I have a model where two types of households have the same consumption structures, though their distribution parameter across different types of consumption goods (A,B,C) are different. I end up having aggregation issues.
The consumption of the two households follow the same CES that bundles three different goods, but the distribution parameters are different for the two households inside the CES.
The conventional way to aggregate the consumption of good A with identical preferences is:
Good A= share of household 1 * Consumption of good A of household 1 + (1- share of household 1) * Consumption of good A household 2. And so on for the two other goods that are bundled with the good A in the CES function.
However, this does not work when the two households have different distribution parameters into their CES bundle of consumption goods.
Is their an easy way to aggregate ?
I thought of assuming same preferences and parameters for the CES of consumption for both households, and to add a “preference” parameter in the budget constraint for the goods. This one is straight forward to treat with aggregation. The new equation for aggregation would be
Good 1 =share of household 1 * preference parameter good A* Consumption of good A household 1 + (1- share of household 1) * preference parameter good A* Consumption of good 1 * household 2. And so on for the two other goods. Household 1 could have a higher preference for good A than household 2, and so on for other goods.
With the preference parameters attached to the consumption expenses in the budget constraint and ending up in the FOCs for consumption.
I am not sure I understand the problem. What you describe sounds like each household has different preferences over goods. Aggregation for each good will simply be the sum of goods consumed over all agents. It seems you are trying to define a “final” good that is a CES bundle. But that will not work, because different preferences imply different “production” processes for this final good entering utility.
Yes, exactly @jpfeifer . They have different preferences over goods because of different distribution parameters in their CES consumption function:
Here the alpha_x is different between the households. If it is not, and I aggregate X, SD and C via the standard aggregation rules (e.g. X = omega*X_household1+(1-omega)*X_household_2) it works. But when I make the alpha_x different for the two types of households, this aggregation rules does not work and I have an error in my aggregation equations.
So, should put instead of X = omega*X_household1+(1-omega)*X_household_2, the following aggregation rule: X = X_household1+ X_household2? But then you are saying that I cannot have one final good only in my economy because of different preferences.
You must be careful in terms of notation. Nothing in your equation indices what is specific to households and which variables market clearing involves. If \alpha_x^i differs for two households i, the resulting C_t^i are different goods. You cannot aggregate them. Similarly, different \alpha_x^i will imply different expenditure shares on the inputs like X_t^i.
All right, so in that case, I have to assume that I have two different final good producers for my two households, right @jpfeifer ?
For the aggregate resource constraint, can I say that I have a final good aggregator that aggregates the two different final goods used by the two households ? Or should I just write that:
Y = omegga_1*Y_1 + (1-omegga)*Y_2, Y_i being the production of final good i for household I ? Because at the end, the production technology for Y_1 and Y_2 would be the same (same CES structure, same inputs, same parameters).
Is there another way to bypass this, maybe by having the same CES consumption structure and different “preference” shocks on goods expenses for the two households ? Pretty sure I’m gonna still bump into aggregation issues
I think you need to get your setup straight. Whether an agent consumes a final good or bundles the intermediates within the preference structure with an aggregator is a matter of preference. The “final good producer” is essentially just a bundler for you. Aggregation in the various markets will simply depend on the setup you choose.
I’ll go with the the second option where the agent bundles two consumption intermediates (X and SD) within the preference structure with the aggregator C. Preferences would be different between the two households with different distribution parameters. That means now, with this structure, I have to assume two final good producers, one that produces the intermediates for household 1 and the other one that produces the intermediates for household 2, am I getting this right ?
Again, you have to get your setup straight. X and SD seem to be different intermediate goods that households use to consume. But which goods do firms use for investment? What does the government consume? And what is exported.
You now need to be careful in your wording. In most models, intermediate goods like X and SD are bundled into one final good that is used for consumption, investment, government spending, and net exports. So there is only one final good. But now your consumers have different aggregation structures for the intermediates, so there are essentially two different goods. But that does not turn G or I into intermediates. You still need a proper setup to produce these goods.
Yes I understood that part. I was just saying that my economy is populated by only one final good firm that produces one final good. This final good is used by the government for public consumption (G), invested as capital goods by one of the household (Inv_k), exported abroad (net of imports) (NX), and consumed by the two households (C). These two households consume this final good via a CES function C that bundles non-durable goods and semi-durable goods. These two consumption goods are also produced by the final good firm.
However, I assume now that these two households bundles non-durable goods and semi-durable goods in a non-standard way. Each household has different distribution parameters values in the CES for good consumption that bundles X and SD, so different preferences. Such that I cannot aggregate in the standard way. I think I’m just going to stick with the standard way of treating consumption: same preferences across households. Because in this consumption bundle, I actually have more goods than just non-durables and semi-durables goods (6 in total). Else, I would have to add 6 different producing sectors for each type of consumption goods I’m considering.
This is the one that I’m following but adding on top of non^durables, durables, also semi-durables. I add two different households too. Of course, this is ,not a same good. But this goods are produced by one final good firm.