Appendix C: Equations and Parameters | RDP 2020-04: The Apartment Shortage

RDP 2020-04: The Apartment Shortage Appendix C: Equations and Parameters

Keaton Jenner and Peter Tulip

August 2020

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This appendix explains the equations and parameter values for the costs of building up and out in Figures 7 and 8 and the efficient heights in Figure 9.

We start by rewriting Equation (1) for average construction cost, ACC, in abbreviated but hopefully obvious notation:

(C1)

A C C = B a s e + S l o p e \times H e i g h t

As discussed in Section 4.2, for Sydney Base = $316,337 and Slope = $2,291. We compound developer's margins, finance, managerial and professional fees. Values for these terms (as percentages) are in Table 3. Their product as a ratio, 1.37, is represented by $θ$ . We enter Infrastructure charges additively. This gives average variable costs, AVC, the blue line in Figure 8.

(C2)

A V C = (B a s e + (S l o p e \times H e i g h t)) \times θ + I n f r a s t r u c t u r e c h a r g e s

We multiply by number of apartments (= Units per storey × Height) to get total cost. Differentiating with respect to height gives the marginal cost of supplying apartments by raising height. We then divide by Units per storey (assumed to be constant) to express on a per apartment basis.

(C3)

M C = (B a s e + 2 \times (S l o p e \times H e i g h t)) \times θ + I n f r a s t r u c t u r e c h a r g e s

which is the black line in Figures 7 and 8. The average total cost of building out per apartment is:

(C4)

A T C = L a n d c o s t p e r a p a r t m e n t + A V C

(C5)

\begin{array}{l} A T C = (\frac{L a n d c o s t_{p e r s q m} \times L a n d a r e a r e q u i r e d}{U n i t s p e r s t o r e y \times H e i g h t}) \times γ + (B a s e + (S l o p e \times H e i g h t)) \\ \times θ + I n f r a s t r u c t u r e c h a r g e s \end{array}

In Figure 7, Land cost = $4,033 per square metre and Land area required = 2,397 square metres are from Table 4. Units per storey = 11.2 is apartments per building = 117, from Table 4, divided by predicted height in 2018 = 10.5, from Figure A1, after rounding. (Note that units per building in Table 8, 11.4, is for 2013 to 2018.) The land component of average costs is scaled by $γ$ , equal to 1.5. This represents similar factors as $θ$ but is larger, reflecting the greater uncertainty (and therefore larger margins and cost of debt) that exists at the beginning of a project.

The ‘efficient’ or lowest-cost building height is where marginal cost (Equation (C1)) equals average cost (Equation (C2)). That is

(C6)

\begin{array}{l} (B a s e + 2 \times (S l o p e \times H e i g h t)) \times θ + I n f r a s t r u c t u r e c h a r g e s \\ = (\frac{L a n d c o s t_{p e r s q m} \times L a n d a r e a r e q u i r e d}{U n i t s p e r s t o r e y \times H e i g h t}) \times γ \\ + (\begin{array}{l} (B a s e + (S l o p e \times H e i g h t)) \\ \times θ + I n f r a s t r u c t u r e c h a r g e s \end{array}) \end{array}

Re-arranging for height yields the expression:

(C7)

H e i g h t = \sqrt{\frac{L a n d c o s t_{p e r s q m} \times L a n d a r e a r e q u i r e d_{s q m} \times \frac{γ}{θ}}{U n i t s p e r s t o r e y \times S l o p e}}

In Figure 8 we hold all the right-hand side variables in Equation (C7) constant except the land cost, which we calculate from the CoreLogic data as the average value of houses sold for a given SA3, divided by the average land area of those properties.