An elemental macroeconomic model for applied analysis at undergraduate level



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ECONOMICS


AN ELEMENTAL MACROECONOMIC MODEL

FOR APPLIED ANALYSIS

AT UNDERGRADUATE LEVEL


by

Rod Tyers

Business School

The University of Western Australia and

ANU Research School of Economics


DISCUSSION PAPER 11.11

An Elemental Macroeconomic Model

for Applied Analysis at Undergraduate Level*
Rod Tyers
UWA Business School and

ANU Research School of Economics

May 2011

Working Papers in Economics

UWA Business School
DISCUSSION PAPER 11.11

* Thanks are due to Ben Smith and Matthew Benge, former colleagues at the ANU, with whom the original representation of the model presented was originally worked out and trialled since the 1990s. Thanks are also due to Juerg Weber and William Coleman for useful discussions about applications to the third year and honours levels.



Abstract:
A graphical representation is offered based on a fairly standard formulation of an underlying comparative static model for applied undergraduate analysis. Contrary to standard practice the approach takes Walrasian equilibrium as its starting point and considers market failures that include nominal rigidities as special cases. It therefore builds intuition that centers on price rather than quantity adjustment following shocks. Beyond this, its advantages include that it offers comparative ease of representation of external shocks, which are particularly important in small open economies, it uses intuitive demand-supply market diagrams throughout and it provides the ability to set as clear targets for monetary policy the exchange rate, the CPI and the GDP price. Finally, it allows for forward expectations so that it offers useful insights for students into the economic consequences of financial shocks.
1. Introduction:
Macroeconomics is a very old sub-discipline to which approaches have been many and varied. The challenge is to capture key characteristics of the economy as a whole, which presses toward general equilibrium analysis and economic dynamics in ways that are too technically advanced for the average undergraduate. To avoid disguising fundamental ideas about macroeconomic behavior behind advanced technique, the undergraduate teaching of macroeconomics has tended to emphasize short run comparative statics. Most commonly, the approach uses aggregate demand and aggregate supply, combined with Hicks’ (1937, 1980) IS-LM representation.1 Fixed price levels are commonly assumed in the Keynesian tradition and there is most often a “Keynesian cross” diagram the subterranean message from which is to emphasize that, in determining GDP, failures of labour, product and financial markets are the default case, with output being driven by unintended inventory changes.2 Awkwardly, this treatment is usually presented to undergraduates immediately following a semester of microeconomics that is dominated by economic stories about optimizing agents and clearing markets.
The approach described here was conceived to offer an alternative, one that avoids the difficult intuition of the equilibrium loci used in AD-AS and IS-LM and the default conception of the economy crippled by nominal rigidities. It was conceived jointly with Ben Smith (2000) originally as a teaching tool to offer an alternative representation to the standard used in principles and intermediate economics courses.3 It is centered on a flexible price level Mundell (1963) - Fleming (1962) framework with imperfect financial capital mobility, in which an open financial capital market experiences inflows and outflows depending on yield differentials and exchange rate expectations but does not achieve uncovered interest parity.
The key advantages it offers are 1) the ease of representation of external shocks, which are particularly important in small open economies, 2) the use of more intuitive demand-supply market diagrams throughout, 3) the endogeneity of the price level, 4) the Walrasian starting point (reference case) with wage stickiness offered as a special case, 5) the ability to set as clear targets for monetary policy the exchange rate, the CPI and the GDP price, an advantage over the AD-AS IS-LM representation in which the money supply must be exogenous, and 6) it incorporates expectational variables that allow the analysis of optimism and pessimism shocks stemming from information failures in financial markets.
The graphical approach has been used in support of wider macroeconomic analysis by Tyers (2001), Roberts and Tyers (2003) and Rees and Tyers (2004). The model generalizes readily into one suitable for applied analysis in more advanced classes and a two period, multi-product version is available for computer simulation using the freeware Rungem, a component of the Australian Gempack modeling package.4
The next section summarises the key assumptions, the labour market is presented in Section 3 and consumption and saving are discussed in Section 4. The small open economy financial capital market is presented in Section 5, the Foreign Exchange market in Section 6 and the market for money in Section 7. The complete model is assembled in Section 8 and money neutrality is demonstrated in Section 9. Shocks with nominal wage rigidity are considered in Section 10 and, for illustration, an unexpected bond-financed fiscal expansion is analysed in Section 11 and a financial crisis in Section 12. Links with the more standard AD-AS IS-LM representation are formalized in Section 13 and conclusions are offered in Section 14.

1. Key assumptions:
Short run: The relevant length of run is within the gestation period of new investment, so that investment places demands on the capital goods sector but does not add significantly to the productive capital stock in the current period (though it does in future periods, if modelled) so .
Expectations: Much depends on expectations and these are exogenous in the comparative static graphical representation. Agents form expectations over the GDP price level, PY, the exchange rate, E, the level of future private disposable income, YF, and the real rate of return on installed capital net of depreciation, rC. These then become shifters in the graphical representation. If these are left inactive, the model generates changes in price levels, exchange rates and interest rates but they are not anticipated.
No steady state: The real interest rate in the capital market, , generally differs from the net rate of return on installed capital, :

(1)

Financial assets are bonds and money: risk considerations are not modeled explicitly, so assets are consolidated into bonds of short maturity (monetary instruments) and of long maturity (the bulk of the collective asset portfolio and the key instruments in saving and investment), the yields on which are separated via the segmentation theory of the yield curve.
No uncovered interest parity: In general, financial capital is imperfectly mobile internationally on the grounds that home long maturity bonds are differentiated from their foreign counterparts by considerations of risk and transaction cost. Yet responses to shocks to yield differentials and exchange rate expectations do move toward uncovered interest parity.
An initially clearing labour market: The reference case is Walrasian, in which all represented markets clear. The price level is flexible (endogenous) and, at least initially, the nominal wage is assumed to adjust to clear the labour market. Once changes in the price level are determined, the effects of wage stickiness can then be resolved. So employment is initially constant at . Initial output in volume units is therefore:

(2)

where A is the exogenous total factor productivity coefficient.


2. The labour market:

Labour supply is inelastic on the grounds that empirical studies find its short run elasticity to be small, and because it is intuitive for students to derive the labour supply curve from the trade-off between consumption and leisure with Cobb-Douglas preferences.5 In this case, income and substitution effects offset one another, leaving the labour supply curve vertical.

Labour demand is driven by production technology with the standard condition equating the money wage, W, in $/worker year, to the value of the marginal product of labour, or the real wage, w, to the marginal physical product of labour:

(3)

This equilibrium is illustrated in the labour market diagram of Figure 1.

3. Investment depends on the net rate of return on installed capital:

The real gross rate of return on an already installed machine is just the value of its marginal product, PYMPK, which is the annual increase to revenue generated by the machine, divided by the market price of the machine, PK. The net rate of return just subtracts the rate of depreciation, . This is what drives investment decisions.

(4)
Figure 1

w LS



w0=W0/PY

LD: )

L

In the short run the capital stock is given, so firms do not necessarily have the optimal quantity. In the long run steady state, however, investment takes place until the real net rate of return on capital falls to the level of the real financing cost or the opportunity cost of invested funds (the real rate of return that can be earned on financial instruments, or that must be paid if bonds are issued to finance the investment, or the market real interest rate), r. The real rate is preferred here to the nominal rate on the assumption that capital gains are expected to accrue when the price of capital goods inflates at the same rate as home goods-services. So in the steady state, but not in general, .

So, in the length of run to be considered, expected changes in drive investment:


  • if the return will exceed the real financing cost,

  • if the investment will not be made.

This expected future value of this rate of return which is then exogenous.

4. Consumption and saving:

Here a simple income tax rule is used with a Keynesian consumption equation in present and expected future disposable income, measured in real volume equivalents. All units are product volumes. Government spending, G, is a real exogenous policy variable.

(5) Elemental tax rule:

(6) Real private consumption

(7) Real private saving:

(8) Total real domestic saving:

To reflect consumption smoothing behaviour, private saving responds positively to current disposable income and negatively to expected future disposable income. The substitution effects are assumed to dominate when changes in bond yields occur, so that private saving also responds positively to the real yield on saving instruments (long maturity bonds), r.

5. The small open economy financial capital market:

This is the market that matches saving to investment. Units are equivalent volumes of goods and services.

(9) Real investment (break-even + net):

Here the ratio of the expected net rate of return on installed capital and the current real long bond yield is closely related to Tobin’s Q, since the numerator determines the current market value of capital assets and the denominator determines the real cost of financing their replacement.

(10) Net foreign investment demand:

(11) Capital account net inflows:

Where represents the net provision by foreigners of funds to finance a surplus of imports over exports (current account or CA deficit), SNF is net foreign saving, or the private component of these inflows, dependent on the foreign bond yield, r*, the home bond yield, r, and any (exogenous) expectation of a real exchange rate change, . The real exchange rate here has the financial definition, so it rises with appreciations. The change in the sock of official foreign reserves, , is an exogenous monetary policy variable and, consistent with the small open economy assumption, the foreign bond yield, r*, is also exogenous.

This dependence implies something short of the real version of uncovered interest parity . Indeed, SNF is influenced by departures from uncovered interest parity, the latter applying only in a steady state:

(12) .

The size of the slope of this curve, bSF, then determines how quickly the solution approaches uncovered interest parity. Small values suggest capital controls and large values suggest high levels of financial capital mobility. Allowing departures from uncovered interest parity admits reality in a way that avoids the application inflexibility of approaches that impose it.6

The small open economy financial capital market equilibrium is then obtained by finding the home bond yield, r, that renders:

(13)

Once the supply side equilibrium has been determined, GDP, or Y, is known. All the other variables in this equilibrium condition are exogenous, except the home long bond yield, r, which is readily solved for. From this, then, the real value of net financial inflows on the capital account, KA, also emerges.

Critically, with clearance of the labour market, the capital account balance is seen to depend only on financial variables, not on trade and the current account. The equilibrium in this small open economy financial capital market is illustrated as in Figure 2, with the curves showing only their shifters in parentheses.

Figure 2

r

KA(ΔR,,r*)

r0

I-SD =



KA- KA0 0 KA+
6. The market for foreign exchange:

First define the exchange rates. The real exchange rate is the number of representative foreign product-service bundles that can be acquired in exchange for a single home product-service bundle. It is therefore the common currency ratio of the price of home goods to that of foreign goods, best represented by the respective GDP prices, and . But is expressed in foreign currency and so requires conversion at the nominal exchange rate, E. Thus, if that is also defined financially (appreciating when it rises) we have that:

(14) .

Because the capital account7 must balance the current account and the latter is linked to the values of imports and exports in home $, the balance of payments in home $ is:

(15)

where N is net factor income from abroad. Imports, M, and exports, X, both depend on the real exchange rate.

Consistent with the practice adopted in representing the financial capital market, this should also be expressed in equivalent real volumes of home product. Since N depends mainly on past investments abroad, consider hereafter that N is an exogenous constant, expressed in real terms and so in units of foreign output. Redefining the X and M in home product equivalents, the balance becomes:

(16)

Exports are foreign demands for home goods, which, like any product demand, depend positively on foreign income and negatively on the relative price of home goods or the real exchange rate. Imports are home demand for foreign goods and so will depend positively on home income, or GDP, and negatively on for relative price of foreign goods (the inverse of the real exchange rate). For an intuitive diagram, express these in terms of the inverse of the real exchange rate – the relative price of foreign goods. So the current account deficit is:

+ - + +


(17)

Net import demand, or excess demand for goods, is therefore unambiguously downward sloping in the relative price of foreign goods or positively dependent on the real exchange rate. Net factor income has the opposite dependence, however, though this is a pure valuation effect, whereas the trade flows combine valuation with volume adjustment effects. We will assume the trade account to dominate, so that the current account deficit can be written:

+ - -

(18)



The current account measures the net inflow of payments associated with current transactions, here measured as the equivalent volume of home goods. A deficit means a net outflow.

This is the equivalent of a net outflow of foreign exchange, matched by a net inflow of foreign exchange arising out of financial flows in the financial capital market. But we know from that market that the net inflow of foreign exchange associated with asset transactions is KA. And, so long as the labour market clears, the equivalent quantity of home products is already bolted down. So the real exchange rate adjusts to ensure that the balance of payments does actually balance:

(19)

The only endogenous variables in this relationship are Y and , but Y has already been determined in the labour market. So this relationship reveals the real exchange rate, as shown graphically in Figure 3. Knowing Y and eR, M and X can be obtained by back solving if needed.



7. Demand side, nominal part:

Since the maturity of financial assets matters, a note on the yield curve is useful at the outset. Financial assets are here generalised as bonds. No risk considerations are explicit in the model, except to the extent to which they differentiate home from foreign bonds and therefore cause the failure of uncovered interest parity, leading to imperfect financial capital mobility even where no capital controls are present. But central banks dominate trade in “money market instruments” of maturity less than a year, while the collective portfolio comprises mainly long term instruments, which might be thought of here as 10 year bonds. The yields on short and long instruments do not move together through time, with short rates being altered by central banks in the management of home business cycles while long rates move more smoothly through time and their markets are more integrated across countries, reflecting their role in equating the global supply of saving to global investment demand.


For these reasons, the home real long bond yield is the equilibrating variable in the financial capital market. And, since long instruments dominate the collective portfolio the nominal long bond yield is the opportunity cost of holding money. The two are related as , where is the expected rate of inflation.

A conventional cash in advance constraint is assumed to underlie the demand for real money balances, which then depends on GDP, representing transactions demand, and the nominal long term bond rate.

+ - - +

(20)


Figure 3

(from financial capital market)

1/eR

1/eR



CA+ CA0 0 CA-

Where is an exogenous risk premium that may be shifted upward by pessimism about the performance of financial instruments other than money. Thus, when financial instruments appear riskier, the opportunity cost of holding money may not be as large as the current long term real yield even after adjustment for expected inflation.

The supply of money begins with the monetary base, MB, which central banks control by committing to trade in short term instruments until excess demand at a designated short yield is reduced to zero. The relationship between short rates and MB need not be made explicit here, however. The money multiplier then links the monetary base to the nominal money supply and thence to the supply of real money balances.

(21)

Where c is the public’s cash to deposit ratio and is the reserve to deposit ratio of banks and other financial institutions. So the supply of real money balances can then be equated with its demand in a clearing money market.

(22)

If the monetary base is considered an exogenous policy instrument, determined by declared levels of iS in a separate process, the expected inflation rate is exogenous, as are the parameters c, and , the only endogenous unknown is the price level, PY, or the exchange rate between home money and home goods-services.

The diagram in Figure 4, unconventionally, uses the real interest rate on the vertical but, conventionally, real money balances on the horizontal. GDP, as a proxy for transactions demand, expected inflation, , and the risk factor, , are then shifters of the money demand function, with the latter two always exogenous. As above, the money market equilibrium is where:

+ - + -

(23)

The only variable that remains to be calculated is the nominal exchange rate. If the foreign price level, , is exogenous, consistent with the small country assumption, then the nominal exchange rate, E, can be obtained by transforming the standard expression (14) for the now known real exchange rate in terms of the known home price level:

(24) .



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