´ des Sciences Sociales de Toulouse Universite MPSE ´e universitaire 2004-2005 Anne DEA Macro´economie 2 — Cours de Franck Portier Homework 3 Solutions Problem I – Technological shocks, Preference shocks and the endogeneity of TFP
We consider here a variation of the simple analytical RBC model. We study a model economy A populated with a representative household and a representative firm. The firm has a Cobb-Douglas technology: Yt = Zt Ktγ Nt1−γ
(1)
where Kt is capital, Nt labor input, and Zt the stochastic Total Factor Productivity (TFP). All profits of the firm are distributed to the household. Capital evolves according to Kt+1 = It
(2)
where It is investment in period t. The representative household works Nt and consumes Ct . Preferences are given by U = E0
∞ X
β t [log Ct − χt Nt ]
(3)
t=0
where χt is a preference shock. Capital is accumulated by the household and rented to the firm. Let κ denote the real rental rate of capital, P the price of the final good and W the nominal wage. 1 – Write down the budget constraint of the household and the profit function of the firm wt N t + κt K t pt for the budget constraint, and for the profit: wt Πt = Yt − Nt − κt It−1 pt Ct + It ≤
with Kt = It−1
(a)
(b)
2 – Derive FOCs of the utility and profit maximization For the household, the Lagrangian is L0 = E0
∞ X
β
t
log Ct − χt Nt + λt
t=0
and FOC are
wt Nt + κt Kt − Ct − It pt
1 − λt = 0 βt Ct wt t β −χt + λt =0 pt β t λt + Et β t+1 λt+1 κt+1 = 0
(c) (d) (e)
For the firm, the problem is to maximize Πt s.t. Yt ≤ Zt Ktγ Nt1−γ , and the FOC are (1 − γ) (γ)
Yt wt = Nt pt
Yt = κt Kt
3 – Define a competitive equilibrium of this economy It is a sequence of quantities {Ct , Nt , It } and prices {pt , wt , κt } such that 1
(f ) (g)
1. Those quantities maximize profit and utility at those given prices; 2. markets clear. 4 – Solve the model and show that the equilibrium process of output is yt = zt + γyt−1 − (1 − γ) log χt (1) (dropping constants and with the notation x = log X) The model is solves in a way similar to the analytical model of the course. Using (g), (e) and (c), on obtains 1 1 Yt+1 1 Ct+1 + It+1 = βEt γ = βEt γ Ct Ct+1 Kt+1 Ct+1 Kt+1 which implies It = βγ + βγEt Ct
It+1 Ct+1
and solving forward It βγ = Ct 1 − βγ from which we obtain Ct = (1 − βγ)Yt , It = βγYt and Kt = It−1 = βγYt−1 . From (c) and (d), one gets C1t = χt wptt . Using (f ) and the above result, we get Nt =
1−γ 1 1 − βγ χt
Plugging those results in the production function, taking logs and dropping constants, yt = zt + γyt−1 − (1 − γ) log χt NOTE THAT IN THE TEXT, I HAVE USED IMPROPERLY THE SAME LETTER χ FOR THE PREFERENCE SHOCK AND FOR ITS LOG. IN THE SOLUTION, I USE χ AND log χ 5 – Assume y−1 = 0, zt = 0 ∀t, log χt = 0 ∀t, except log χ0 = 1. Draw the time path on χ, z and y. Explain why y is persistent. See Figure 1.
? θ
We now consider an economy B, in which the TFP is not exogenous at the aggregate level, but given by Zt = Y t Xt , θ where X is the exogenous part of TFP and Y t act as an externality. More precisely, Y is taken as given by firms and households, but one has at the competitive equilibrium Y = Y NOTE : I ASSUME THAT θ ∈]01[ 6 – What is the economic interpretation of this externality? Learning-By-Doing is a possible interpretation 7 – Solve for the competitive equilibrium and give the equilibrium process of output (again in logs). Comment As we have added the effect of Y on Z as externality (meaning that agents do not take that into account when they take their decisions), nothing changes in the individual behaviors an in the definition of the equilibrium, so that one still has yt = zt + γyt−1 − (1 − γ) log χt but we have know that zt = θyt + xt , so that yt = (
1 ) (xt + γyt−1 − (1 − γ) log χt ) 1−θ
Note that we have an amplification of the effect of the shocks (compared to the previous case). 8 – Assume that an economist observes the economy B, with externality, but thinks that he is observing economy A, and is therefore using equation (1) to understand the data. Draw the response of observed TFP zt , of y and χ if y−1 = 0, xt = 0 ∀t, χt = 0 ∀t, except χ0 = 1. See Figure 2. 9 – How to interpret the positive correlation between observed TFP and output? Could such an economist believe (wrongly) that technological shocks are driving part of the response of the economy? Discuss. Here, z is an incorrect measure of technological shocks, as it is partially endogenous (it moves when a preference shock hit the economy). An economist interpreting z as technology would conclude wrongly that the economy is driven by technology shocks. 2
Figure 1: Impulse Response to a χ shock
Figure 2: Impulse Response to a χ Shock in the Economy with Externalities
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Problem II – Labor Market Dynamics with Sticky Prices This problem is taken from a 1999 AER paper of Jordi Gali (Technology, Employment, and the Business Cycle: Do Technology Shocks Explain Aggregate Fluctuations?). See pages 251 to 255 of the paper, that is on the course web page.
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