ABSTRACT

Morgan (1983) guaranteed that VSS dominated both FSS and SSR. But it is difficult to calculate the optimal sample size and the optimal reservation price both without recall and with full recall. As VSS without recall is a simplification of VSS with full recall, we will present on appendix a VB30 program that calculates only the full recall case. As known, on VSS, the search is sequential and in each period the sample size is variable.

As normal, we will extract sellers prices from F(x) that is common knowledge, temporal horizon is T, goods are homogenous, no discount and consumer buys once just one unity of goods.

1. WITHOUT RECALL

1.1. DETERMINATION OF THE SAMPLE SIZE, N(T)*.

Buyer must calculate how many prices to ask. In the beginning of each period, he will chose n that minimizes expected value of expenditure, Pi + K(n).

(1)

We can see that there is a reservation price, P(T)*.

(2)

Using, on equation (1), M (x) instead of Min(P1…Pn) and P(T)* instead of , it gives the next equation.

(3)

Using expectation we will have the next equation, that we will minimize using n as variable.

(4)

We can see on Fig 1 the evolution on the optimal size with the diminution of the time horizon, Prices from 10,20, uniform, and K(n)= 0.15n.

Fig 1- Sample Size Without Recall

1.2. DETERMINATION OF THE RESERVATION PRICE, P*

Now that we know the optimal sample size, n*, we have the reservation price, P(T+1)*, as the minimum of equation (4).

Equation (4), that we show with p(T+1)* on next equation, will permit calculate, by backward induction, the reservation prices and sample size at all periods. Being , the marginal gain of search, P(T+1)*-P(T) will be the cost of losing one opportunity.

(5)

We can see on Fig 2 the evolution on the reservation price with the diminution of the time horizon, prices from 10,20, uniform, and K(n)= 0.15n.

Fig 2- Reservation Prices Without Recall

We have SSR and FSS on Fig 1 and Fig 2 to show that VSS dominate them.

2. WITH PERFECT RECALL

Apparently, this is a very complex problem as the reservation price changes with the best price found till now, Z, but changes too with the present price, Pi, as we can see on next equation (we use n* = 1 on expressions to simplify them with no lost).

(6)

Using expectation, we will have a variable of integration as limit of the integration as owe can see on next equation (P(x, T)*<Z with no lost).

(7)

We will call P(Z,T) the Reservation Function because it changes with Z.

The reservation price will exists only under the next condition, Lippman and McCall (1976, eq. 24).

(8)

We see on Fig 3 that the reservation function is convex to origin, and P(0,T) is positive, so reservation price exist as the solution to the next equation.

(9)

The reservation price, Pr(T)*, don't depends on Z or on Pi, and is the limit of integration on equation (7)

2.1. DETERMINATION OF THE SAMPLE SIZE, N(T)*.

If Z Pr(T)*, the expected value of expenditure, known Z, is given by the next equation.

(10)

If Z > Pr(T)*, we will have instead the next equation.

(11)

The sample size minimizes equations (10). or (11). We can see on Fig 4 the relation between sample size, best price till now, and time horizon. Prices on 10,20, rectangular, and K(n)= 0.15n.

Fig 4 - Sample Size function of Z and T, with Perfect Recall

2.2. DETERMINATION OF THE RESERVATION PRICE, PR(T)*

Has we have seen the reservation price is the solution to being P(x, T, n)* the minim of equations (10). or (11).

We can see on Fig 5 the relation between the reservation function, the reservation price, the temporal horizon and Z.. Prices on 10,20, rectangular, and K(n)= 0.15n.

The reservation price is constant all periods but the last (Gal e al. 1981).

APPENDIX

A program written in VB30 to calculate the VSS with recall.

We used a array with 101 points (0 to Resol)of the function P(Z,T) and, by backward induction, we calculate al periods. N_Optim is a variable global that is the optimal sample size.

Recursivity is computationally inefficient.

By running Reserv_Function we calculate P(Z,T+1). For example, to calculate P(Z,10) we will do the next routine.

Initialise

For I=1 to 10

Reserv_Function

Next i

And then we may print Pact(i). To print the optimal sample size we need to do it inside routine Reserv_Function.

´ Declarations

Const c = .15, Resol=100

Dim N_Optim As Integer, Pant(0 To Resol), Pact(0 To Resol)

Sub initialise ()

'P(Z,T) for T = 1

Dim i As Integer

For i = 0 To Resol

Pant(i) = 20

Next i

End Sub

Function g (N As Integer, x)

'Distribution of minimum prices on a sample size n

Dim fp, Fg

'fp(x) - uniform on [10 e 20]

If (x >= 10) And (x <= 20) Then

If (x = 10) Or (x = 20) Then fp = .05 Else fp = .1

Else

fp = 0

End If

'Fg(x)

If (x >= 10) Then

If x < 20 Then Fg = -1 + x / 10 Else Fg = 1

Else

Fg = 0

End If

'g(n,x)

g = N * (1 - Fg) ^ (N - 1) * fp

End Function

Sub Reserv_function ()

'Calcul Resol+1 points from the reservation function

Dim Z, LimInt, i As Integer

LimInt = Pr()

For i = 0 To Resol

Z = 10 + i *10/ Resol

Pact(i) = Point_Resev_Function(Z, LimInt)

'Here print variable N_Optim

Next i

For i = 0 To Resol ' Pant is the P(Z,T) and Pact is the P(Z,T+1)

Pant(i) = Pact(i)

Next i

End Sub

Function Pr ()

Dim L1 As Integer, L2 As Integer, L3 As Integer

Dim V2, X1, X3

L1 = 0: L3 = Resol

If (Pant(L3) - (10 + L3*10 / Resol) >= 0) Then

Pr = 20

Exit Function

End If

While L3 - L1 > 1

L2 = Int((L1 + L3) / 2 + .01)

V2 = Pant(L2) - (10 + L2 *10/ Resol)

If V2 >= 0 Then L1 = L2 Else L3 = L2

Wend

X1 = 10 + L1*10 / Resol

X3 = 10 + L3 *10 / Resol

Pr = X1 - (X3 - X1) / (Pant(L3) - X3 - Pant(L1) + X1) * (Pant(L1) - X1)

End Function

Function Point_Reserv_Function (ByVal Z, LimInt)

Const L3 = 20

Dim L1, L2, ConstZ, N As Integer

Dim V1, V2

'Case Z<=Pr* or otherwise

If Z <= LimInt Then

L1 = Z: L2 = Z

ConstZ = Z

Else

L1 = LimInt: L2 = Z

ConstZ = Pvss_inter(Z)

End If

'Minimization

N = 1

V1 = Integ(N, L1, L2, ConstZ) + N * c

V2 = Integ(N + 1, L1, L2, ConstZ) + (N + 1) * c

While V1 >= V2

N = N + 1

V1 = V2

V2 = Integ(N + 1, L1, L2, ConstZ) + (N + 1) * c

Wend

N_Optim = N

Point_Reserv_Function = V1

End Function

Function Integ (ByVal N As Integer, L1, L2, ConstZ)

Const Delta = .01, L3 = 20

Dim Integral, x

Integral = 0

For x = 0 + Delta / 2 To L1 Step Delta

Integral = Integral + x * g(N, x) * Delta

Next x

For x = L1 + Delta / 2 To L2 Step Delta

Integral = Integral + P_inter(x) * g(N, x) * Delta

Next x

For x = L2 + Delta / 2 To L3 Step Delta

Integral = Integral + ConstZ * g(N, x) * Delta

Next x

Integ = Integral

End Function

Function P_inter (Z)

Dim V1, V2, L1, L2

Dim i As Integer

i = Fix((Z - 10) * Resol / 10)

If i = Resol Then

P_inter = Pant(Resol)

Else

L1 = 10 + i *10 / Resol

V1 = Pant(i)

L2 = 10 + (i + 1) *10 / Resol

V2 = Pant(i + 1)

P_inter = V1 + (V2 - V1) * (Z - L1) / (L2 - L1)

End If

End Function

REFERENCES

Gal, S., M. Landsberger and B. Levykson (1981). "A Compound Strategy for Search in the Labour Market." International Economic Review, Vol. 22, No. 3, pp. 597-608.

Lippman, Steven A. and John J. McCall (1976). "The Economics of Job Search: A Survey." Economic Inquiry, Vol. 14, pp.155 - 89.

Morgan, Peter B. (1983). "Search and Optimal Sample Sizes." Review of Economic Studies, Vol. 50, pp. 659-75.