Advanced marcoeconomics test

ECON 401 Advanced Macroeconomics

Homework 1 Suggested Answers

Fabio Ghironi

University of Washington

Spring 2022

Problem 1

Consider the problem that a benevolent social planner would solve in the real business cycle model (RBC) we have been studying. The notation below is the same as in the slides. The planner would maximize the representative house- hold’s expected intertemporal utility:

Et

∞∑ s=t

βs−t C1−γs 1 −γ

where 0 < β < 1 and γ > 0, subject to the sequence of constraints:

Ct + Kt+1 = (1 − δ) Kt + Aαt K 1−α t ,

0 < α < 1, one such constraint in each period. Following the approach described in slides 20-23 on the RBC model,

1. set up the planner’s maximization problem,

2. obtain the Euler equation for capital accumulation,

3. explain it intuitively,

4. and explain why it coincides with the equation that would be implied by the market outcome.

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Part 1

Observe that the fact that the constraint holds in each period allows us to rewrite it as:

Cs + Ks+1 = (1 −δ) Ks + Aαs K 1−α s

for every period s. Hence:

Cs = (1 −δ) Ks + Aαs K 1−α s −Ks+1.

Substitute this for Cs in the representative household’s expected intertem- poral utility function to obtain:

Et

∞∑ s=t

βs−t [ (1 − δ) Ks + Aαs K1−αs −Ks+1

]1−γ 1 −γ

.

The planner chooses the sequence {Ks+1} ∞ s=t (the sequence of capital stocks

from t+ 1 to the infinite future) to maximize this expected intertemporal utility function. Note that the planner’s choice begins with Kt+1 because Kt is given: It is the capital stock with which the economy began period t, determined by choices that happened in the past (at t− 1).

Part 2

The structure of the problem is such that we can just focus on the choice of Kt+1, which shows up only in the first two terms of the summation above:[ (1 − δ) Kt + Aαt K

1−α t −Kt+1

]1−γ 1 −γ

+βEt

[ (1 − δ) Kt+1 + Aαt+1K

1−α t+1 −Kt+2

]1−γ 1 −γ

+...

Wedonotneedtheexpectationoperatorbefore thefirst termbecauseeverything in it is known at time t, including Kt+1, which is chosen at t. If you wrote the next term in the summation, it would contain only Kt+2 and Kt+3, but no Kt+1, and so on. Take the derivative with respect to Kt+1 of the expression above and set it

equal to 0:

−(1 −γ) [ (1 − δ) Kt + Aαt K

1−α t −Kt+1

]−γ 1 −γ

+β (1 −γ) Et

{[ 1 − δ +

( At+1 Kt+1

)α] [ (1 −δ) Kt+1 + Aαt+1K

1−α t+1 −Kt+2

]−γ 1 −γ

} = 0.

Note that the constraint implies

(1 − δ) Kt + Aαt K 1−α t −Kt+1 = Ct

2

and (1 −δ) Kt+1 + Aαt+1K

1−α t+1 −Kt+2 = Ct+1.

Hence, the first-order condition above becomes:

−C−γt + βEt {[

1 −δ + ( At+1 Kt+1

)α] C −γ t+1

} = 0,

or:

C −γ t = βEt

{[ 1 − δ +

( At+1 Kt+1

)α] C −γ t+1

} ,

which is the same Euler equation implied by the market economy. (The choices of Kt+2, Kt+3, etc. would be governed by similar equations.)

Part 3

The Euler equation balances the cost to the household of giving up one unit of consumption and increasing the capital stock by one unit today (the marginal utility of today’s consumption) with the expected discounted benefit of doing so: the expected discounted utility value of increasing consumption tomorrow by the return the investment will generate. The return is given by the undepreciated portion of one unit of capital and the marginal product of capital tomorrow, and multiplying it by the marginal utility of tomorrow’s consumption gives us the utility value of that return.

Part 4

The planner’s Euler equation coincides with the market outcome because there is no distortion in this model: Markets are perfectly competitive, there is no issue of asymmetric information, no rigidity in prices, etc. Therefore, the planner cannot do better than the market.

Problem 2

Consider again the basic RBC model, but now allow for government spending shocks as a source of fluctuations. The representative household maximizes:

Et

∞∑ s=t

βs−t C1−γs 1 −γ

,

where 0 < β < 1 and γ > 0, subject to the constraint:

Ct + It + Xt = r̃tKt + wt

in each period. In this constraint, Xt is exogenous lump-sum taxation, which we assume is equal to government spending. The rest of the notation is as in the slides.

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The law of motion for capital is:

Kt+1 = (1 − δ) Kt + It, 0 < δ < 1,

in each period. The production function is:

Yt = A α t K

1−α t , 0 < α < 1.

(a) Set up and solve the maximization problem of the household to obtain the Euler equation for capital accumulation. Does the presence of government spending affect it? Why do you think that happens?

Answer:

The law of motion for capital implies:

It = Kt+1 − (1 − δ) Kt.

Substitute this in the budget constraint to obtain:

Ct + Kt+1 − (1 − δ) Kt + Xt = r̃tKt + wt.

Hence, in every period s:

Cs = r̃sKs + ws −Ks+1 + (1 −δ) Ks −Xs.

and substituting this in the expected intertemporal utility function implies that the household will choose the sequence {Ks+1}

∞ s=t to maximize:

Et

∞∑ s=t

βs−t [r̃sKs + ws −Ks+1 + (1 − δ) Ks −Xs]

1−γ

1 −γ .

Proceeding exactly as in Problem 1 or as in slides 20-23 (focusing on the first two terms of this summation, taking the derivative with respect to Kt+1, and setting it equal to 0) yields the Euler equation:

C −γ t = βEt

[ (1 − δ + r̃t+1) C−γt+1

] .

This is exactly the same Euler equation as in the model without government spending. Therefore, the presence of Xt does not affect it. Why? Because we assumed that the government is financing its government spending using lump- sum taxation. In essence, we assumed that there is a government spending GOVt that is financed with a lump-sum tax Xt so that:

GOVt = Xt.

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Suppose that we change this assumption, and we assume that GOVt is financed with a proportional tax XPt on capital income:

GOVt = X P t r̃tKt.

The household’s budget constraint in this case is:

Ct + It = ( 1 −XPt

) r̃tKt + wt.

If you solve the household’s optimization problem in this case, you find that the Euler equation becomes:

C −γ t = βEt

{[ 1 − δ +

( 1 −XPt+1

) r̃t+1

] C −γ t+1

} .

Now the Euler equation is affected by future taxation. Why? Because taxation now affects the net return to investment that is left to the household. I of course did not expect you to know all this, but you should make sure

you understand this point.

(b) Show that, since factors are paid their marginal products in this model, the following equation holds:

Ct + Kt+1 + Xt = (1 −δ) Kt + Aαt K 1−α t . (*)

Answer

Factors being paid their marginal products implies that

r̃t = ∂Yt ∂Kt

= (1 −α) ( At Kt

)α .

Moreover, all output is exhausted in payments to factors:

Yt = r̃tKt + wt,

implying that

wt = Yt − r̃tKt = Yt − (1 −α) ( At Kt

)α Kt

= Yt − (1 −α) Aαt K 1−α t .

Therefore, the budget constraint of the household can be rewritten as:

Ct + It + Xt = (1 −α) Aαt K 1−α t + Yt − (1 −α) A

α t K

1−α t = Yt,

or: Ct + It + Xt = Yt

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(not surprisingly, all output is either consumed, invested, or consumed by the government), and, by virtue of the production function:

Ct + It + Xt = A α t K

1−α t .

From the law of motion for capital accumulation, we have:

It = Kt+1 − (1 − δ) Kt.

Hence, these two equations imply:

Ct + Kt+1 − (1 − δ) Kt + Xt = Aαt K 1−α t ,

from which you have equation (*).

(c) Assume Āt+1/Āt = X̄t+1/X̄t = G and solve for the balanced growth path. (Here, treat X̄t/Ȳt as an exogenous variable– i.e., do not try to solve for this ratio, just treat it as exogenously given.).

Answer

Everything here is as in slides 31-32 and 35 except for the difference implied by the presence of X̄t in the steady-state version of equation (*). This is:

C̄t + K̄t+1 + X̄t = (1 −δ) K̄t + Āαt K̄ 1−α t .

Dividing both sides of this equation by K̄t yields:

C̄t K̄t

+ K̄t+1 K̄t

+ X̄t K̄t

= 1 − δ + ( Āt K̄t

)α .

K̄t+1/K̄t = 1 +g (using G ≈ 1 +g and replacing ≈ with = to simplify notation), and the expression for Āt/K̄t is in the slides. Observe that:

X̄t K̄t

= X̄t Ȳt

Ȳt K̄t

= X̄t Ȳt

( Āt K̄t

)α ,

because of the production function. Therefore,

C̄t K̄t

+ 1 + g + X̄t Ȳt

( Āt K̄t

)α = 1 −δ +

( Āt K̄t

)α ,

or: C̄t K̄t

=

( Āt K̄t

)α ( 1 −

X̄t Ȳt

) −g − δ.

Use the expression for Āt/K̄t in the slides, and you have:

C̄t K̄t

=

( r + δ

1 −α

)( 1 −

X̄t Ȳt

) −g − δ,

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where we are treating the trend-level of the ratio between government spending and GDP as exogenous (as per the hint I had given you). Notice that the presence of government spending reduces the consumption that households can obtain for given capital stock.

(d) Assume log-normality (so that log (EtXt+1) ≈ Et (log Xt+1) + (1/2) vart (log Xt+1) for any variable X) and homoskedasticity. Log- linearize equation (*), the Euler equation, and the expression of the return to capital accumulation around the steady state.

Answer

The log-linearizations of the Euler equation and the return to capital accumu- lation are identical to slides 44-50. As for equation (*), apply the differentiation operator to obtain:

dCt + dKt+1 + dXt = (1 − δ) dKt + αdAtĀα−1t K̄ 1−α t + (1 −α) Ā

α t dKtK̄

−α t ,

where we used the rule for differentiation of a product and we evaluated levels around which we are differentiating at trend. Hence,

C̄t dCt C̄t

+K̄t+1 dKt+1 K̄t+1

+X̄t dXt X̄t

= (1 − δ) K̄t dKt K̄t

+α dAt Āt

Āαt K̄ 1−α t +(1 −α) Ā

α t

dKt K̄t

K̄1−αt .

Divide both sides of this equation by K̄t and recall that, for every variable X, xt ≡ dXt/X̄t. Hence:

C̄t K̄t

ct + K̄t+1 K̄t

kt+1 + X̄

K̄t xt = (1 − δ) kt + αatĀαt K̄

−α t + (1 −α) ktĀ

α t K̄ −α t ,

or:

C̄t K̄t

ct + (1 + g) kt+1 + X̄t Ȳt

Ȳt K̄t

xt = (1 − δ) kt + ( Āt K̄t

)α [αat + (1 −α) kt] ,

from which:

C̄t K̄t

ct + (1 + g) kt+1 + X̄t Ȳt

( Āt K̄t

)α xt = (1 − δ) kt +

( Āt K̄t

)α [αat + (1 −α) kt] .

Substituting in this equation the expressions for C̄t/K̄t and Āt/K̄t from part (c), rearranging and collecting terms, and defining the coeffi cients λ1, λ2, and λ4 as in the slides (and in part (e) below for λ4) yields the equation:

kt+1 = λ1kt + λ2at + λ4xt + (1 −λ1 −λ2 −λ4) ct.

Combined with

Et (ct+1 − ct) = λ3 γ Et (at+1 −kt+1)

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(which you get from combining the log-linearized Euler equation and the log- linearized return to capital) and with the shock processes

at = φaat−1 + εa,t

and xt = φxxt−1 + εx,t,

you have the four-equation log-linear system to which you have reduced the model.

(e) Assume at = 0 ∀t (there are no percentage deviations of tech- nology from the steady state). Assume xt = φxt−1 + εt, Et−1εt = 0. Show that the model reduces to:

kt+1 = λ1kt + λ4xt + (1 −λ1 −λ2 −λ4) ct,

Et (ct+1 − ct) = − λ3 γ kt+1,

xt = φxt−1 + εt, Et−1εt = 0,

with:

λ1 ≡ 1 + r

1 + g , λ2 ≡

α (r + δ)

(1 −α) (1 + g) ,

λ3 ≡ α (r + δ)

1 + r , λ4 ≡−

(r + δ) X̄t/Ȳt (1 −α) (1 + g)

.

Answer:

This just follows from setting at+1 = at = at−1 = 0 in the system you obtained in part (e), observing that the expectation operator before kt+1 in the log-linear Euler equation is not necessary because kt+1 is known at time t, and noting that xt = φxt−1 + εt is identical to xt = φxxt−1 + εx,t once you no longer need to differentiate notation with the technology shock.

(f) The solution for consumption and capital has the form:

ct = ηckkt + ηcxxt,

kt+1 = ηkkkt + ηkxxt.

What is the intuition for this solution?

Answer:

The set of variables kt and xt is the most parsimonious description of the state of the economy when households and firms take their decisions: kt (the endoge- nous state) is the capital with which the agents begin the period, and xt (the exogenous state) is the shock they observe just before taking their decisions.

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The solution maps the description of the state into the behavior of agents with regard to consumption and capital accumulation. The solution is linear because the system we are solving is linear. Extra exercise for you: How would the solution look like if we put the tech-

nology shock back in the picture alongside the government spending one? What would be the set of variables that describes the state of the economy in this case?

(g) Briefly describe how these equations and xt = φxt−1 + εt can be used to trace the responses of capital and consumption to a govern- ment spending shock.

Answer

Suppose the economy was in steady state up to and including period −1. Then there is an innovation to government spending ε0 in period 0. No other innova- tion occurs. All three variables (kt+1, ct, and xt) have values of 0 in period −1, because

the shock has not happened yet. The process for government spending implies the following values for xt in periods 0 and 1: x0 = ε0 and x1 = φε0. Capital does not move in period 0 because k0 was determined at time −1,

before the shock happened. Hence, k0 = 0 and k1 = ηkxx0 = ηkxε0. Since k0 = 0, the initial movement of consumption depends only on gov-

ernment spending. We have c0 = ηcxx0 = ηcxε0 and c1 = ηckk1 + ηcxx1 = ηckηkxε0 + ηcxφε0. And so on in subsequent periods. The Excel file I posted in Canvas gives

you a numerical example of this process for the case of a technology shock. You should practice constructing a similar file for the case of a government

spending shock.

(h) Is it desirable for government spending to exist in this model? Why?

Answer

No. As you studied, a planner cannot do better than the market in this model. In the specific case we are looking at, government spending is a pure waste of resources. It is as if the government were taking a portion of output and throwing it into the ocean. (Notice: One of the assumptions of the model is that government spending does not enter the utility function of the representative household, i.e., the model does not capture welfare gains that the household may obtain from, say, provision of services like public schools or road maintenance, etc.)

(i) Briefly: Describe the possible calibration approaches to evalu- ate the empirical performance of models.

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Answer

There are two basic approaches. One consists of selecting some moments of the data (say, variances of consumption and output, their covariance, and their first-order autocorrelations– a measure of how persistent their movements are, like φ in the process for government spending above) and picking the values of parameters so that the simulated model matches those data moments exactly. If the required parameter values are reasonable given existing findings in the literature, and if your model matches suffi ciently well moments other than those you picked to determine the parameter values, then you have a good model (for the purposes for which you are using it– you should never forget that models are written to study specific questions, they may well be very inappropriate for other uses). The second calibration approach goes in the other direction: Suppose you set

the values of all the parameters at reasonable numbers (but you do not pick any of them to match any feature of the data exactly). Suppose you then simulate your model. Does it produce impulse responses and moments for the data that are consistent with empirical evidence? If so you have a good model (again, for the purposes for which you are using it).

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