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Feb 20, 2024
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Problem Set #1: Answers Q1a: [5 points]
Utility for
Steak and Eggs is 13 [ BEST Option!]
Kale Salad is -1.5
Cookies is 10 Q1b: [5 points]
A miracle pill is discovered that halves the negative health impact of cookies. How does this impact your diner’s choice?
Solution: Utility for
Steak and Eggs is 13
Kale Salad is -1.5
Cookies is 15 [ BEST Option!] Q1c:
[5 points]
What effect does the miracle pill have on the diner’s health H? Interpret
this result. Does this mean the diner would be better off without the miracle pill? Solution: the consumer chooses the least healthy option and achieves a lower health level than under b (-5 versus -1) despite the fact that the pill intends to promote health. The utility is still higher than under a [15 versus 13] so the consumer is better off. Q1d: [optional not graded and no points] Solution: One intuitive way to approach this problem is to calculate the life-time consequence of a meal choice in period 0. Suppose the consumer chooses Steaks and eggs in t=0, then the life time utility payoffs can be summarized in the table below: t 2*Z_t ½*H_t U_t 0 14 -1 13 1 0 -1 -1 2 0 -1 -1 . 0 -1 -1 . 0 -1 -1
The key observation is that leisure only enters in period 0, whereas the health part matters for every period thereafter as well. Lifetime utility of a meal choice in t=0 is then given by: 𝑈
0
= 2𝑍
0
+
1
2
∗ ∑ 𝛿
𝑡
𝐻
0
∞
𝑡=0
= 2𝑍
0
+
1
2
∗
1
1 − 𝛿
H
0
For 𝜹 = ?. ??
we have ?
?
= ?𝒁
?
+ ?𝑯
?
: Utility for
Steak and Eggs is 10 [ BEST Option!]
Kale Salad is 6
Cookies is -20 For 𝜹 = ?. 𝟗
we have ?
?
= ?𝒁
?
+ ?𝑯
?
Utility for
Steak and Eggs is 4
Kale Salad is 21 [ BEST Option!]
Cookies is -80 So as 𝜹
becomes sufficiently large, consumers go for the “healthiest” food option.
Question 2: Grossman Model Qa:
[25 points]
Solution: A first observation is that time productive, T_p=24-T_s. Filling in the third equation from above we have T_p=24-T_s=24-(
?? − 𝑯)
=H. Furthermore, we have T_p=T_H+T_z. Together we can rewrite the first equation as 𝑯 = ?
𝒑
= ?
𝑯
+ ?
𝒁
= ? ∗ √?
𝑯
+ ?
− ?
Solving for T_Z we have the PPF ?
𝒁
= ? ∗ √?
𝑯
+ ?
− ? − ?
𝑯
Point A is now T_H=T_Z=0. Point E=T_H=24 and T_Z=0. To find C, we search for the maximum of T_Z. We construct the first derivative of the PPF with respect to T_H and set this to zero. This yields 𝒅?
𝒁
𝒅?
𝑯
=
?
√?
𝑯
+ ?
− ? = ?
Solving for T_H we have T_H=8 and T_Z=4. The figure below shows this graphically:
Qb: [25 points]
: At the optimum, the slope of the IC curve equals the slope of the PPF. The slope of the IC curve is -1/4. There are two ways to get here. Either solve the utility function for T_z or construct the marginal rate of substitution, which equals: MRS= -dU/dT_H/ dU/dT_z=-1/4. The slope of the PPF was derived above which we not set equal to -1/4 ?
√?
𝑯
+ ?
− ? = −
?
?
Solving for T_H, we get T_H=15 and then T_Z=3. This is illustrated via point F below.
Question 3: ### Stata code [35 points] ******************************************************************************** **** Analysis Files ******************************************************************************** . *** load data . . use nsch_2022_topical.dta, clear . . *insheet using nsch_2022_topical.csv, clear . . *** Question a construct obesity . . tab bmiclass Body Mass Index, Percentile | Freq. Percent Cum. ----------------------------------------+----------------------------------- Less than the 5th percentile | 2,831 8.61 8.61 5th percentile to less than the 85th pe | 20,293 61.68 70.29 85th percentile to less than the 95th p | 4,783 14.54 84.83 Equal to or greater than the 95th perce | 4,992 15.17 100.00 ----------------------------------------+----------------------------------- Total | 32,899 100.00
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