Two agents: two Eulers?

Hi everyone.

This question has already appeared on the forum in many forms but I cannot really get an answer for my simple case.

I would like to build a two sectors-two households model in which labor is fully specialized but consumption isn’t (each family supplies labor to one sector but consumes both the goods produced by the two sectors).

In the toy model I am attaching, only labor is used in production. I have the following equations concerning consumption quantities:

  • Intertemporal (Euler) condition household A

  • Intratemporal condition household A

  • Budget constraint household A

  • Market clearing good A

  • Intertemporal (Euler) condition household B

  • Intratemporal condition household B

  • Budget constraint household B

  • Market clearing good B

By Walras’ law I only take 3 out of 4 equations among market clearings and budget constraints (I have checked everything is invariant with regard to this selection).

Anyway, I have to choose just one out of the two Euler equations. Depending on that choice, dynamics show some differences (the steady state remains the same).

So my question is: is a model like that meaningful? How can the two households’ intertemporal choices be at work if just one Euler is present?

Many thanks!

Heterogeneity.mod (5.7 KB)

Why exactly is that? There should be one redundant equation, but you suggest there are two.

Dear Professor Pfeifer,

my reasoning was that:

starting from the Euler equation A

\frac{1}{c_{AA,t}} =E_t \frac{\beta R_t}{\Pi_{A,t+1}}\frac{1}{c_{AA,t+1}}

one could use the two market clearing conditions and one budget constraint to get that

c_{AA,t} = p_{B,t}\frac{c_{BB,t}}{a_{J,t}}

which implies the Euler B

\frac{a_{J,t}}{c_{BB,t}} = E_t \frac{\beta R_t}{\Pi_{B,t+1}}\frac{a_{J,t+1}}{c_{BB,t+1}}\frac{a_{J,t+1}}{a_{J,t}} .

Above, a_{J,t} is just a preference autoregressive term and p_{B,t} is the relative price of the two goods \frac{P_{B,t}}{P_{A,t}}. Indeed, I have checked that the two give the same dynamics.

Leaving the two Euler equations would imply having one equation in excess. I am a bit puzzled and I do not know whether I am missing something. What do you think?

In that case, you are eliminating c_{AA} from the model by substituting for it. You should have one equation less, but also one variable less.

Actually I substituted for c_{AA} just in Euler A and kept it in the other equations. By substituting it everywhere in the model I have to comment one other equation in addition to Euler B and to remove c_{AA} from the list of variables. Even in that case results do not change.

I do not understand if this model makes sense economically. It seems like consumption loses a degree of freedom in the two agents model, in the sense that it cannot evolve independently from the other household’s choice. In other words, it seems like the optimization problem by which the two household make two independent intertemporal choices is ill-posed. Is that right and is there any intuition behind?

Again, I don’t understand what you are doing. Why do you have two more equations that unknowns in your model? You are saying that when you substitute for c_{aa} in all equations, you are losing one equation and one variable. But that does not answer the question why there are two equations too much.
The question why the consumption choices may or may not imply each other is then the second matter.

I see. Sorry for that. I will try to describe the model so that maybe you can help me spotting the source of imbalance (if this is doable).

The variables of the model are:

  • the amounts of consumption by household i of good j, c_{ij}

  • the hours worked in the two sectors n_j

  • production in the two sectors, Y_j

  • wages in the two sectors, w_j

  • inflations in the two sectors, \Pi_j

  • the two markups, X_j

  • \Pi_{aggregate}

  • the relative price of good B in terms of good A, p_B

  • the interest rate R

The two households maximize utility and this gives

  • the two labor supply schedules
  • the two intratemporal conditions stating \frac{c_{iA}}{ c_{iB}} = p_B
  • the two intertemporal conditions equating the marginal utility of today’s consumption to discounted tomorrow’s expected marginal utility

I assume that both households borrow from two independent institutions that supply bonds in zero net supply at the same rate.

Then I have the two production functions, the two labor demand equations and the two Phillips curves.

After that I have the two market clearings and the two budget constraints, among which I take the two market clearings and the first budget constraint.

Then the model is closed by the Taylor rule, the aggregate inflation definition and the law of motion of the relative price \frac{p_{B,t}}{p_{B,t-1}} = \frac{\Pi_{B,t}}{\Pi_{A,t}}.

So, you see that there are 17 variables and 18 equations.

Do you spot anything?

Why do you have two equations saying exactly the same thing:

Sorry, but I cannot see why these two equation say the same thing.
One equation is for household i=A


and the other is for household i=B


They set the optimal ratios of consumption of the two goods for each of the two households, so they do not seem redundant to me. Or am I missing something?

By the way, thanks for your help.

Maybe they say the same thing in the sense that \frac{C_{AA}}{C_{AB}} = \frac{C_{BA}}{C_{BB}} = p_B. Thus, the optimal ratios of consumption are the same for each of the two household. Isn’t it?

I missed the i-index. Yes, you are right that those are two equations. Have you tried breaking down the problem to a very basic model without market power and price rigidity? That should reduce the dimensionality.

@kofiemma Hi. Sure, but they are needed anyway.

@jpfeifer Here there is the simplest model I can think of in order to settle the thing:

Heterogeneity3.mod (4.1 KB)

one could use the two market-clearing conditions and one budget constraint to get the Euler B. Thus, cAA + cBA = YA, cAB + cBB = YB and cAA + pB*cAB = + wA*nA + retail_profits_A implies ucBB = BETTA*R/dpA(+1)*ucBB(+1).

And at the same time, these 3 same equations: cAA + cBA = YA, cAB + cBB = YB and cAA + pB*cAB = + wA*nA + retail_profits_A implies cBA + pB*cBB = pB*wB*nB + retail_profits_B;.

So you drop both ucBB = BETTA*R/dpA(+1)*ucBB(+1). and cBA + pB*cBB = pB*wB*nB + retail_profits_B;.

Strange to me. Anyway, can you attach the full model? Like pdf containing the solved model. By the way is ‘= +’ a typo in this equation: cAA + pB*cAB = + wA*nA + retail_profits_A?

@Carlo But the Heterogeneity3.mod works flawlessly.

@kofiemma Here I have prepared a pdf with the equations:

ModelHeterogeneity.pdf (125.8 KB)

@jpfeifer I agree that the model seems flawless, but I cannot still understand whether this is really a well posed one. Why does the intertemporal choice depend just on one household?

If the second Euler is obtained by substitution as @kofiemmaand and I suggested, the dynamics are independend of the choice of the Euler to be commented out.
But when the two Euler are written directly from the optimization, results change for some variables.
If you run the file HeterogeneityPDFmodel.mod, which is exactly the one in the pdf file, you will see that, by commenting out first one Euler and then the other, dynamics are different.

HeterogeneityPDFmodel.mod (4.0 KB)