1 Satiation in money & the Friedman rule (50 points, 12.5 points each subpoint) Consider the Sidrauski model from class, but now suppose that the household has a per capita utility function of the form u(c(t), m(t) ) = c(t) 1-0 - 1 1 - 0 to m (t ) 1 - v - 1 1 - v - mum (t ) where m > 0 represents a "satiation level" of real money balances. (a) Find an expression for the marginal utility of real money. Draw the marginal utility of real money in a figure with m(t) on the horizontal axis and marginal utility on the vertical axis. Explain what the figure says in words. Hint: to properly draw the graph, it is useful to find the value of m(t) such that the marginal utility is zero and find the limit of the marginal utility for m(t) -> co.] (b) Find the household's optimality condition with respect to real money. Explain what this condition says in words. Hint: the household's Hamiltonian is very similar to the one in class, lecture note 9 page 5. The only difference is that now you have that new m term in the utility. If you do everything right you should find equation (6) exactly as in the lecture note. You should also find another equation, very similar to equation (7) but that accounts for the m term. Combining them you should find an equation very similar to (10) in the lecture note but, again, that accounts for the new m term.] The government targets a growth rate of nominal money per person of y. We assume that the real side of the economy is at its steady state. Here, I work out for you the fundamental equation of the Sidrauski model (these are thesame calculations did in class in section 3.1 Fundamental equation, but specific for the utility considered in this exercise). The government targets: dis (t) 1 dt ms(t) by money market clearing, ms(t) = p(t)m(t) dms(t) 1 dt ms(t) dp(t) 1 dm(t) 1 dt p(t) dt m(t) - 1 = 1 (t) + gm(t) Solving for 7(t) and substituting into the optimality condition derived in (b), 2 (m (t ) " - m-" ) c (t ) - 0 = r(t) +y - gm (t ) Solving for gm(t) and imposing r(t) = r* = p, c(t) = c* (steady-state condi- tions), it implies that: Im ( t ) = (p+ 7) - 2(c*)(m(t) -" - m-" This is the new fundamental equation (this is the equivalent of equation (13) derived in class, but specific for the utility introduced in this exercise). And here is the new Sidrauski diagram: * * ( ( ) ( mlt ) - - m " " ) p+ 8 in mix ) Recall that m* (as indicated in the diagram) is that value such that gm(t) = 0 (please read section 3.2 Diagram in Lecture note 9).(C) (d) The government wants to make the household happy. Based on the diagram and on the fundamental equation provided above, what value of qr should it choose? Explain. [Hint: it mayr be negative. Also, consider that for m(t) > m, additional real money actually provides disutility (makes the household anhappy!)] From wikipedia: \"The Friedman rule is a monetary policy rule pro- posed by Milton Friedman. Essentially, Friedman advocated setting the nominal interest rate at zero. According to the logic of the Fried man rule, the opportunity cost of holding money faced by private agents should equal the social cost of creating additional at money. It is assumed that the marginal cost of creating additional money is zero (or approximated by zero). Therefore, nominal rates of interest should be zero. In practice, this means that the central bank should seek a rate of deation equal to the real interest rate on government bonds and other safe assets, to make the nominal interest rate zero\". What nominal interest rate t) (recall the Fisher equation t) = r(t) + ir(t)) is associated with the optimal monetary policy you found in (c)? This is known as the Friedman rule. Please explain the economic intuition for why this is socially optimal. [Hint: remember that, at the steady state, %51 = 0 => 'r(t) : p. Also, recall from lecture note that qr = 7r(t) + gm(t). What is the social marginal cost of printing money?]