Poor Household’s Demand for Cheap Dietary Staples

You will predict the impacts of changes in prices of dietary staples.

• You will calibrate a model of the demand for cheap dietary staples of the extreme poor.

• You will simulate the demand to extract some predictions.

Predictive Analytics:
Poor Household’s Demand for Cheap Dietary Staples

Ramses Y. Armendariz

University of Illinois, Urbana-Champaign

Last Revised FALL 2020 Predictive Analytics (Armendariz) 1

What is the Final Project?

• You will predict the impacts of changes in prices of dietary staples.
• You will calibrate a model of the demand for cheap dietary staples of the extreme poor.

• You will simulate the demand to extract some predictions.

• The demand model is given to you: 𝑏 =
𝑖 𝛽𝛼𝑏𝑝𝑚−𝑝𝑏𝛼𝑚 +𝑝𝑚𝑝𝑏 1−𝛽 ҧ𝑐

𝑝𝑏 𝛼𝑏𝑝𝑚−𝑝𝑏𝛼𝑚
• The parameters in this model are ҧ𝑐,𝛽,𝛼𝑏, and 𝛼𝑚.
• The economic statistics in this model are 𝑖,𝑝𝑚,and 𝑝𝑏.

• You will calibrate for the values of the parameters to fit this model to the data (Jensen and Miller, 2008).
• You will use the method of moments to calibrate the parameters.

• Once you have calibrated for the parameters, you will simulate some predictions:
• You will graph the demands for cheap dietary staples of the extreme poor.

• You will graph the Income-Consumption Curves and Engel Curves.

• You will graph the Indifference Curves of the household.

• You will estimate the Willingness To Pay for a policy that drops the price of the staple by 1%.

Last Revised FALL 2020 Predictive Analytics (Armendariz) 2

The Model

• This model characterizes a household that faces the following economic problem:

• max
𝑏,𝑚

𝛼𝑏𝑏 + 𝛼𝑚𝑚 − ҧ𝑐
𝛽 𝛿𝑚 1−𝛽 s.t.: 𝑝𝑏𝑏 + 𝑝𝑚𝑚 ≤ 𝑖 and 𝑏,𝑚 ≥ 0.

• The following two equations characterize the solution to that economic problem

• 𝑏 =
𝑖 𝛽𝛼𝑏𝑝𝑚−𝑝𝑏𝛼𝑚 +𝑝𝑚𝑝𝑏 1−𝛽 ҧ𝑐

𝑝𝑏 𝛼𝑏𝑝𝑚−𝑝𝑏𝛼𝑚
𝑚 =

𝑖−𝑝𝑏𝑏

𝑝𝑚

• Intuition of the model:
• Households have Cobb-Douglas preferences for hunger satiation (𝛼𝑏𝑏 + 𝛼𝑚𝑚 − ҧ𝑐) and flavor (𝛿𝑚).

• Households must go to the markets to buy a cheap dietary staple, 𝑏, and a superior composite good, 𝑚.
• Households want to avoid consuming a quantity of calories ҧ𝑐.

• This model enables us to do the following:
• Extrapolate the demands for dietary staples of the extreme poor.

• Extrapolate the demands for total caloric intake.

• Extrapolate the preferences (indifference curves) for food of households.

• Estimate the Willingness To Pay for subsidies on dietary staples.

Last Revised FALL 2020 Predictive Analytics (Armendariz) 3

The Project

• Your project report must be between 6 to 10 pages.
• The point of you report is to explain how to replicate your results.

• Cite all the sources you used and the people you worked with.

• I will add some instructions on how to calibrate the model and generate some predictions in Compass.
• The instructions I share in Compass are to create the whole project using Excel.

• Our student TA, Francis, has created instruction on how to simulate the model using R.

• The due date for the project is the date of the final exam: December 12th, 2020 at 11:00 PM.

• To know more about this project, you can click on the paper located in this link:
• https://www.drarmendariz.com/research

Last Revised FALL 2020 Predictive Analytics (Armendariz) 4

mailto:ramses@illinois.edu

https://www.drarmendariz.com/research

2

>

Sheet

1

1

1

7

0

2

6

7

1

3

2

6

4

6

8

5

4

2

5

6

2

7

8

9

9

4

5

1

3

1

5

12

1

13

3

1

6

14

8

7

5

5

2

1

17

5

18

3

3

-0.527

1

4

5

21

0.537

22

0.556

3

3

24

0.037

5

26

27

7

0

30

31 0.512 0.092

32

0

-0.623

33

34

35 0.536

36 0.542

37

38 0.554

39

40

41

0

42

43

44

45

46

0.815

47

0.360

48 0.615

49

0.815

50

51

52

0.863

53

54

55

56 0.663

57

58

59

60

0.717

61

62

63

0.748

64

0.746

65

66

67

68

69 0.742

70 0.748

71

72

0.815

73

74

0.640

75

76 0.785 0.092 0.480

77 0.791

78

79

80

0.264 -0.499

81 0.815

82

83

84

85 0.839

86

87 0.851

88

89 0.863

90

91

-0.623 0.000

92

93 0.888

94 0.894

-0.007

95

96

97

98

0.072

99

-0.828 0.111

i	x_i	y_i	std_hi_i	std_low_i
0.	3	30	-0.	5	7	0.5	6	-1.	60
0.	33	-0.	4	17	0.	61	-1.	45
0.	34	-0.3	26	0.	67	-1.	32	8
0.3	48	-0.	24	0.7	36	-1.	22
0.	35	-0.	18		0.	78	-1.	14	9
	0.360	-0.	13	0.8	12	-1.0	75
0.3	66	-0.0	77	0.8	37	-0.9	91
0.3	72	-0.0	21	0.8	57	-0.	89
0.378	0.0	27	0.8	68	-0.	81
10	0.	38	0.0	64	0.	86	-0.	73
11	0.	39	0.091	0.	84	–	0.6	63
0.3	97	0.110	0.	82	-0.602
0.	40	0.118		0.	79	–	0.	55
0.409		0.111			0.	74		-0.	52
15	0.	41			0.0	92	0.	69	–	0.	51
16	0.	42	0.0	62		0.640	-0.515
0.427		0.037	0.	59	-0.521
0.	43	0.018	0.	56
19	0.439	0.004	0.	54	-0.	53
20	0.	44	-0.002	0.533	–	0.537
0.451		0.000	-0.537
0.457	0.010	-0.535
23	0.	46	0.024	0.	58	–	0.536
0.469		0.615	–	0.542
25	0.	47	0.048	0.6	49	–	0.554
0.482	0.058	0.6	85	-0.5	70
0.4	88	0.0	65		0.	71	-0.588
28	0.4	94	0.069		0.746	-0.608
29	0.	50	0.075	0.773			-0.623
0.506	0.082	0.7	96	-0.6	31
				0.815	-0.631
0.518	0.103	0.	83
0.524	0.117	0.841	-0.607
0.530	0.135		0.851	-0.581
0.159			0.863	-0.545
0.188	0.8	76		-0.4	99
0.548	0.220		0.888	-0.448
0.248	0.8	93	-0.397
0.560	0.273	0.892	-0.346
0.566	0.293	0.885	-0.299
0.572	0.306	0.	87	-0.259
0.578	0.315	0.854	-0.223
0.585	0.326	0.840	-0.189
0.591	0.337	0.831	-0.158
0.597	0.345	0.822	-0.131
0.603	0.352	-0.110
0.609	0.812	-0.092
0.368	0.811	-0.075
0.621	0.3	80	-0.056
0.627	0.396	0.825	-0.034
0.633	0.416		0.839		-0.007
0.639	0.445	0.027
0.645		0.480		0.894	0.066
0.651	0.519	0.931	0.108
0.657	0.559	0.971	0.148
0.599	1.013	0.185
0.669	0.637	1.057	0.217
0.675	0.671	1.099	0.242
0.682	0.6	98	1.138	0.258
0.688	1.170		0.264
0.694	0.732	1.198	0.265
0.700		0.742	1.222	0.261
0.706	1.243	0.253
0.712	1.253	0.239
0.718	0.735	1.252	0.218
0.724	0.714	1.237	0.1	90
0.730	0.678	1.204	0.152
0.736	0.626	1.151	0.102
0.562	1.079	0.044
0.489	0.9	95	-0.017
0.754	0.417	0.905	-0.070
0.760	0.350	-0.114
0.766	0.284	0.725	-0.156
0.772	0.221	-0.199
0.778	0.157	0.558	-0.245
-0.296
0.032	0.412	-0.348
0.797	-0.022	0.354	-0.399
0.803	–	0.072	0.305	-0.449
0.809	-0.118
-0.159	0.229	-0.547
0.821	-0.196	0.201	-0.593
0.827	-0.234	0.175	-0.644
0.833	-0.275	0.151	-0.700
-0.316	0.128	-0.759
0.845	-0.359	0.107	-0.824
-0.406	0.085	-0.897
0.857	-0.458	0.062	-0.977
-0.512	0.040	-1.064
0.869	-0.568	0.019	-1.155
0.875	-1.246
0.882	-0.673	-0.010	-1.335
-0.715	-0.013	-1.417
-0.750	-1.493
0.900	-0.779	0.005	-1.563
0.906	-0.802	0.021	-1.626
0.912	-0.819	0.043	-1.680
0.918		-0.828	-1.729
0.924	-1.766
100	0.930	-0.810	0.167	-1.787

Sheet1

Price Elasticity
95% Confidence Interval
95% Confidence Interval
Demand Elasticity as a Function of Initial Caloric Share

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

ECON 490: Predictive Analytics
Final Project: Simulating Counterfactuals

The objective of this project is to estimate a structural model of a poor household demand for dietary staples, test the model against the
data, and use the estimated version of the model to extrapolate some predictions. In these notes, I will explain how to use Excel to
simulate counterfactuals. As always, you are welcome to use any computer language to solve this part of the project.

From the first set of notes, we found that the estimated version of the model is

𝑏(𝑝$; 𝑝&, 𝑖) =
𝑖[0.059𝑝& − 𝑝$] + 𝑝&𝑝$(0.941)4,951.367

𝑝$(𝑝& − 𝑝$)

Now, we can employ this model to simulate some counterfactuals. That is, we will use the model to predict what will happen in
environments that are not existent (yet) as, for example, plausible changes in policy. We must keep in mind that there are plenty of
policies that can be evaluated using this model. In these notes, though, we will only perform 2 simulations:

1. We will extrapolate the whole demand function for the average household in the experiment.
2. We will extrapolate the income elasticity of the staple as a function of price.

Simulation 1: Extrapolating the demand function

This simulation is simple. We just need to graph the quantity demanded for bread as a function of price. To generate this graph, it is
useful to set a range of prices of the staple that starts at 0.40 and goes to 1.40, with a space between numbers of 0.01.

Quantity Demanded of the Staple
as a Function of Its Price

This graph characterizes the demand for bread. In the y-axis, we find the quantity demanded for the staple measured in calories. In the
x-axis, we find the price of the staple measured as a proportion of the price of the staple before the experimental treatment was
introduced. Notice that the axes are flipped relative to how we usually teach demand in introductory courses.

According to this simulation, the demand for the staple satisfies the Law of Demand for all the prices lower than 65% of the price before
the treatment. For all the prices that are higher, the demand is Giffen.

Simulation 2: Extrapolating the income elasticity as a function of the staple price

We will extrapolate the arc-elasticity of income as a function of price in this simulation. Recall that the formula of this arc-elasticity is

𝑏8 − 𝑏9
(𝑏9 + 𝑏8)

2;
𝑖8 − 𝑖9

(𝑖9 + 𝑖8)
2;

3000.000

3200.000

3400.000

3600.000

3800.000

4000.000

4200.000

0.
40

0.
44

0.
48

0.
52

0.
56

0.
60

0.
64

0.
68

0.
72

0.
76

0.
80

0.
84

0.
88

0.
92

0.
96

1.
00

1.
04

1.
08

1.
12

1.
16

1.
20

1.
24

1.
28

1.
32

1.
36

1.
40

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

This graph looks like this:
Income Elasticity as a Function of Price

Other possible Simulations

As I mentioned, there are many other simulations that can be performed. Here, I list some of them:

1. Cross-price elasticity of the staple as a function of income.
2. Income elasticity of the staple as a function of the Household Staple Calorie Share (HSCS)
3. Quantity demanded of calories as a function of the price of the staple

-1.500

-1.400

-1.300

-1.200

-1.100

-1.000

-0.900

-0.800

-0.700

-0.600

0.
40
0.
44
0.
48
0.
52
0.
56
0.
60
0.
64
0.
68
0.
72
0.
76
0.
80
0.
84
0.
88
0.
92
0.
96
1.
00
1.
04
1.
08
1.
12
1.
16
1.
20
1.
24
1.
28
1.
32
1.
36
1.
40

Ramses Y. Armendariz, Ph.D.

ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

ECON 490: Predictive Analytics

Final Project: Testing the Model

The objective of this project is to estimate a structural model of a poor household demand for dietary staples, test the model against the

data, and use the estimated version of the model to extrapolate some predictions. In these notes, I will explain how to use Excel to test

the model by comparing its predictions with the data. As always, you are welcome to use any computer language to solve this part of

the project.

From the previous set of notes, we found that the estimated version of the model is

𝑏(𝑝𝑏; 𝑝𝑚, 𝑖) =
𝑖[0.059𝑝𝑚 − 𝑝𝑏] + 𝑝𝑚𝑝𝑏(0.941)4,951.367

𝑝𝑏(𝑝𝑚 − 𝑝𝑏)

Where the value of 𝑖 for the average household in the experiment is 𝑖 = 8,209.194.

Now, we are going to simulate the model to compare its predictions with the data reported in Jensen and Miller (2008, AER). I have

uploaded the data to Compass, in the project section.

Using the spreadsheet from the data, we will create 6 columns. We will use columns from H to M. We will label the columns as follows:

• H: p_m

• I: b_0

• J: b_1

• K: m_0

• L: HSCS

• M: elasticity

The spreadsheet will look like this:

And we will type the following in each cell:

• H2: 1.855

• I2: We will predict the quantity demanded for bread using the estimated version of the model and evaluated at
▪ 𝑖 = 8,209.194
▪ 𝑝𝑏 is equal to 1.
▪ 𝑝𝑚 is equal to the value reported in H2.

• J2: We will predict the quantity demanded for bread using the estimated version of the model and evaluated at
▪ 𝑖 = 8,209.194
▪ 𝑝𝑏 is equal to 1.01.
▪ 𝑝𝑚 is equal to the value reported in H2.

• K2: We will predict the quantity demanded for all the other sources of food using this formula: 𝑚 = (𝑖 − 𝑝𝑏𝑏0) 𝑝𝑚⁄
▪ 𝑖 = 8,209.194
▪ 𝑝𝑏 is equal to 1.
▪ 𝑏0 is equal to the value attained in I2.
▪ 𝑝𝑚 is equal to the value reported in H2.

• L2: We will predict the Household Staple Calorie Share following this formula: 𝐻𝑆𝐶𝑆 = 𝑏0 (𝑏0 +𝑚0)⁄ .
▪ 𝑚0 is going to be equal to the value reported in K2.
▪ 𝑏0 is equal to the value attained in I2.

• M2: We will predict the arc-elasticity using the standard formula.
▪ 𝑏1 is going to be the value reported in J2.
▪ 𝑏0 is going to be the value reported in I2.

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

▪ 𝑝1 = 1.01
▪ 𝑝0 = 1.00

• H3: In this cell, we will type “=$H2+0.005”

After feeling all the previous cells, the spreadsheet looks like this:

Next, we will drag all these columns to the raw 1400. The spreadsheet will look like this:

…

Notice that we have predicted plenty of values. Now, we need to compare these predicted values against the data. For these, we will

develop another column, labeled as “model.” We will use columns F. Now, we will type the following:

• In cell F1, we will type “model.”

• In cell F2, we will type “=VLOOKUP($B2,$L$2:$M$1400,2,TRUE)”

• Now, we will drag cell F2 all the way to raw 101.

The spreadsheet will look like this:

…

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics
Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

Now, we are basically done. We just need to report our results. We will do this with a graph. We will graph columns C, D, E, and F.

Elasticity as a Function of HSCS

*The dashed lines are the 95% confidence intervals from the data.

The pointed line is the point estimate from the data.

The solid line is the model prediction.

The graph we have produced compares the values that the model predicts with the values measured in the experiment. Notice how the

model performs relatively well. For most of the points, its prediction lies inside the 95% confidence interval.

ar DGPs.

-2.000

–

1.500

–

1.000

–

0.500

0.000

0.500
1.000
1.500

0.330 0.391 0.451 0.512 0.572 0.633 0.694 0.754 0.815 0.875

Ramses Y. Armendariz, Ph.D.

ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

ECON 490: Predictive Analytics

Final Project: Model Estimation

The objective of this project is to estimate a structural model of a poor household demand for dietary staples, test the model against the

data, and use the estimated version of the model to extrapolate some predictions. In these notes, I will explain how to use Excel to

estimate the model; however, you may use any computer language to estimate this model. Later, I will upload notes on how to test the

model against the data and other notes on how to use the model to make predictions.

In this project, we propose that a poor household’s demand for dietary staples follows this Data-Generating Process (DGP):

𝑏 =
𝑖[𝛽𝑝𝑚 − 𝑝𝑏] + 𝑝𝑚𝑝𝑏(1 − 𝛽)𝑐̅

𝑝𝑏(𝑝𝑚 − 𝑝𝑏)
+ 𝑒

The variables in this model are the following:

• Endogenous variable: quantity demanded for the staple measured in calories (𝑏).
• Exogenous variables:

o Price of a calorie from the staple (𝑝𝑏).
o Price index of a calorie from all the other sources of food (𝑝𝑚).
o Income of the household (𝑖).
o Quantity of caloric intake that makes the feeling of hunger become unbearable (𝑐̅).
o The importance of satiating hunger in the preferences of the household (𝛽).

• 𝑒 is a random variable. We ignore its probability distribution. All we know is that 𝐸(𝑒) = 0.

Estimation Process

Step 1: Transform the model to match it to the data

From Table 1 in Jensen and Miller (2008, AER), we know that

• average caloric intake per household before the experimental treatment took place was 1,805(2.85) = 5,144.

• the caloric share of the staple before the experimental treatment took place was 0.64.

We are going to use these numbers to calculate the income of the household, 𝑖 = 𝑝𝑏𝑏 + 𝑝𝑚𝑚, according to the economic model proposed
in this project. Specifically, we are going to use these numbers to substitute for 𝑏 and 𝑚. After the substitution, we find that the income
of the household before the experimental treatment took place is

𝑖 = 𝑝𝑏1,805(2.85)0.64 + 𝑝𝑚1,805(2.85)0.36 = 1,805(2.85)[0.64𝑝𝑏 + 0.36𝑝𝑚]

Now, we can substitute for income in the demand:

𝑏 =
5,144.25(0.64𝑝𝑏 + 0.36𝑝𝑚)[𝛽𝑝𝑚 − 𝑝𝑏] + 𝑝𝑚𝑝𝑏(1 − 𝛽)𝑐̅

𝑝𝑏(𝑝𝑚 − 𝑝𝑏)

Step 2: Choose the moments that the model is going to match

First, we are going to normalize the model in terms of the price of the staple before the experimental treatment was introduced. Thus,

we are going to set the original price of the staple equal to 1 (𝑝𝑏 = 1). Second, we need to find values for the remaining variables in the
last equation: 𝑝𝑚, 𝛽, 𝑐̅. Thus, at minimum, we need three moments.

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics

Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

1. The average consumption of 𝑏 before the experimental policy was introduced.

5,144.25(0.64 + 0.36𝑝𝑚)[𝛽𝑝𝑚 − 1] + 𝑝𝑚(1 − 𝛽)𝑐̅

(𝑝𝑚 − 1)
= 2.85(1,805)0.64

2. The average consumption of 𝑏 after its price has increased by 1%.

5,144.25(0.64 + 0.36𝑝𝑚)[𝛽𝑝𝑚 − 1.01] + 𝑝𝑚1.01(1 − 𝛽)𝑐̅

1.01(𝑝𝑚 − 1.01)
= 2.85(1,805)0.64(1 + 0.00235)

3. The average consumption of 𝑏 after its price has decreased by 1%.

5,144.25(0.64 + 0.36𝑝𝑚)[𝛽𝑝𝑚 − 0.99] + 𝑝𝑚0.99(1 − 𝛽)𝑐̅

0.99(𝑝𝑚 − 0.99)
= 2.85(1,805)0.64(1 − 0.00235)

Notice that the price of the staple in the first moment is equal to 1 (because that is the price we normalized), in the second moment is

equal to 1.01, and in the third moment is equal to 0.99.

The values in the right hand of the equations characterize the ATE that Jensen and Miller (2008, AER) document in Table 3. In

particular, they document that the price elasticity of the staple is + 0.235, making it a Giffen good. 0.00235 is the price elasticity after

it has been divided by 100.

Step 3: Solve the system of equations defined in step 2.

It is necessary to use a computer to approximate numerically the solution to this system of equations. I will explain how to do it using

Excel; however, it can be approximated with other software as well (e.g., Matlab or Python).

To solve this system of equations in Excel, it is necessary to install solver in it. To install solver, follow these instructions:

https://support.office.com/en-us/article/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-

e24772f078ca#OfficeVersion=Windows

Then, create a spreadsheet that looks like this:

Note that this spreadsheet contains four sections: Parameters, Function, Objective, and Population Moments. The section of

Parameters contains the values that the computer will find. Those are the three parameters we are looking for. The section Function

https://support.office.com/en-us/article/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-e24772f078ca#OfficeVersion=Windows

https://support.office.com/en-us/article/load-the-solver-add-in-in-excel-612926fc-d53b-46b4-872c-e24772f078ca#OfficeVersion=Windows

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics
Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

contains the demand functions that we are going to match to the sample moments. The section objective contains the Objective

function that we are going to make the computer minimize in order to find the values of the parameters. Finally, the section Population

Moments contains the values of the sample moments we want to match.

PARAMETERS

We need to state some initial guesses for all these parameters. These initial guesses will give the computer an initial point to start its

approximation. We want, therefore, to set “educated guesses” rather than just purely arbitrary ones. A good guess will save computing

power. A bad guess, on the other hand, can make the computer approximate a point that is not the solution we are looking for. So, let’s

state the following guesses:

• beta: All we know is that 𝛽 ∈ [0,1], so we should set beta = 0.50, which is half way in between.
• c_bar: Average caloric intake per household in the data is 1,805(2.85) = 5,144. Let’s use this number as an initial guess.

• p_m: This price index should be higher than the price of the staple. Thus, let’s guess it is twice the price of the staple: 2.

FUNCTION

In this section, we need to set the functions from Step 2. Specifically, we need to type in cell A12 of the spreadsheet the left hand side

of equation 1 in Step 2. Then, we need to type in cell C12 the left hand side of equation 2. Finally, we need to type in cell B12 the left

hand side of equation 3. Make sure that the 𝛽 from each function is linked to cell A3, the 𝑐̅ from each function is linked to cell B3, and
𝑝𝑚 from each function is linked to cell C3.

POPULATION MOMENTS

• Plug 2.85(1,805)0.64 in cell A17.
• Plug 2.85(1,805)0.64(1 − 0.00235) in cell B17.
• Plug 2.85(1,805)0.64(1 + 0.00235) in cell C17.

Notice that these are the values of the right hand of the equations from Step 2. Strictly speaking, these values are the sample moments

that Jensen and Miller collected with their experiment. However, I call them Population Moments because the Method of Moments

assumes that they are equal.

OBJECTIVE

In cell G12, we need to type

= (𝐴12 − 𝐴17)2 + (𝐵12 − 𝐵17)2 + (𝐶12 − 𝐶17)2

We are going to ask Solver to minimize this function. This is because the solution to the system of equations that characterize the

method of moments minimizes the value of this function.

After having followed all these steps, the screen should look like this:

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics
Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

USING SOLVER

1. Click on cell G12.
2. Launch Solver. Solver is located in the Data tab after you have installed it.
3. If you use Excel for Macbook, Solver will look like this:

4. In the box that says “Set Objective,” type “$G$12.”
5. Choose “Min” in the option that says “To:”.
6. In the space that says “By Changing Variable Cells:”, write “Sheet1!$A$3:$C$3.”
7. In the section for constraints, you will add 6 constraints. For each constraint, you need to click the add button.

a. Cell Reference: “Sheet1!$A$3.” Then, choose “>=”. Then, in Constraint write “0”. Finally, click “Add.”
b. Cell Reference: “Sheet1!$A$3.” Then, choose “<=”. Then, in Constraint write “1”. Finally, click “Add.” c. Cell Reference: “Sheet1!$B$3.” Then, choose “>=”. Then, in Constraint write “0”. Finally, click “Add.”
d. Cell Reference: “Sheet1!$B$3.” Then, choose “<=”. Then, in Constraint write “10000”. Finally, click “Add.” e. Cell Reference: “Sheet1!$C$3.” Then, choose “>=”. Then, in Constraint write “1.5”. Finally, click “Add.”
f. Cell Reference: “Sheet1!$C$3.” Then, choose “<=”. Then, in Constraint write “4”. Finally, click “OK.”

8. In “Select a Solving Method” choose “GRG Nonlinear.”
9. Click “Solve.”

After point 8., the screen from solver should look like this:

Ramses Y. Armendariz, Ph.D. ECON 490: Predictive Analytics
Department of Economics Spring 2020 University of Illinois, Urbana-Champaign

And, after point nine, the spread sheet should look very similar to this:

ESTIMATED VALUES

The estimated values of the parameters are �̂� = 0.059, 𝑐̅̂ = 4,951.367, �̂�𝑚 = 2.655. Notice that the values that are approximated may
change slightly each time we approximate. Thus, the estimated version of our model is the following:

𝑏(𝑝𝑏; 𝑝𝑚, 𝑖) =
𝑖[0.059𝑝𝑚 − 𝑝𝑏] + 𝑝𝑚𝑝𝑏(0.941)4,951.367

𝑝𝑏(𝑝𝑚 − 𝑝𝑏)

Or, we can also present the demand model in terms of the price of the staple only:

𝑏(𝑝𝑏) =
8,209.194[0.156 − 𝑝𝑏] + 12,370.272𝑝𝑏

𝑝𝑏(2.655 − 𝑝𝑏)

In this later estimated version of the model, I substituted for the income using this formula: 𝑖 = 𝑝𝑏𝑏 + 𝑝𝑚𝑚, where the value of 𝑏 is
2.85(1,805)0.64, the value of 𝑚 is 2.85(1,805)0.36, the value of 𝑝𝑚 is 2.655, and the value of 𝑝𝑏 is 1. When you substitute all these
numbers, you find that 𝒊 = 𝟖, 𝟐𝟎𝟗. 𝟏𝟗𝟒. This is the average real income of a household in the experiment. This real income is
measured in terms of staple calories. This is one of the main results of this estimation process. It provides an alternative technique to

measure real income in terms of nutrition.

Talking about the results

The numerical approximation found the values of 𝛽, 𝑐̅, and 𝑝𝑚 such that it made the prediction of the model be equal to the sample
moments. This can be seen by comparing the screen shot of the spread sheet before we used Solver with the screen shot of after. In the

screen shot before we used solver, the predicted values of the model were around 5,000 calories. These numbers are shown in the section

FUNCTION of the spreadsheet. On the other hand, the values of the sample moments (those are the values in the section POPULATION

MOMENTS) are around 3,300. Therefore, the prediction of the model was not close to the numbers reported by the data. In the screen

shot after we used solver, the predicted values almost matched those values of the sample moments. That is, solver found the values of

the parameters that make the model predict the values reported by the data.

Now, you have an estimated structural model of a poor household’s demand for dietary staples ready to be used to make active

predictions! Notice that this model is not linear, which makes it impossible for us to employ the estimation techniques we have learned

for linear DGPs.