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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.
a_O	a₁	Y	N_1-5	N_6-10	N_11-15	N_16-20	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

a_O	a₁	p	N_1-5	N_6-10	N_11-15	N_16-20	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

Public fields
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.
Public methods
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.
Enumerations
Anonymous enum 1		State space dimensions.

Fertility

Enumerations
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.

WolpinJPE1984.ox

The Model

Replication

Table 5. Estimated and Replicated Birth Probabilities

Effects of ln Y = a00+a01t

Table 9. Effects of ln (p/1-p) = a10+a11t

Fertility

Fertility

aa

ab

alph

b

bet

c

delt

EMax

FeasibleActions

gam

M

Mortality

n

PD

prow

psi

Replicate

Sbar

TimeEffect

Utility

Yrow

Effects of ln Y = a₀₀+a₀₁t

Table 9. Effects of ln (p/1-p) = a₁₀+a₁₁t