Dear all,

There’s a mathematical oddity that I don’t quite understand. When we have non-separability in utility derived from non-durable and durable consumption, we can use the following CES aggregator:

which becomes the Cobb-Douglas equation when \eta goes to 1

Now, we assume there’s a housing preference shock so we can shock the \gamma parameter. Since I wanted to see what’s going on, I linearized the equation and we get:

where g_t is the shock. Now, what could happen here is that both C and H go down in deviation from steady state, but X goes up. Mathematically this is possible. Yet I find it hard to interpret it. X is the consumption basket so to speak that gives utility to the household, which consists of non-durable goods and housing services. But I find it hard to explain what drives the increase in X even though both of its components decline.

Thanks in advance.

Are you sure the linearization is correct? Can you also provide your derivation?

Sure, please find the pdf attached. I did in both level and log-linearization. Thanks.

X Bundle.pdf (85.0 KB)

The linearization is correct, but the sign of the effect of \epsilon_t depends on \ln(H/C). If H is bigger than C, you get a positive sign. That makes intuitive sense. Take a numerical example

```
gamma=2;
epsilon=1;
C=1;
H=2;
X=C^(1-gamma*epsilon)*H^(gamma*epsilon)
```

You will get X=4.

Now use

```
epsilon=1.01
X=C^(1-gamma*epsilon)*H^(gamma*epsilon)
```

and you will get X = 4.0558. The reason is that increasing \epsilon will put more weight from C=1 to H=4 in the aggregate, increasing its value. That is the first order effect the linearization shows.

Hi Johannes,

Mathematically this makes sense indeed. But what doesn’t make sense to me is the story behind it, from an economic perspective. Here’s a picture of my IRFs.

So the story is that just because the value of the housing weighting factor goes up, when both C and H go down in deviation from the steady state, X goes up. So if X is the consumption basket, just increasing preferences for one of the goods in the basket increases the value of the basket, despite the fact that both of its components go down? For me this wouldn’t make sense.

Why not? You are changing the preferences of the agent by altering the weighting in a weighted average. If the increase in weight for the relatively abundant good increases by more than its actual values, the household will feel better off.

Sure, but in the weighted average both of the components decrease, regardless of which good is more abundant. Even if we make the weight on housing 1, X should in theory experience a fall, since housing falls…or I’m overlooking something…

Say you current weight on food and housing is 0.5 and you have a rather large flat and enough to eat. but you don’t value that large flat too much. Then a kid arrives and your utility weight on housing increases to 0.7 despite your housing actually going down a bit because you need space for a crib and you have less to eat for yourself. Say it decreases 10%. Your overall utility aggregate may still increase in this case. Take numbers of C=1 and H=2 and make the aggregate just linear. 0.5\times 1 + 0.5\times 2=1.5. After the kid arrives 0.3\times 0.9 + 0.7\times 1.8=1.53. The reason is that you derive more utility from housing despite its actual amount going down.

I see, I guess what confuses me is the fact that X is not utility, but rather an aggregate consumption bundle. The Utility function is just a function of X and labour. So the interpretation of X is somewhat interesting. I guess one can say, your aggregate consumption goes up as a result of a higher preference for housing, in whatever abstract units we measure it…