Shadow price

A company manufactures X, Y and Z. The market demand for these products are 4,000, 2,000 and 1,500 respectively.

Here are the details of each product.

X:
Timber cost $5
Contribution $8

Y:
Timber cost $15
Contribution $17.50

Z:
Timber cost $10
Contribution $16

The cost of timber is $2 per m2.

The timber limited in supply to 20,000 m2 per annum.

The optimum production plan is calculated to be:
4,000 X ; 333 Y ; 1,500 Z.

Calculate the maximum price which should be paid per m2 for timber in order to obtain extra supplies.

My answer was:
Since the market has been satisfied for X and Z, we are left with product Y whose market has not been satisfied.

The shadow price can therefore be calculated by:

2.5 (4,000) + 7.5 Y + 5 (5,000) = 20,001 m2
Y= 333.467 — (1)

Old Contribution of optimum production plan:
8 X + 17.5 Y + 16Q = $61,827 — (2)
Sub (1) into (2)
New contribution= 8 (4,000) + 17.5 (333.467) + 16 (1,500) = $61,835.57

The shadow price is therefore: $8.67

Maximum price is $8.67 + $2 = $12.67.

However,
The solution given was:
The present situation is that demand for X and Z is fully satisfied from existing resources, but there is some in satisfied demand for Y. Thus any additional timber would be used to manufacture more Y.

Based on the current input cost of $2 per m2, each m2 of timber earns a contribution of $2.33. This the maximum price to be paid is the sum of these values = 4.33 per m2.

However, there is no benefit in obtaining more timber that can be used to satisfy the total demand for benches, so this shadow price of $4.33 per m2 only applies for up to 12,500 m2 of timber. Thereafter there is no use for the timber, so it’s shadow price is nil. (I understood this part)

Is my method wrong? Should I use this method to calculate shadow prices now?

Reference: 2014 Becker education Corp revision kit QNS 12.