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Functions & Models

Functions are relationships between variables. Models are collections of functions, just enough functions to make a system, so that a prediction can be made.

Functions

Functions specify that one variable depends on certain other variables. The important thing about functions is not what they say something depends on, but what it doesn't depend on. In reality, every variable depends on every other variable. But when we make a model we choose to ignore certain effects, so, for example, the Keynesian demand function should be read not as saying Consumption depends on income, but rather Consumption is independent of everything except income.

Functions are based on two things (a) observed correlations in the data; and (b) theoretical arguments. Most of academic work called "macroeconomics" is argument (both empirical and theoretical) over whether a function is justified, often trying out a new function in a model.

In general the function relating one variable to another could be of any shape (quadratic, or sine, or jumping around), but when we need to calculate numerical solutions, instead of just qualitative solutions, we generally assume the function is linear, so instead of y=f(x), we just use y=mx + c. This means that the answer will be wrong, but hopefully it will be a good approximation.

Keynesian Consumption Function: C = c₀ + c₁(Y - T)

• c₁ is between 0 and 1, meaning that you don't consume all of your income.
• This seems to be confirmed in the data in the cross-section, though not over time.

Investment Function: I = d₀ + d₁Y - d₂i

• Investment increases with income, because people don't consume all of their income (they invest some).
• Investment decreases with the interest rate, because investment becomes less profitable as the interest rate rises.

Money Demand: M^d/P = L(i,Y) = m₀ + m₁ Y - m₂ i

• As income increases you want to hold more money, but as the interest rate increases the opportunity cost of holding money (instead of savings) is higher, so money demand falls. (data?)

Money Supply (1): M^s = H * 1/(c + θ(1 - c ))

• This function comes straight from the definitions of the variables, but it becomes useful for predictions if we assume that θ and c stay constant, in which case we can determine the effect of H on M (i.e., how the central bank can affect the money supply through high-powered money).

Money Supply (2): M^s = m₀ + m₁i

• The money supply may be increased by the interest rate because banks want to make more loans, so increase their lending multiple. This could be represented in the equation above by making θ a function of i (where i lowers theta, the reserve ratio), but it is simpler to just write the relationship directly between i and M.

Wage Function: W / P^e = F(u,z), dF/du < 0, dF/dz > 0

• This relationship can be justified by either of two different models: (a) wage negotiation; (b) efficiency wages. Both models predict a trade-off between benefits of having the job (the wage), and costs of not having the job (unemployment rate). In negotiation, you can secure a certain total amount of benefit from the employer, which is a mix of wage and protection from unemployment. Likewise in the efficiency wage model the employer wants to make their workers satisfied, and they do this by a mixture of wage and protection from unemployment.
• Note that wages are generally set in long-term contracts (e.g., 1 year long), so the expected price level is important. If expectations are correct then P^e=P.

Monopolistic Price Setting: P=(1+μ)MC

• MC is the marginal cost of production, under a simple production function where output equals labour (Y=N) then MC=W. Under perfect competition price equals marginal cost, so P=W. But if we allow some monopoly power they will mark-up prices, so P=(1+μ)W, where μ represents the degree of monopoly power. Higher monopoly power will cause lower real wages, W/P, because everything becomes more expensive, so you can buy less for the same wage.

Phillips Curve: Y = Y* + α(P - P^e)

• This is a clear pattern in the data, but there are different explanations for it. Also the parameters seem to change over time, so it's dangerous to rely on it.
• There are a few different theoretical justifications: (a) sticky prices, (b) sticky wages, (c) imperfect information.

Okun's Law: ΔY = 3.5 - 2*Δu

• (Mankiw, p256).

Neoclassical Aggregate Production Function: Y=F(K,L)

• Properties: dF/dK > 0, dF/dL > 0, d²F/dK² < 0, d²F/dL² < 0, F(λK,λL)=λF(K,L)
• Often a specific function is used which satisfies these properties, the Cobb-Douglas: Y=AK^αL^1-α

Models

Keynesian Cross (IS, Investment Savings)

• Uses (1) Keynesian demand function; (2) Investment function; (3) definition of GDP.
• Get: Y = (1-c₁ - d₁)^-1(c₀ - c₁T + d₀ - d₂i + G)
• Note: the first term is the multiplier, representing the effect of the right hand side variables on output, when holding the interest rate fixed. We have to assume that c₁ + d₁ < 1.

Money Demand (LM, Liquidity Money)

• Uses (1) Money demand function; (2) Money supply = Money demand.
• Get: M/P = m₀ + m₁Y - m₂i

IS-LM

• Uses (1) IS; (2) LM.
• You can derive a complicated formula for the equilibrium values, Y* and i*.
• Fiscal policy: higher G has a direct proportional effect on Y. There are also indirect effects through C and I, making the effect larger. Finally there is some leakage into the interest rate, making the effect smaller, because higher output will raise money demand, raising the interest rate, so lowering investment.
• Monetary policy: Higher money supply will lower the interest rate, which stimulates investment. The investment effect then goes through the same multiplier process, as with fiscal policy.
• You can derive the relative effectiveness of monetary and fiscal policy: `(dY//dG )/(dY//dM)=m_2/d_2`. If m₂ is high, money demand is very sensitive to the interest rate, so monetary policy is ineffective (only a little movement in the interest rate will accommodate the extra money), and fiscal policy is effective (there is little effect on the interest rate, so little crowding out). If d₂ is higher, investment is more sensitive to the interest rate, so monetary policy is more effective (a lower interest rate will stimulates investment), and fiscal policy is less effective (the increase in interest rate will crowd out investment).

AS-AD

• Keynesian Consumption + Investment Function + Liquidity Demand.
• Short-run: P fixed. Medium-run: Phillips curve. Long-run: Y fixed.

Structural Unemployment

• Wage Function + Monopolistic Price Setting.

Frictional Unemployment

• Beveridge Curve + Labour demand.

Mundell-Fleming

• Keynesian Consumption + Investment + Liquidity Demand + Exports
• Fixed: r