Part 2 Four-Quadrant Model

Mathematically, the four-quadrant diagram (Lectures, and Dipasquale and Wheaton) represents a solution to a system of four simultaneous equations that describe the operation of the property (space) and asset (capital) markets.

In this question, we consider the property and asset market for 2-bedroom rental apartments. An example of four plausible equations for the four-quadrant model could be the following (these are slightly different than in the reading and the lecture):

D = 2-bedroom apartment demand in 1000s of square feet

E = number of childless couples working from home (in 1000s)

R = annual rent in $/square foot

P = asset price in $/square foot

i = long term cap rate for real estate

f(C) = replacement cost of real estate

C = new construction supply in 1000s of square feet

S = stock of space in 1000s of square feet

= long run depreciation

Its January 2020, pre-COVID, the values of the exogenous parameters are as follows:

E = 10

i = 0.10

= 0.02

Solve for the equilibrium solution for all of the endogenous variables (R, P, C, S) given the values of the exogenous variables above. What is the total square feet per couple assuming all 2-bedroom apartments are exactly the same size?

Use the sketch of the four-quadrant model below to depict the equilibrium.

For parts (c)-(e), use the above equations to solve for how each of the following exogenous changes affect conditions in the space and asset markets and lead to a new equilibrium state. In each scenario, describe how and why rents, asset prices, construction, and stock of space, change. Note: each scenario builds on the previous one, i.e. in (e) assume (c) and (d) have already happened.

1. Fast forward to April 2020, and the pandemic has caused a surge in the number of childless couples working from home. E increases from 10 to 15. Calculate the short run values of rent and price given stock and construction do not change. Make sure to describe how/why P and R change.

2. In the long run, construction and stock adjust. What happens to price and space/couple? Compare this long run equilibrium to the equilibrium found in part (a) using a four-quadrant model figure

3. Months into the pandemic, urban apartment real estate is seen as a risky investment so potential holders of the asset demand a higher return (in order to hold the asset). The cap rate increases from 0.10 to 0.12. What happens to price in the short run given that stock is fixed at the level found in (d) (explain in words as well). What happens in the long run if this new cap rate persists and construction and stock adjust accordingly?

4. Compare this long run equilibrium to the equilibrium found in part (d) using a four-quadrant model figure.