Economics 345. We know that a money demand function can be described as:

Transcription

1 conomics 345 The Arithmetic of Monetar Growth, Inflation and the xchange Rate in the Monetar Approach We know that a mone demand function can be described as: αr ( M / ) =. e where M is the nominal quantit of mone; is the price level, is the level of real income, and R is the nominal interest rate. For the moment we will ignore the interest rate so that the mone demand equation can be written as: 2. ( M / ) = This is a most simple form of mone demand 1 that allows us to see what happens when the rate of the mone suppl growth is increased. Rewrite 2 so that the initial relationship is: 3. M = Now assume that M, and are each displaced b a small amount, M,, and. This leads to a new version of 3: 4. (M + M)=( + ).( + ) which can be multiplied out to leave: 5. M + M= Now, first notice that from 3, M =, so that the first term on both sides of 5 cancel. Second, if and are both small, then multipling two small things together leads to a product that is ver small so small in fact that we shall assume that it disappears! (That is, assume that. =.) This leaves us with 6: 6. M= As a challenge, suppose that real balances were described b M σ αr = e in which σ is the income elasticit of mone demand and α is the (semi) elasticit of the interest elasticit of mone demand. Work out the effects of such an assumption on the algebra of price and exchange rate determination. (Implicitl, σ=1 in the text.) 1

2 Now recall that M =. Divide the term M b M on the left hand side (LHS), and both terms on the right hand side (RHS) b. (This is OK because the are equal.) Now what we have is 7: M But since the cancels in the first term on the RHS, and the cancels in the second term on the RHS, we are left with the expression: M Since each of these terms can be read as a percentage change, equation 8 reads as: the percentage change in the mone suppl is equal to the percentage change in prices plus the percentage change in real income. If we now divide b a change in time,, so that the change in M, M is per period,, then we have an equation like 9 where everthing is expressed in rates of growth per ear (or per month or whatever time period ou like): 1 M quation 9 reads as: the percentage rate of growth of the mone suppl is equal to the percentage rate of growth of prices plus the percentage rate of growth of real income. If the rate of growth of the mone suppl is called, ρ (rho), and the rate of growth of the price level (otherwise known as the rate of inflation) is called, π (pi), and the rate of growth of real income is called, λ (lambda), then we can write the relationship binding the long-run growth rates of mone, income and prices as: 1. π=ρ-λ. This is a ver important relationship. We have a simple expression ting the inflation rate to variables about which we know something. It is a long-run relationship insofar as the price level is variable. A Technical Aside (for those with calculus) 2

3 A second wa to show the relationships among the rates of change is to take the logarithms, and then take the derivative of both sides of 3. This is a particularl quick wa to get the percentage changes since the derivative of the logarithm of a variable is the percentage change in that variable: 2 For example, M= so that lnm=ln+ln. Now find changes in lnm: dlnm=dln+dln. The use of the d means take the derivative which means to take the change in the variable. From footnote 1 that establishes that for an variable, d ln x =, or that the x derivative of the log of a variable, d(ln(x)), is the percentage change in x, this ensures that: dm d d d ln M = = d ln + d ln = +. M Or that, as before in 1, thinking of all of the changes as per unit time: π = ρ λ Adding the Nominal xchange Rate, The relationship to the xchange rate comes from urchasing ower arit. The exchange rate reflects the price levels between two countries: 11. = is the anadian price level and is the uropean price level. The exchange rate is, as usual, the dollar price of a uro. This can be rewritten as: = Taking the rates of change as in 4-9 give us a relationship between changes in the exchange rate and changes in the price levels: 2 The exponential function has a defining characteristic: the change in the function is equal to the function itself. That is if or, b rearranging, that x = e, then d = d = x. Notice that since x exponential function. This means that d ln =. Since we know that e = e, this also means that d = e x =. We also know that the natural logarithm is the inverse of the ln = x d =. Taking the derivative of both sides we have, then we have established that d ln = d. Since the percentage change in, we have established that the derivative (the change) in the logarithm of a variable is the percentage change in the variable itself. d is 3

4 t 12. = π π In 12, the expression t is the percentage change in the exchange rate or the rate of depreciation of the anadian dollar. (For those following the footnotes, dln= t.) Using 12 and recognizing that 1 is the underling structure generating the rate of growth of the price level in each countr, we now have a theor of the long-run determination of the exchange rate. Labeling each index appropriatel we can write the exchange rate as a function of the rates of change of income and the mone suppl: t t 1 13 = π π = ρ ρ + λ λ t 1 This means that increases in the rate of growth of the domestic mone suppl will lead to a depreciation of the dollar, ceteris paribus. An increase in real income growth, in contrast, will lead to an appreciation of the anadian dollar. The Real xchange Rate As an exercise, determine the variables underling the behaviour of the real exchange rate and find the equivalent of 13 for. 4

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