# Spring 2021 Topics in Macroeconomics Solution to Homework Assignment #2

University of Southampton | Chiara Forlati |

Economics Department | Office: 58/3027 |

Spring 2021
Topics in Macroeconomics |

**Solution to Homework Assignment #2**

due by 23:59 Thursday April 22, 2021

- (
*50 points*)**An Endogenous Monetary Policy Rule in the Basic New Keynesian Model**

Consider a version of the basic New Keynesian model with no preference shocks such that:

with) and the notation being as in the lectures. Assume that *a _{t }*is an exogenous TFP shock that follows an

*AR*(1) process with

*ρ*∈ [0

_{a }*,*1). Moreover, suppose that monetary policy is set according to the following interest rate rule

*i*=

_{t }*ρ*+

*φ*+

_{π}π_{t }*φ*ˆ

_{y}y*+*

_{t }*υ*

_{t}where *φ _{π }*≥ 0,

*φ*≥ 0 and

_{y }*υ*is as well an

_{t }*AR*(1) process such that:

with *ρ _{υ }*∈ [0

*,*1).

- Obtain the canonical representation of the system of equilibrium equations above.

[10]

**Ans:**

First consider that:

because *E _{t}*{

*a*

_{t}_{+1}} =

*ρ*and

_{a}a_{t }*E*{

_{t}*a*} =

_{t}*a*. Then, substituting the interest rate rule into DIS we obtain:

_{t}where

Then using the NKPC we obtain:

Then, the dynamic system of equilibrium equations can be rewritten:

where the second equation is recovered by substituting in the NKPC the process of the output gap found above.

- Using of the method of the undetermined coefficients, derive the coefficients of

the policy functions of inflation and output gap, *ψ _{π }*and

*ψ*. [20]

_{y}**Ans:**

We need to solve the system of equations found in (a). As a first step we guess the following solution:

*y*˜*t *= *ψ**yu**t π**t *= *ψ**πu**t*

We substitute our guesses into the system of equations above to obtain

This system of two equations in two unknown allows determining the coefficients of the policy functions *ψ _{y }*and

*ψ*. To see how we rewrite the first condition as follows:

_{π}where Λ ≡ *σ *+ *φ _{y }*+

*κφ*. Solving this last equation for

_{π}*ψ*we obtain:

_{y }which can be substituted in the second equation above to get:

This last equation leads to the following condition:

which allows us to conclude that:

(c) Assume that *υ _{t }*= 0 for all

*t*. Suppose also that

*α*= 0 and

*ρ*= 0. What is the effect of a rise in the TFP (

_{a }*a*) on employment,

_{t}*n*? Is this response consistent with the VAR evidence on TFP shocks discussed in the lectures? Make sure you show the derivations and provide an intuition for your results. [20]

_{t}**Ans:**

If *α *= 0 and *ρ _{a }*= 0 then

*y*ˆ* _{t }*=

*y*˜

*+*

_{t }*y*ˆ

*ˆ*

_{t}^{n }n*=*

_{t }*y*ˆ

*−*

_{t }*a*

_{t}where ˆ . As a result given the condition found in (b)

Hence, the effect of a TFP shock on employment is ambiguous. If *σ *≥ 1 or if *φ _{y }*is large enough a technological shock induces a decline in employment consistently with Gal´ı (1999).

As *σ *rises, an additional unit of wealth increases current utility by less since households are more prone to transfer part of it to future periods in order to smooth consumption over time. Everything else equal, a lower value of an additional unit of income makes households less willing to work and push them to reduce their labour supply. When *σ *≥ 1, in response to an increase in the real wage as the one generated by a TFP shock employment might fall, since the wealth effect prevails over the substitution effect.

As *φ _{y }*increases the central bank reacts more aggressively to changes in the output gap. A stronger response of the nominal interest rate leads to a stronger response of the real interest rate. The higher

*φ*, the more households change their consumption-saving behaviour and hence their labor decisions.

_{y}- (
*50 points*)**The Optimal Production Subsidy in a Monopolistic Competitive Setting**

Consider an economy in which agents maximize:

subject to the sequence of budget constraints:

*P**tC**t *+ *Q**tB**t *≤ *B**t*−1 + *W**tN**t *+ *D**t*

where the variables are as defined in the lectures. Firms are monopolistic competitive, each producing a differentiated good whose demand is given by the.

Each firm has access to a linear production technology:

*Y _{t}*(

*i*) =

*A*(

_{t}N_{t}*i*)

where:

with *ρ _{a }*∈ [0

*,*1) and

*ε*is i.i.d with zero mean. Prices are fully flexible.

_{a,t }(a) Write down the planner problem. Find the optimality conditions and interpret

them. [7]

**Ans:**

The social planner chooses *C _{t}*(

*i*),

*N*(

_{t}*i*) and

*N*so as to maximize

_{t }where subject to:

The Lagrangian of this maximisation problem can be written as

The first order conditions with respect to *C _{t}*(

*i*),

*N*(

_{t}*i*) and

*N*are:

_{t }At the solution one can guess that it is optimal to consume and produce the same quantity of all goods:

*C _{t}*(

*i*) =

*C*

_{t}*N _{t}*(

*i*) =

*N*

_{t}for all *i*. Intuitively, we can expect that the planner chooses *C _{t}*(

*i*) =

*C*since each good enters symmetrically in the utility function.

_{t }Moreover, if *C _{t}*(

*i*) =

*C*:

_{t}In words, the marginal rate of substitution between consumption and labour is equal to the corresponding marginal rate of transformation.

(b) Find the optimality conditions of the household and firm problems. [7] **Ans:**

Agents choose *C _{t}*,

*N*and

_{t }*B*so as to maximize:

_{t }subject to the sequence of dynamic budget constraints:

*P**tC**t *+ *Q**tB**t *≤ *B**t*−1 + *W**tN**t *− *T**t*

The first order conditions with respect to *C _{t}*,

*N*and

_{t }*B*can be read as:

_{t }*C**t*−1 = *λ**tP**t*

*N**tϕ *= *λ**tW**t*

*Q**tλ**t *= *βE**t*{*λ**t*+1}

As a consequence:

Firms maximize profits

*P _{t}*(

*i*)

*Y*(

_{t}*i*) −

*W*(

_{t}N_{t}*i*)

subject to the technological constraint:

*Y _{t}*(

*i*) =

*A*(

_{t}N_{t}*i*)

and tanking account the demand function. If we substitute both these constraints in the profits function we obtain:

According to the first order condition with respect to *P _{t}*(

*i*):

Hence

- Find the equilibrium process for output and employment. [12]

**Ans:**

According to the goods market clearing conditions

*C**t *= *Y**t*

According to the labour market clearing condition:

Combining these two last conditions we get:

Taking logs we we obtain:

- Does the decentralized equilibrium implement the Pareto efficient allocation? Ex-

plain why or why not. [12]

**Ans:**

The decentralized equilibrium is not Pareto efficient. This can be easily inferred by comparing the labour market clearing condition above (i.e., *C _{t}N_{t}^{ϕ }*=

,) against the condition resulting from the planner’s problem that equates the marginal rate of substitution between consumption and labour to its marginal rate of transformation (i.e., *C _{t}N_{t}^{ϕ }*=

*A*). The intuition behind this result is quite straightforward: in the decentralized equilibrium (in which prices are flexible and

_{t}*C*(

*i*) =

*Y*(

_{t}*i*) =

*C*for all

_{t }*i*) firms have market power since their demand is imperfectly inelastic. For this reason they set their prices above their marginal cost. As a consequence consumption, employment and output are below their efficient levels.

(e) Suppose that the government uses a production subsidy financed with lump-sum taxes. What is the optimal level of this subsidy that maximizes household welfare?

Provide an intuition for your answer. [12]

**Ans:**

By using the production subsidy, the government can eliminate the effects of the monopolistic distortion and restore the Pareto efficient outcome. To see how, consider the firm maximization problem in the presence of a production subsidy is modified as follows. Each firm *i *maximises the following profit function

*P _{t}*(

*i*)

*Y*(

_{t}*i*) − (1 −

*τ*)

*W*(

_{t}N_{t}*i*)

subject to the technological constraint:

*Y _{t}*(

*i*) =

*A*(

_{t}N_{t}*i*)

and tanking account the demand function. According to the first order condition w.r.t. *P _{t}*(

*i*):

This last condition represents the optimal pricing rule of each firm *i *in the presence of a production subsidy. The higher is the subsidy the lower the price and then the higher is the demand for each good and the higher *Y _{t}*. If the government sets the subsidy so that ) = 1 and

*P*(

_{t}*i*) = C

^{0}(

*Y*), then the distortion due to monopolistic competition can be completely eliminated. Subsidizing production allows demand and output to reach the efficient level at which

_{t}*C*=

_{t}N_{t}^{ϕ }*A*.

_{t}