The Basic Classical & Keynesian ModelsFrom Sargent, 1979, Macroeconomic Theory Business Cycles Prof. Manoj Pradhan Department of Economics SUNY, Stony Brook

The following version of the Classical and Keynesian models as well as the analysis behind them is primarily from the seminal work of Sargent. This is a macroeconomic text that is used by graduate students all over the country. Thomas Sargent is one of the best recognized and most respected names in macroeconomics and is currently at Stanford University. In the mid-80s, he wrote another nastier book (Dynamic Macroeconomic Theory) that ensured the doom of any graduate student who had been brave enough to wade through Macroeconomic Theory even though solutions to the problems in these books are available. Note: for notation/definitions, look at the description at the end of this document. What are we after? In this exercise, we are looking for simple characterizations of the Classical and Keynesian general equilibrium models. These models can (in your more advanced studies) be modified and solved explicitly. A solution method for performing simple theoretical experiments is described towards the end of the paper. A quick look at the models is often startling because both models look very similar yet the results from these models represent polar extremes in macroeconomic ideology. The differences are subtle and should not be missed. Also, dont be misled by the apparent simplicity of the model it is designed to be simple since it appears in the first few chapters of the book. By the time most graduate students are done with the book, these models seem like a swimming pool after wandering around the Sahara for a year without a camel.

Classical Model The Classical model can be represented by a set of simultaneous equations, each equation either a market clearing condition or behavioral relationship. Remember that a system of n equations allows us to solve for at most n unknowns. The mechanics of setting this economic model are beyond simply counting variables and equations since the question of whether to make a particular variable endogenous or exogenous is an important issue.

Wage Equation W/P = FN(K, N) Real Wage = Marginal Product of Labor (Partial derivative of the production function with respect to labor/employment). Labor Market Equilibrium Condition N = NS(W/P) (NS) = d(NS)/d(W/P) > 0

Labor demand (N) = Labor supply (NS) where labor demand is derived from the behavior of firms and the supply curve is obtained from the labor force. The condition indented to the right states that labor supply is an increasing function of the real wage.

Aggregate Production Function Y = F(K, N) FK > 0, FN > 0, FKK < 0, FNN < 0 Output depends on the stock of capital and labor. Note that no restrictions (constant, increasing or decreasing returns to scale) have been imposed on the production function. The usual restriction is that that the production function has constant returns to scale. This also implies that the production function is homogeneous of degree one. In that case, if you multiply all inputs by a factor , then the output will increase by a factor as well. Mathematically: F(K, N) = [F(K, N)] = Y. FK is the partial derivative of F with respect to K, i.e., the marginal product of capital and FN is the marginal product of labor (both are assumed to be positive over all ranges of capital and labor). FKK and FNN are the rates of change of the marginal products of capital and labor respectively. FKK, FNN < 0 imply that the marginal product of capital keeps decreasing with each unit of additional capital or labor employed. Consumption Function C = C((Y, T, K, ((M + B)/p), I), i ) C1 > 0, C2 < 01

1

C1 is the partial derivative of the consumption function with respect to its first argument perceived disposable income. Similarly, C2 is the partial derivative with respect to the second argument the real interest rate.

This can be rewritten as C = C(YD, i ) where YD is perceived disposable income2. In the classical model, disposable income is defined as the rate at which households can consume while expecting that their real wealth is being left intact. Perceived disposable income varies positively with output (Y) and Investment (I) Perceived disposable income varies negatively with taxes (T), th amount of output used to replace depreciated capital (K), deficit of the Government financed by money creation (M) and sale of bonds (B) whose real value will be [(M+B)/P].

Overall, consumption increases if perceived disposable income (YD) increases (i.e., C1 < 0). Consumption depends negatively on the real interest rate (i.e., C2 < 0).

Investment function

dK F (i + ) I = I K dt i

or

I = I(q(K, N, i , )),

I > 0

We assume the real interest rate (r ) is always positive. Then, I will be positive (i.e., the time derivative or the growth rate of K will be positive) if FK (the marginal product of capital) exceeds the real cost of capital (i + ) which includes the cost of borrowing (i ) as well as the cost of replacing depreciated capital. The ratio that is the argument of the Investment function may be seen as a relative price (the ratio of the benefit of procuring capital to letting funds remain idle in some interest bearing account that represents the opportunity cost). It is important to note that this is a Keynesian investment function since the original classical model was equipped with an unrealistic condition that the second-hand capital goods market is continuously cleared (i.e., no excess demand or supply exists). The Investment function is redefined as q which, in turn, is a function of K, N, i and . Finally, note that q depends on K and N only because of the presence of FK which is the marginal product of capital (which depends on K and N).

National Income Identity linking Aggregate Output and its Components Y = C + I + G + K Thus, aggregate output is made up of household consumption, investment, government spending and K the amount of capital stock that has depreciated and needs to be replaced.

2

Obviously, YD = YD(Y, T, K, ((M + B)/p), I)

Money Market Equilibrium M/P = m(i, Y) mi < 0, mY > 0 Money supply (M/P) = money demand (m(i, Y)). The money demand function (m) is shown to have a speculative motive (mi < 0) and a transactions motive (mY > 0). Keynesian Model There is no discussion of arguments/equations/functions since they are essentially similar to the Classical model. Aggregate Production Function Y = F(K, N) Wage Equation W/P = FN Consumption Function C = C(Y T K ((M + B)/P), i ) This is a simpler version of the consumption function in the Classical model. There is a discussion below. Investment Function I = I(q(K, N, i , )) National Income Accounting Identity C + I + G + K = Y Money Market Equilibrium M/P = m(i, Y) Two differences between the Classical & Keynesian

Inessential Difference: The Keynesian model uses a simpler Consumption function one that depends on a simpler concept of disposable income than the one used in the Classical model. The Keynesian consumption function ignores any effects that changes in equity and reproduction cost of capital may have on consumption. I.e., it ignores changes in the relative

price of investment [FK (i )] compared to the opportunity cost (i ) of putting funds into production. This simpler definition really doesnt make that much of difference to the results of the model.

Essential Difference: The Keynesian model does not include the labor supply curve that was combined with the labor market equilibrium condition in the Classical model. This means that there are only six equations in the Keynesian model and it can therefore solve for a maximum of six variables. To close the model (i.e., make it general equilibrium), the Keynesian model takes nominal wages W to be exogenous. Note that this does not mean the W does not change over time, but that the change in W is not determined by changes in the model. Note that the Classical model includes the equilibrium condition N = NS (where N is labor demand and NS is labor supply) which is made up of the labor demand schedule (derived from the needs of the firm) and the labor supply schedule NS = NS(W/P) where NS increases with an increase in W/P. Essentially, the Keynesian model is saying that the labor market is driven by demand conditions so that the equilibrium in the labor market is not an equilibrium in the sense that demand need not equal supply all the time. In fact, labor market conditions are determined by labor demand rather than labor supply because of which we observe the phenomenon of involuntary unemployment (i.e., people look for jobs but cant find them). In the Classical model, the labor market equilibrium condition postulates that we are always at the intersection of the labor demand and supply curves there can be no involuntary unemployment. It is this difference that magnifies the role of policy in the Keynesian model while relegating policy to a non-entity in the Classical model.

Solution method

The entire set of equations is first totally differentiated (an example is provided in a technical appendix towards the end of this handout). This means that the variables in the equations will now be dW or dP or dY instead of W or P or Y (respectively). The fully differentiated equations are then solved as a system of equations. Relationships between endogenous variables and other endogenous and exogenous variables are examined from the solution. Thus, from solutions, expressions (comparative statics) are obtained which tell us about behavioral relationships within the economy.

Technical Appendix: Total Differentiation Consider an expression/equation (that is part of a system of equations) where x and y are endogenous variables, z is an exogenous variables and , and are parameters.

x = y zTotally differentiating this equation gives us:

dx = dy dzOf course, if the variables were observed multiplicatively, the other rules of calculus would have been applicable. If I totally differentiate all the equations in the system in this manner, I will have a system whose variables are now dx, dy and dz instead of x, y and z. The system can then be solved as suggested above. Notation/Definitions W = nominal wage P = aggregate price level Pi = price set by firm i Pi/P = relative price of firm i (relative to other prices) N = employment/employed workforce K = stock of capital Y = aggregate output NS = Supply of labor F = aggregate production function

= rate of depreciation of capital stock (given exogenously)C = consumption I = Investment (net of replacement of depreciated capital) M = money stock B = stock of bonds

= inflation (no distinction is made of ex-ante and ex-post inflation in this simple model)