Prof. Stefania Albanesi The Ohio State University 8723 Macroeconomic Theory IIA Spring 2016 Assignment 2, due February 10, 2016.

Question 1 Consider a representative agent economy in which the household’s preferences are given by ∞ X −t (1 + ρ) U (ct ) t=0

where

1 1−σ c . 1−σ t The household is endowed with one unit of time, which can be used either to produce goods or to carry out transactions. Total output yt satisfies yt ≤ nt ≤ 1, where nt is time devoted to production. Call st the fraction of time devoted to transacting. The amount of output available for consumption is: U (ct ) =

ct = (1 − st ) yt . Households begin period t with Mt units of money, out of which they pay or receive a lump sum dollar tax Xt . The price level is Pt . In addition to holding money the household can hold bonds. The time t dollar price of a bond which yields Bt+1 dollars at time t + 1 is Bt+1 / (1 + Rt ). So the cash flow budget constraint is given by Mt+1 +

Bt+1 = Mt − Xt + Pt [(1 − st ) yt ] − Pt ct + Bt . 1 + Rt

There exists a transactions constraint relating household holdings of real balances, mt , and the amount of household time devoted to transacting, st , to the spending flow that the household carries out, ct . This constraint takes the form 1/α

ct = mt st where α > 1.

(i) Display the household’s first order conditions for ct , mt+1 and Bt+1 . (ii) Use the first order conditions to derive a portfolio balance or money demand equation of the form mt = L (ct , Rt−1 ) . 1

(iii) Suppose that Xt /Mt = x for all t. Define a sequence of markets equilibrium for this economy. Show how the rate of inflation, Rt and mt depend on x.

Question 2- Endogenous Money Demand This question is motivated by empirical evidence on comovement among M1 velocity and the relative size of the banking sector in different countries. Specifically, in high inflation countries, movements in velocity and the relative size of the banking and credit sector tend to parallel movements in inflation. Consider the following economy where there are both cash and credit goods. These types of goods are perfect substitutes in consumption and investment but differ in their production technologies. The production of credit goods is a resource using activity. Production Sector Total output Yt∗ is produced using capital Kt and labor Nt with a constant returns to scale production function F. Total output can be used to produce goods, Yt , for investment and consumption on a one-to-one basis or can be used to produce credit services St where one unit of credit services requires qst units of total output. Therefore: Yt∗ = F (Kt , θt Nt ) = Yt + qst St , where θt is a labor augmenting technology shock. The cost of producing credit services qst is assumed to follow a stationary stochastic process. The output of goods is assumed to be uniformly distributed across a continuum of types indexed by i ∈ [0, 1]. A unit of type z good. One unit of type z good can be used in one of two ways. It can be used to produce one unit of the cash goods on a one to one basis or it can be combined with R (z, εt ) units of credit services and used to produce a unit of the credit good, where R is strictly increasing in z. The random variable εt is assumed to follow an exogenous stationary stochastic process that is independent of θt and qst . A unit of type z good (either cash or credit good) can be consumed or used for gross investment. Let it and it (z) be gross investment and the amount of type z good used for gross investment, then the technology for gross investment is the following: it = inf {it (z)} . z

A) Let p1t and pst denote the prices of cash and credit services. Also, let wt and rt denote the wage rate and the rental rate on capital in units of cash.

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Let p2t (z) denote the price of type z credit good. Show that firm optimization implies: pst = qst , p1t wt = θt F2 (Kt , θt Nt ) , rt = F1 (Kt , θt Nt ) , p2t (z) = p1t [1 + qst R (z, εt )], and that the price of credit good is increasing in z. Households There is a representative infinitely lived household which has one unit of labor endowment available in each period. The household consumes ct (z) of type z good in each period and supplies the amount nt of labor input in each period t. Preferences are given by: E0

∞ X

β t u (ct , 1 − nt ) ,

(1)

t=0

with β ∈ (0, 1) and ct = inf z {ct (z)} is composite consumption. The household purchases χt (z) units of type z good and uses these goods for consumption and gross investment. In view of the Leontief type aggregators, in equilibrium: it (z) = it , ct (z) = ct , χt (z) = χt , with χt = it + ct = ct + kt+1 − (1 − δ) kt

(2)

(δ is the depreciation rate). The household is subject to a cash in advance constraint and a dynamic budget constraint. Total household purchases χt will in part be in the form of cash goods and in part be in the form of credit goods. Since the cash price of goods is constant across types and the price of credit goods is increasing in the type index z, there will be a cut off index zt∗ such that the household will optimally purchase goods with index below zt∗ as credit goods and purchase goods with indices above zt∗ as cash goods. Thus, credit good purchases are given by χt zt∗ and cash good purchases are equal to χt (1 − zt∗ ) . The household begins the period with mt units of currency and bt units of nominal bonds. The household receives a nominal lump sum transfer from the government, Xt . There is financial market, which takes place before the goods market, in which households can rearrange their portfolio between currency and bonds. On the goods market, the cash in advance constraint is: mt + Xt bt bt+1 + − ≥ χt (1 − zt∗ ) , p1t p1t Rt p1t 3

(3)

where the left hand side is the amount of cash that the household has available at the close of the financial market. The household’s budget constraint is as follows: mt + Xt bt + +wt nt +rt kt ≥ χt (1 − zt∗ )+ p1t p1t

Z

zt∗

χt 0

p2t (z) mt+1 bt+1 dz+ + . (4) p1t p1t Rt p1t

Note that wage and rental income is not available for consumption in the current period. The household maximizes (1) subject to (2), (3) and (4). Government The government sets the money growth rate xt = (Mt+1 − Mt ) /Mt in such a way that xt follows a stationary stochastic process that is independent of {θt , qst , εt }. B) Show that the solution to the household optimization problem is characterized by the following first order conditions: Ult wt = , Uct 1 + τt Uct = βEt [Uct+1 (p1t /p1t+1 ) Rt+1 / (1 + τt+1 )], 1 + τt Uct = βEt [Uct+1 (1 − δ + rt+1 / (1 + τt+1 ))], Rt =

p2t (zt∗ ) , p1t

where 1 + τt = (1 − zt∗ ) Rt +

Z

zt∗

p2t (z) dz/p1t , 0

and Ult , Uct denote the marginal utility of consumption and leisure, respectively. Provide intuition for each of the first order conditions. In equilibrium, the following equations must also hold: F (Kt , θt Nt ) = Yt + qst St , χt = Yt = Ct + Kt+1 − (1 − δ) Kt , Z zt∗ Yt R (z, εt ) = St , 0

(kt , nt , ct , mt ) = (Kt , Nt , Ct , Mt ) . C) Show that in equilibrium i) Rt = 1 + qst R (zt∗ , εt ) , ii) this condition determines the cut off index, so that for goods with index lower than zt∗ are purchased as credit goods, and those with index above are purchased as cash goods, iii) other things equal, and increase in Rt leads to a decrease in the 4

fraction of goods purchased with cash, iv) the inverse money demand function is: Rt = 1 + qst R (1 − (Mt+1 /p1t ) /Yt , εt ) . D) Let φt = qst St /Yt∗ denote the value added of credit services in GNP. Let: Z at =

zt∗

qst R (z, εt ) dz. 0

Show that: at = φt / (1 − φt ) , and that this implies a positive relation between inflation and the relative size of the credit sector in the economy. E) Suppose that we consider a non-stochastic steady state where the growth rate of money is equal to µ for all t. What is the relationship between µ and the size of the credit sector?

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