Hello,
I have some general questions concerning dynamic stochastic general equilibrium models:

1. Can somebody explain the difference between the stochastic and the deterministic steady state or name a reference where I can read something about it? Does the stochastic steady state change if the volatility of the shock terms change?

2. In case of a 2nd order approximation, the volatility of the shock term matters. But how does it matter? Assume a standard RBC-model with technology shocks: Given a shock of the same size, are the impulse response functions of variables different if we assume different volatilities of the shock term?

3. Finally, I find the following contradiction: Assume again a standard RBC model with bonds which yield a certain (gross) return R_b(t), which is known in period t and paid in period t+1. The household can decide between buying a bond or investing in physical capital. The usual household optimization problems of the household yields two Euler equations with respect to bonds and physical capital:

MU(t) = beta * E[MU(t+1)] * R_b(t)
MU(t) = beta * E[MU(t+1)*R_k(t+1)]

where MU denotes marginal utility of consumption. With E[MU(t+1)*R_k(t+1)] = E[MU(t+1)] * E[R_k(t+1)] + cov[MU(t+1),R_k(t+1)], we have

R_b(t) = E[R_k(t+1)] + cov[MU(t+1),R_k(t+1)] / E[MU(t+1)].

The covariance should be negative: If there is a negative technology shock, the return on capital is low and consumption as well and therefore marginal utility high. Thus,** R_b(t) < E[R_k(t+1)]** which makes sense at first sight: The return on physical capital is uncertain and depends on the realization of the technology shock. The household is risk averse, i.e. the expected return on physical capital should be higher than the certain return on bonds, otherwise the household would not be indifferent between both types of investments.
Yet, R_b(t) < E[R_k(t+1)] does not seem plausible from another point of view: Assume the economy (in period t) is in the steady state, where the return on bonds equals the realized return on physical capital, i.e. R_b(t)=R_k(t). Now, the mean of the shock term is assumed to be zero as usual. That is, the household neither expects a positive technology shock nor a negative technology shock in t+1. However, if there is neither a positive nor a negative technology shock, then it must be that the realized return on capital equals the realized return on capital from last period, i.e. R_k(t)=R_k(t+1). In other words, the household expects neither a positive nor a negative technology shock such that the expected return on physical capital in t+1 is equal to the return on physical capital today. Putting all this together implies R_b(t)=E[R_k(t+1)] which contradicts the above statement.

Where is the error?

I would be really glad and thankful if someone could answer these questions.

best regards,
Niklas

Your problem is that you are ignoring Jensenâ€™s Inequality that arises due to the rational expectations and the ensuing expectations operator. If would recommend taking a look in for example homepage.newschool.edu/het//essays/uncert/aversion.htm

1. For the deterministic steady state, you assume that all shocks are identically 0 for all time, i.e. everything is deterministic. The term stochastic steady state is only loosely defined and not really precise. Most of the time it refers to the mean of the ergodic distribution of the stochastic system, i.e. the mean values of the variables if let the system run forever with shocks hitting the system according to the specified distributions. Of course the ergodic distribution changes if the volatility of the shock terms changes. However, there are special cases when this is not the case, e.g. if everything (and particular utility) is linear.

2. In case of a 2nd order approximation the volatility matters, but it only shows up as a constant term (see columbia.edu/~mu2166/2nd_order/2nd_order.pdf). In a first order approximation, everything is linear and certainty equivalence holds because the expectations operator is linear. In a second order approximation, the curvature of the functions are taken into account. Hence, precautionary motives due to risk aversion may play a role, which are neglected if everything is linearized. Regarding the IRFs, they will in general be different in Dynare. Due to a quadratic term in the second order approximation it matters where the simulation is when the shock is added to the system. As the ergodic distribution differs for different volatilities, this starting point will differ in the two cases.

3. The error is that the Euler equation is in terms of E[MU(t+1)R_k(t+1)] and not E[R_k(t+1)]. Intuitively, Agents do not care about the return on assets per se, but they care about consumption. Hence, it is the covariance of the asset return with their consumption that matters. Risk averse agents want to be compensated with a higher expected return for assets that deliver low returns in a bad state with low consumption. More formally, if utility is not linear, Jensens Inequality tells you that E[MU(t+1)R_k(t+1)] is unequal E[MU(t+1)]E[R_k(t+1)]. You argue that in your two states E[R_k(t+1)]=0.5r_low+0.5r_high=R_b. Say e.g. r_low=0.8 and r_high= 1.4, and R_b=1.1. Even if this is the case, E[MU(t+1)R_k(t+1)]=0.5MU(t+1)_lowr_low+0.5MU(t+1)_highr_high=0.4MU(t+1)_low+0.7MU(t+1)_high. Only if utility is linear, will this linear combination give the agent the same utility as a sure return of R_b, i.e. be equal to MU(t+1)_R_b * R_b=1.1*MU(t+1)_R_b . Or put differently, Jensens Inequality tells you that for risk averse agents the utility of the expected value is higher than the expected value of the utility (see expected utility hypothesis or St. Petersburg Paradoxon on Wikipedia)

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Dear jpfeifer,
thanks a lot for your help. This was very helpful. Concerning question 3), I still have some problems of understanding:

So R_b(t) < E[R_k(t+1)] must be true. But then, imagine that the economy is in a situation where there is (by chance) repeatedly neither a positive nor a negative technology shock (in effect, like the non-stochastic steady state). As a result, the realized return on capital must be the same every period, i.e. R_k(t)=R_k(t+1), and equal to R_b(t). But this implies that the expected value of the return on capital is repeatedly higher than the realized. i.e. E[R_k(t+1)]>R_k(t+1). How can this be true? Can the agent in the model expect/require a higher return than it would result in case of the mean shock, i.e. neither a positive nor a negative shock. This can be formulated more formally in the standard RBC model, where the net return r_k(t) is equal to r_k(t)=Y(t)/K(t)*alpha, with alpha being the share of capital, (which follows from the optimal behaviour of the firm):

E[R_k(t+1)]=E[1-delta+r_k(t+1)]=1 - delta + alpha*E[Y(t+1)]/K(t)

Thus, it must be that E[Y(t+1)]>Y(t) although the household expects no technology shock. Can the household with rational expectations expect a too high output (or interest rate). In other words, the household takes on risk in period t by investing in physical capital, but earns no risk premium in period t+1 (since R_b(t)=R_k(t+1)). Where is my mistake?

best regards,
Niklas

Your mistake is that you are confusing perfect foresight with a stochastic system under rational expectations. Your example imposes that the agent perfectly knows all shocks and that they are identically 0. In this case, capital is not risky, but rather pays fixed risk-free return. Capital then is a perfect substitute for the risk free bond and hence has the same return.
Things change if the return on capital becomes risky due to the presence of shocks. Note also that the ex-post realized return can of course be different from the one expected ex-ante. What matters is the ex-ante riskiness.

Dear jpfeifer,

Then, maybe everything boils down to the following question: Are the values of the variables belonging to the mean of the ergodic distribution (i.e. the values of the variables when the economy is in its ergodic mean) equal to the values of the variables in the non-stochastic steady state? If yes, we can make a thought experiment: The stochastic economy is in exactly this state, the ergodic mean. Then, the agent expects ex-ante a risk premium for risky capital, i.e. the ex-ante expected return on capital is different from the certain return on the bond. But then imagine that the realized shock term in the next period is equal to its mean value (which was expected). What happens is that the economy stays in its ergodic mean. The consequence is that the realized return on capital is the same as the return on the bond (since non-stochastic steady state and ergodic mean yield equivalent values of the variables). Thus, the realized return on capital ex-post is lower than the ex-ante expected return on capital although the shock term exactly coincides with its mean. I am not saying that the expected ex-ante return must always equal the realized ex-post return. But the point is that if the shock term turns out be as expected (zero mean), then the ex-post return should be equal to the ex-ante expected return, or not? Otherwise, we wouldnâ€™t have rational expectations (?).
In the extreme case, canâ€™t we imagine a stochastic economy where the shock term always realizes its mean value, i.e. the economy always stays in the ergodic mean of the system, i.e. the allocation is exactly the same as in the non-stochastic steady state. However, the agent does not know this, the economy is stochastic, the agent requires ex-ante a risk premium for risky capital. But ex-post, the realized return is always equal to the certain bond return. This is not what I understand under rational expectations.

Best regards,
Niklas

If your model is not linear, the ergodic mean is not identical to the deterministic steady state. However, you do not compute the solution to this nonlinear model but to an approximated version. For a first-order approximation, the approximated model is again linear and there is no difference. For higher-order approximations, the two do not coincide in general.

Your thought experiment is wrong. If you assume that the shocks are always at their mean of 0, your shock distribution is degenerate (all other points have mass zero) and the model is deterministic. Higher-order approximations are about the higher moments of the data. Your thought experiments sets them all to 0, including the variance. If you do this, the model is deterministic and the ergodic mean is equal to the determinstic steady state. Hence, there is no risk premium required as the return on capital is safe.

Dear jpfeifer,

thanks a lot. The last question from my side (hopefully)â€¦
So, if I compute a second order approximation to my non-linear model and conpute the ergodic mean (by setting the shock term to its mean value), then the return on capital must be different from the safe return on bonds? This return on capital also coincides with the expected return on capital?