# Housing policy analysts debate the best way to increase the number of housing units available to low

Feb 3rd, 2012
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a- With a price ceiling at PC, the quantity bought and sold is QC, indicated by point A. The ceiling at PC is eliminated and the rent returns to the market equilibrium E1, with an equilibrium rent of P1. The quantity supplied increases from QC to the equilibrium quantity Q1. At the same time, you should expect the quality of rental housing to improve. One of the inefficiencies caused by price ceilings is inefficiently low quality.

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1. Prove :?x?y(StrongPref(x,y)? StrongPref(y,x))2. Prove :?x?y?z((Indiff(x,y)?StrongPref(y,z))?StrongPref(x,z))using (for questions 1 and 2)?x?y(WeakPref(x,y)?WeakPref(y,x))?x?y?z((WeakPref(x,y)?WeakPref(y,z))?WeakPref(x,z))?x?y(StrongPref(x,y)? WeakPref(y,x))?x?y(Indiff(x,y)?(WeakPref(y,x)?WeakPref(x,y)))3. Prove :?x?y(StrongPref(x,y)? Indiff(x,y))4.Prove :?x?y?z((Indiff(x,y)?StrongPref(z,x))?StrongPref(z,y))Using: (for questions 3 and 4)?x?y(StrongPref(x,y)? StrongPref(y,x))?x?y?z((StrongPref(x,y)?StrongPref(y,z))?StrongPref(x,z))?xIndiff(x,x)?x?y(Indiff(x,y)?Indiff(y,x))?x?y?z((StrongPref(x,y)?Indiff(y,z))?StrongPref(x,z))?x?y(StrongPref(x,y)?Indiff(x,y)?StrongPref(y,x))?x?y(WeakPref(x,y)?(StrongPref(x,y)?Indiff(x,y)))First think about what you have to proofand make an informal proof for it.If you want something of the form VxVyVz(formula<1> -> formula<2>)Then use the following "outer module":P1 | . . . . . . . premmise 1P2 | . . . . . . . premmise 2P3 | . . . . . . . premmise 3P4 | . . . . . . . premmise 4---|-------------------------------------------1. | |___________a . . . . . . . . . . . . . . . new variable for Universal-intr

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