# Conceptual question about Inequality Constraints

I have a conceptual question. When a model has borrowing constraints say something like LTV (loan to value) rule as in Gerali and Neri or Alpanda and Zubairy which looks like the following

(1+r)L <= \phi_{ltv} E [ q_{t+1}h_{t}\pi_{t+1}]

where r is the mortgage interest rate, \phi_{ltv} is the loan to value ratio, q is the housing price, h is housing stock and \pi is the inflation rate. I have observed these constraints are never entered into the model block. But when I solved these kind of models, I have always treated them as binding and worked out steady state values of either r or \phi_{ltv}. and included them in model block. My questions are
(1) Is it wrong to work out steady state values from such inequality constraints by treating them as binding.
(2) Should such constraints ever be included in the model block?

Steve

The answer depends on whether you consider the constraints to be always binding or occasionally binding. Only if they are always binding you can use perturbation techniques with a linearization around the steady state. Otherwise, you need to use solvers like Occbin.

Thanks. I consider them as binding.

But then can I use these constraints to solve for r (mortgage interest rate) in steady state? Can I include them in the model block (I have not seen people doing that though).

I am not sure I understand. If the constraint holds with equality, you can use the resulting equation both for computations and as a model equality constraint.

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The law of motion of phi_ltv is given as

\phi_{ltv} = (1-rho)\phi_{ltv}(-1) + rho\phi_{ltv}_bar + epsilon

where phi_ltv_bar is the value of phi_ltv in steady state.

When a paper says lets evauate the impulse responses to a 5 pp decline in \phi_{ltv}, what would that mean. Because in the shocks block we are evaluating response to one sterr shock to epsilon. How do we know if that is equivalent to 5 percentage point or 10 per centage point decline in \phi_{ltv}?

\phi_{ltv} seems to be measured in percentage points. Then simply set the standard deviation of epsilon to 0.05. At first order, this is valid.

Thanks @jpfeifer! How can you say by looking at this equation that \phi_{ltv} is set in per centage points. I have set it at 0.78 in this case. Also what is meant by â€śat first order this is validâ€ť. Does that mean if I do a second or third order approximation, it wont work?

1. The loan to value-ratio gives the ratio of the loan to the value, i.e. is in percent of the house value.
2. At higher order, certainty equivalence does not hold anymore. In that case, the standard deviation is a fixed number that you are not allowed to arbitrarily scale as it will influence the decision rules. For that reason, you would need to use the simult_-function to simulate the IRF for the desired shock size.
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