hw 2

Macroeconomics: Homework 2
April 3, 2016
Exercise 1: Investment (from DeGregorio)
Suponga que un inversor puede comprar 1 bien de capital por valor Q. El bien paga un
ingreso Z el perodo de compra, y Z(1 + r)/2 en el periodo 2. Es decir que el capital se
deprecia la mitad del total cada perodo. Al fin del perodo 2 el capital se ha depreciado
completamente. Suponga que no hay inflacion y la tasa de interes real es r. El inversor paga
impuestos a una tasa τ sobre las utilidades.
a. Asuma que r = 0. Suponga que se le permite depreciar la mitad del valor del capital
en cada perodo. Calcule el valor presente del proyecto y demuestre que la tasa de
impuesto es irrelevante en cuanto a la decision de realizar o no la inversion.
b. Siga asumiendo que r = 0. Suponga ahora que se le permite depreciar aceleradamente
el capital, imputando el total de su valor como costo el primer perodo. Muestre que
el valor presente es el mismo que el del caso anterior y por lo tanto la decision de
inversion es independiente de la forma en que se permite depreciar el capital.
c. Asuma ahora que r ¿ 0. Calcule el valor presente del proyecto bajo las dos formas de
depreciacion: lineal (un medio-un medio) y acelerada (todo el primer perodo). En que
caso es mas probable que se realize el proyecto? Que puede decir respecto de la forma
en que se tributa la depreciacion y la inversion?
d. Por que si r ¿ 0 o r = 0 hace la diferencia? Para responder calcule el valor presente de
los descuentos hechos por la depreciacion.
Exercise 2: Investment (from DeGregorio)
Considere un proyecto de inversion que requiere invertir 100 en el periodo t. Si la inversion
se realiza en t + 1 el costo tambien seria 100. Sin embargo los retornos llegaran dos perodos
despues (el proyecto se hara en t + 1 y los retornos llegan en t + 2). Suponga r = 10%. Una
vez realizado el proyecto, este paga un unico rendimiento de F el perodo siguiente, y despues
se acaba el proyecto y el valor residual es cero.
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a. Si no hay incertidumbre respecto a F = 130 calcule el valor esperado y diga si conviene
o no hacerlo. Conviene postergar el proyecto?
b. Suponga ahora que el proyecto tiene un retorno incierto, con un retorno de 180 u
80, ambos con la misma probabilidad (1/2 por supuesto). Cual es el valor presente
esperado?
c. Suponga que el inversionista espera un perodo a que se resuelva la incertidumbre, es
decir sabra el siguiente perodo si los retornos futuros seran 180 u 80 (por ejemplo se
puede observar si producto lograra ser exitoso). Cual es el valor presente si ocurren los
flujos altos de 180?Y cual si ocurren los flujos bajos?Que hara en consecuencia el inversionista si se revela que los flujos seran bajos? Basado en la respuesta anterior. Cual es
el valor presente esperado si se posterga un perodo la realizacion del proyecto?Conviene
esperar? Discuta su resultado.
Exercise 3 (Romer)
Consider a firm that produces according to a Cobb-Douglas production function Y =
K α L1−α , the firm’s price is fixed in the short run and it takes the price of the product
P and the quantity Y as given. Inputs markets are competitive, hence, the firm takes inputs
prices as given W and rK .
a. What is the firm’s choice of L given P, Y, W and K?
b. Given this choice, what are the profits?
c. What is the optimality condition for K? Is the second order condition satisfied?
d. Solve the first order condition of K of the previous part as a function of P, Y, W and
rK . How do changes in these values affect the choice of K?
Exercise 4: Fiscal Policy (from DeGregorio)
Consider and economy with Bt−1 = 40 issued at a floating interest rate. GDP in (t-1) was
100, government spending G=20 and tax revenue T=20. in t-1, the interest rate was 5%
a. Compute deficit, total deficit and Bt . Write everything in levels and in % of GDP
Suppose that in period t, GDP falls to 95, tax falls consistently with an elasticity taxrevenues to output of 2, interest rate increases to 15% and the government decides to
raise government spending 3%.
b. Suppose that international markets decide not to lend more than 50% pf GDP. What’s
the maximum G this government can finance without accordingly to its budget constraint?
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c. Suppose that the government (in period t − 1) expects an even worst period t. Expect
a 10% output fall, and an interest rate equal to 20%. If the government wants to
keep G constant. How much should be Bt (as a percentage of GDPt−1 ) such that this
government can keep G constant?
Exercise 5: Fiscal and Monetary Policy ( from Walsh)
Suppose the government’s budget identity is
bt = Rbt−1 + gt + τt yt − st ;
where bt is a one-period debt, R is the gross interest rate, gt are government purchases, τt yt
are income tax receipts, and st is the seigniorage. Assume seigniorage is a function f (πt ),
where pit denotes inflation and gt is exogenous. The government sets time paths for the
income tax rate and for inflation to minimize
Et
∞
X
β i [h(τt+j + k(πt+j )]
j=0
where h and k represent the cost of taxation in terms of distortions. Assume h0 > 0, h00 > 0
and k 0 > 0, k 00 > 0.
a. Find intratemporal optimality conditions that link τt and πt for each t?
b. Find the intertemporal optimality condition
c. Suppose y = 1, f (πt ) = aπt , h(τt ) = bτt2 , and k(πt ) = cπt2 . Evaluate the inter- and
intratemporal
conditions. Find the optimal rules for τt and πt in terms of bt−1 and
P −j
R gt+j .
d. Using your results from part (c), when will optimal financing imply constant planned
tax rates and inflation over time?
Exercise 6: Debt Sustainability (from DeGregorio)
Consider Bt+1 − Bt = Gt − Tt + rBt . Suppose there’s no inflation and, given r, GDP growth
equals to γ, primary surplus in terms of GDP equals to s, write the constraint in terms of
GDP.
Write debt to GDP in the long run (steady state) equals to b∗ . What happens with it if
r increases? why? Suppose there are 2 economies that are alike but γ 1 = 2% and γ 2 = 4%
which one will have larger b∗ and why? Finally, assume r = 6% and s = 1%, compute b∗ for
each of the two growth rates of output.
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Exercise 7: Small Open Economy
Consider the 2 period small open economy seen in class. Suppose Q1 = Qlow < Q2 = Qhigh .
a. Write down the Lagrangian and find the optimal consumption in period 1, 2 and the
level of debt as function of endowments and the interest rate and the initial level of
debt.
b. Suppose a permanent increase in output equals to Σ. Plot, show algebraically and
discuss how does this affect the optimal choices of consumption in each period
c. Suppose a transitory increase in output (in period 1) equals to Σ. Plot, show algebraically and discuss how does this affect the optimal choices of consumption in each
period
Exercise 8: Small Open Economy
Consider the 2 period small open economy seen in class with terms of trade. Suppose
T T1 = T Tlow < T T2 = T Thigh .
a. Write down the Lagrangian and find the optimal consumption in period 1, 2 and the
level of debt as function of endowments, terms of trade and the interest rate and the
initial level of debt.
b. Suppose a permanent increase in terms of trade equals to Σ. Plot, show algebraically
and discuss how does this affect the optimal choices of consumption in each period
c. Suppose a transitory increase in terms of trade (in period 1) equals to Σ. Plot, show
algebraically and discuss how does this affect the optimal choices of consumption in
each period
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