise. Suppose the government provides an in-kind transfer of quantity x1 . How
does the budget set change with prices and income level?
Exercise. Suppose the prices of good 1 and good 2 are $4 and $2. Sandy’s budget is
$18. Now the government introduces an ad valorem subsidy 50% on good 2 and levies
a lump sum tax $6 at the same time.
1. Graph the budget lines before and after the introduction of policies. Label all
intercepts and the intersection point.
2. Consider the goods bundle (2, 4.5). Is it affordable to Sandy initially? Is it
affordable under the new policy? Mark the bundle (2, 4.5) in the above graph.
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