WolpinJPE1984.ox
Replicate Tables 8 and 9 of Wolpin (1982), Yale Economic Growth Center Working Paper No. 424.
This paper was published in the JPE 1984. The published version makes references to these tables but they are not included.
The parameter estimates differ between the working paper and the article. The working paper version are used for the replication,
but the published estimates are given as well.
The Model
Clock: Finite Horizon, \(t = 0, 1, \dots, 19,\dots 29\).
Action Vector: \(\alpha = (n)\), \(n \in \{0,1\}\)
\(n\) is the choice to have another child or not.
Endogenous State Vector: \(\theta = (M)\)
M = number of children at the start of \(t\)
Newborns die with a probability that depends on the mother's age \(t\):
$$p = Prob(death) = Prob(d=1) Logit(a_0 + a_1 t)$$
The stock of children evolves as
$$M' = \cases{ M + 1 & with prob. (1-p)n\cr
M & otherwise.\cr.} $$
Feasible Actions
The woman is fertile for 20 years
\(A(\theta) = \cases{ \{0,1\} &if t< 20\cr
\{0\} &if t= 20\cr}\)
Exogenous State
\(\epsilon = (\psi)\), and \(\psi \sim dN(0,1)\)
Note:The paper solves for cut-off value of \(\psi\) for having a child. This
would be a OneDimensionalChoice() and use of the ReservationValue method.
Instead, the replication simply draws 201 discrete values of \(\psi\) and makes the optimal choice.
Income
$$X = Y - ( b(n-d) + (c_1I_{t=1} + c_2I_{t=2}+ c_{30}+c_{31}t+c_{32}t^2))n.$$
Utility
The model's timing assumes that infant mortality is realized after choosing \(n_t\) but before
the end of \(t\) and before utility is realized. Thus utility is an expectation over \(d_t\).
$$\eqalign{
M* &= M+n-d\cr
U(n,d) &= (\alpha_1+\psi)M* +\alpha_2(M*)^2+ \beta_1X+\beta_2X^2 + \gamma_1M*X+\gamma_2M*S.\cr
EU(0) &= U(0,0)\cr
EU(1) &= (1-p)U(1,0)+pU(1,0)\cr}$$
Replication
Table 5. Estimated and Replicated Birth Probabilities
t Predicted Replicated
Probability Probability
--------------------------------
0 .138 0.13930
1 .360 0.37811
2 .552 0.54229
3 .543 0.53234
4 .530 0.52239
5 .515 0.50746
6 .498 0.49254
7 .477 0.47264
8 .454 0.44776
9 .428 0.42289
10 .400 0.39801
11 .376 0.36816
12 .345 0.33831
13 .313 0.30630
14 .281 0.27363
15 .250 0.23881
16 .216 0.20701
17 .182 0.17413
18 .149 0.14428
19 .122 0.11473
-----------------------------
Effects of ln Y = a00+a01t
Table 8. Working Paper Page 45.
| aO | a1 | Y | N1-5 | N6-10 | N11-15 |
N16-20 | N |
|---|
| 9.35 | O | 11,500 | 2.124 | 2.369 | 1.687 | .867 | 7.047 |
| 8.52 | O | 5,000 | 2.117 | 2.362 | 1.682 | .863 | 7.024 |
| 9.21 | O | 10,000 | 2.121 | 2.367 | 1.685 | .866 | 7.039 |
| 10.13 | O | 25,000 | 2.134 | 2.380 | 1.699 | .873 | 7.086 |
| 10.82 | O | 50,000 | 2.154 | 2.402 | 1.717 | .885 | 7.158 |
| 8.52 | .088 | 11,500 | 2.126 | 2.376 | 1.769 | .875 | 7.146 |
Replicated
Table of Working Paper 8, page 45
----------------------------------
row N1-5 N6-10 N11-15 N16-20 N
1.0000 2.1144 2.3433 1.6844 0.8790 6.5029
2.0000 2.0939 2.3163 1.6614 0.8703 6.0759
3.0000 2.0647 2.2883 1.6368 0.8557 5.6648
4.0000 2.0380 2.2587 1.6156 0.8438 5.2564
5.0000 2.0100 2.2331 1.5920 0.8261 4.8533
6.0000 2.1443 2.3775 1.7088 0.8986 6.9975
Table 9. Effects of ln (p/1-p) = a10+a11t
| aO | a1 | p | N1-5 | N6-10 | N11-15 |
N16-20 | N |
|---|
| 2.78 | O | .94 | 2.124 | 2.369 | 1.687 | .867 | 7.047 |
| 2.09 | O | .89 | 2.078 | 2.326 | 1.614 | .783 | 6.801 |
| 1.67 | O | .84 | 2.032 | 2.282 | 1.540 | .701 | 6.555 |
| 1.33 | O | .79 | 1.979 | 2.230 | 1.453 | .612 | 6.274 |
| 1.05 | O | .74 | 1.927 | 2.175 | 1.367 | .532 | 6.001 |
| 1.05 | 1.81 | .94 | 2.003 | 2.356 | 1.722 | .937 | 7.018 |
Replicated
Table of Working Paper 9, page 47
----------------------------------
row N1-5 N6-10 N11-15 N16-20 N
1.0000 2.1144 2.3433 1.6844 0.8790 6.5029
2.0000 2.1095 2.3380 1.6816 0.8767 6.4888
3.0000 2.1144 2.3433 1.6833 0.8787 6.5019
4.0000 2.1294 2.3578 1.6890 0.8844 6.5380
5.0000 2.1443 2.3728 1.7081 0.8990 6.5956
6.0000 2.1244 2.3532 1.6926 0.8949 6.5400
Fertility
Wolpin 1984 fertility model
|
 aa |
static const |
|
| ab |
static const |
|
| alph |
static const |
Parameters from the
1982 working paper version. |
| b |
static const |
|
| bet |
static const |
|
| c |
static const |
|
| delt |
static const |
|
| EMax |
static |
solution method. |
| gam |
static const |
|
| M |
static |
stock of children |
| n |
static |
decision variable |
| PD |
static |
PanelPrediction. |
| prow |
static |
row of birth prob. |
| psi |
static |
shock to preferences \(\psi\) |
| Sbar |
static const |
|
| Yrow |
static |
row of income table. |
|
 FeasibleActions |
|
Return A(θ). |
| Mortality |
static |
Returns current time-specific transition of number of children. |
| Replicate |
static |
Run the replication, compute predictions. |
| TimeEffect |
static |
Compute \(\pmatrix{1 & t}coeff\). |
| Utility |
|
Utility. |
|
 Anonymous enum 1 |
|
State space dimensions. |
Fertility
|
| Anonymous enum 1 |
State space dimensions. @names dimens
|
T, tau, Ndraws, Mmax |
aa
static const decl aa [public]
-
ab
static const decl ab [public]
-
alph
static const decl alph [public]
-
Parameters from the
1982 working paper version.
b
static const decl b [public]
-
bet
static const decl bet [public]
-
c
static const decl c [public]
-
delt
static const decl delt [public]
-
EMax
static decl EMax [public]
-
solution method.
FeasibleActions
Fertility :: FeasibleActions ( )
-
Return A(θ).
Fertility is not a feasible choice for t>T-1
gam
static const decl gam [public]
-
M
static decl M [public]
-
stock of children
Mortality
static Fertility :: Mortality ( )
-
Returns current time-specific transition of number of children.
Since \(M\) is a RandomUpDown() state variable this returns a row vector of three
probailities for \(M'\) equal to \(M-1,M,M+1\), respectively. In the paper's notation:
$$p = Prob(death) = Prob(d=1) Logit(a_0 + a_1 t)$$
Here the three probabilities are \(\pmatrix{0 & (1-np) & pn}\).
n
static decl n [public]
-
decision variable
PD
static decl PD [public]
-
PanelPrediction.
prow
static decl prow [public]
-
row of birth prob. table.
psi
static decl psi [public]
-
shock to preferences \(\psi\)
Replicate
static Fertility :: Replicate ( )
-
Run the replication, compute predictions.
Sbar
static const decl Sbar [public]
-
TimeEffect
static Fertility :: TimeEffect ( coeff )
-
Compute \(\pmatrix{1 & t}coeff\).
Utility
Fertility :: Utility ( )
-
Utility.
-
$$\eqalign{
X &= Y - ( b(n-d) + (c_1I_{t=1} + c_2I_{t=2}+ c_{30}+c_{31}t+c_{32}t^2))n.\cr
M* &= M+n-d\cr
RU(n,M) &= (\alpha_1+\psi)M* +\alpha_2(M*)^2+ \beta_1X+\beta_2X^2 + \gamma_1M*X+\gamma_2M*S.\cr
EU(0) &= RU(0,M)\cr
EU(1) &= (1-p)RU(1,M+1)+pRU(1,M)\cr}$$
Yrow
static decl Yrow [public]
-
row of income table.
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