A Critical Assessment of Different Schools of Economic Thought

Abstract

The recurrence of economic recessions has stimulated substantial research in this area and has given birth to a number of schools of economic thought. However, different schools have been fighting with each other and this has made macroeconomics a warring zone. Chapter 3 introduces major schools of economic thought and discusses their strong points and their deficiencies.

Appendices

Appendix 1 (for Section 4.8.3.2): The Ways to Obtain the AD Curve

Due to the difficulties in aggregating market demand curves, methods other than direct aggregation are used to obtain the aggregate demand curve, including methods utilizing the AE/AP model, the IS/LM model and the quantity theory of money. However, these methods of obtaining AD curves resulted in much criticism. We describe and assess the four existing methods first and then introduce four new ways of obtaining AD curves by aggregating market demand curves.

• 1. Obtaining the AD curve from the AE/AP model

The difficulty in obtaining an AD curve from the AE/AP model is that there is no explicit price variable in the model. A simple AE/AP model can be expressed as:

Y = I₀ + C₀ + c * Y, or (1 − c)Y = I₀ + C₀, where Y is income, I₀ and C₀ are autonomous investment and consumption, respectively, and c is a parameter indicating the propensity to consume.

To derive the AD function, we can regard the variables in the model as the function of price because price affects Y, I₀ and C₀. As such, the AD/AE model can be written as:

$$ (1 - c)Y(P) = I_{0} \,(P) + C_{0} (P). $$

To gauge the impact of a change in price P on income Y, we differentiate the above equation to obtain:

$$ (1 - c)dY(P)/dp = dI_{0} \,(P)/dP + dC_{0} (P)/dP. $$

Once we have measured the impact of a change in price on autonomous investment/consumption (dI₀ and dC₀), we can use the above equation to obtain the impact on total expenditure/income (dY). This can also be shown by employing a graph of multiplier models. As the left panel of Fig. 4.37 shows, when there is a price hike from P₀ to P₁, autonomous consumption plus autonomous investment decreases from A to B. The total expenditure decreases from Y₀ to Y₁. We have obtained two points for the AD curve: P₀ associated with Y₀ and P₁ associated with Y₁. We can allow more price changes and obtain more data, e.g. P₂ associated with Y₂ at point C, etc. Collecting all the data, we can draw an AD curve.

This approach is criticized for being inconsistent with the general definition of demand curves and for including the supply-side impact. Colander (1995) used a precocious student as a prop to drive his point home. The student first measured that a price decrease by 2 (e.g. P₂ − P₁ = 2) increased the output by 2 (e.g. the size of HG in Fig. 4.37), thus he obtained a slope of AD as −1. Since this increase in output is due to an increase in autonomous consumption/investment, the output increase is not affected by income. Then, the student took care of the multiplier effect and measured the increase of total output by 6 (e.g. the size of Y₁ − Y₂ in Fig. 4.37) when the price decreased by 2, so he obtained a slope of AD of −3. The student was perplexed: Which slope is correct? Colander concluded that the AD curve of the slope of −1 was consistent with the normal definition of a demand curve—the demand change is caused by the price change alone, other things being equal. Following the same reasoning, he thought the AD curve of the slope of −3 was not consistent with the normal definition of a demand curve because the AD curve was obtained from the AE/AP model and thus included the income effect or the dynamic multiplier effect. He further argued that the multiplier effect was caused by the interaction between the supply and demand sides, so the AD curve also embodied the supply-side information.

Fig. 4.37

Deriving the AD curve from the multiplier model

The criticism of inconsistency in the definition of the demand curve largely hinged on the assumption of ‘other things being equal’. When we use this phrase, we mean in practice that ‘other relevant things are equal’. For example, what happened to firms’ management structure is irrelevant to the tastes of consumers and thus irrelevant to the demand curve (one may disagree. Later this is to be discussed further), so a change in firms’ production behaviours will not affect the demand curve. Because income affects demand, it is indeed problematic to allow income change when deriving a demand curve from an AE/AP model. However, in deriving the AD curve from the AE/AP model, income is viewed as a function of price. In other words, we did not allow income to change exogenously. As such, the income change in the model is a partial effect of price change, so it must be included as part of the price-induced demand. If one excluded this part, like the student did in his first calculation, one has only included the autonomous change in consumption/investment, so the calculation is incorrect because it did not include the full impact of the price change on the demand. Needless to say that, due to the nature of the Keynesian consumption function, the AD curve derived from the AE/AP model is not exactly the same as the way we derived Marshallian (with fixed income) or compensated (keeping consumer’s utility constant) market demands, but the reasoning is similar and the derivation is rigorous. Since we can allow different types of demand curves such as the Marshallian and the compensated demand curve, we should also allow a Keynesian demand curve.

The criticism that the AD curve derived from the AE/AP curve contained supply-side information (some people even claim that the derived curve is also an aggregated supply curve) is an unfortunate misunderstanding due to the use of words ‘aggregate output’ or ‘aggregate income’. These two words indeed give the reader the impression that we are dealing with the supply side. However, there is no active supply side in the AE/AP model. Because Keynes assumed excess supply capacity in his theory, the supply side just passively satisfied any requirements of the demand side. In the AE/AP model, the 45^o line makes aggregate income/output equal to planned aggregate expenditure. This equality means that, even if we are talking about aggregate output or income, effectively we mean planned aggregate expenditure. Although we can use the words ‘aggregate income/output’ to connect the model to the passive or latent supply side, we must not be fooled by these words: all information in the AE/AP model comes from the demand side because the supply is purely passive or accommodating up to full-employment output.

• 2. Obtaining the AD curve from the quantity theory of money

The quantity theory of money states that M = kPY or MV = PY. If we fix the money supply M, we have naturally obtained an inverse relationship between price P and output Y and thus produced a downward-sloping curve. The question is: Does this inverse relationship represent a demand curve? Apparently, not all negative relationships between Y and P are a demand function.

The quantity theory of money is essentially an equation indicating that money supply equals money demand. When we fix the money supply M, it becomes exogenous, so the theory is about money demand. Except that a small part of money demand is for speculative purposes as well as for money hoarding due to uncertainty, the majority part of the money demand is for transaction. In other words, demand for money is essentially for the purpose of purchasing goods, so the demand for money by and large reflects goods demand. As a result, although the quantity theory of money is about money demand and money supply, the equation indirectly reflects people’s demand for goods, so the negative relationship derived from the theory can indeed be viewed as an AD curve.

The other criticism about this approach is that the AD curve derived from the quantity theory of money is not related to Keynesian economics and thus it contributes little to neoclassical synthesis. The response to this criticism is that any new theory only has a few unique features so it must use a large body of existing knowledge. In order to reflect Keynesian theory, we do not need to derive everything from its key features. Although Keynes did not agree that the velocity V in the quantity theory of money is constant and reject the claim that increase in money supply would cause price hike in the short run, he did accept this theory in general by saying that ‘This Theory is fundamental. Its correspondence with fact is not open to question’ (Keynes 1923, p. 81). Hence, Keynes would not object to deriving the AD curve from the quantity theory of money. Moreover, the vast majority of economists, including Keynes, accept a downward-sloping demand curve, so the derived AD curve would be consistent with most economic theories, including Keynesian theories.

• 3. Obtaining the AD curve from the IS/LM model

There is no price variable in the IS/LM model, but the price variable is implicitly included in the real money supply M/P embodied in the LM curve. This embedded price provides a way to derive the AD curve.

The IS curve can be expressed as

$$ Y = c(Y - T) + G_{0} + C_{0} + I_{0} - b\,*\,r $$

(4.15)

Where Y stands for income, T for taxes, r for interest rate, c for propensity to consume, b is a parameter, while G₀, C₀ and I₀ are autonomous level (not affected by income level) of government spending, consumption and investment, respectively.

The LM curve indicates that the real money demand must be met by the real money supply M/P:

$$ M/P = kY - hr + A_{0} , $$

(4.16)

Where M is money supply, P is price level, A₀ is the base level of real money supply. In an IS/LM model, that the price P in Eq. (4.16) is fixed, so one can obtain the income Y and the corresponding interest rate which balance money supply and demand. If we relax the assumption of a fixed price and use Eqs. (4.15) and (4.16) to eliminate the interest rate r, we can obtain an equation about Y and P:

$$ (1 - c - bk/h)Y = G_{0} + C_{0} + I_{0} - cT - Mbh^{ - 1} P^{ - 1} + A_{0} bh^{ - 1} $$

(4.17)

Does this equation represent the demand function? The answer is positive. Actually, the LM curve indicated by Eq. (4.16) is an improved version of the quantity theory of money: Eq. (4.16) can be obtained by adding an extra term ‘−hr + A₀’ to the equation for quantity theory of money. Applying the same reasoning used in deriving the AD curve from the quantity theory of money, we can be confident that Eq. (4.17) is an AD curve.

Graphically, the AD curve can be obtained by shifting the LM curve along the IS curve (shown in Fig. 4.38). Suppose that the initial equilibrium is at A with price P₀, interest rate r₀ and income level of Y₀. If the price decreases to P₁ while the total nominal money supply is fixed, the real money supply increases so the LM curve shifts to the right and produces a new equilibrium at B with a lower price P₁, lower interest rate r₁, and higher income level Y₁. Continuing to shift the LM curve to the right in response to a price drop, we can obtain a new equilibrium point C, and so on. Collecting all data about P and Y, we can construct an AD curve that is shown in the right panel.

Fig. 4.38

Deriving the AD curve from the IS/LM model

So far, the aggregate demand curve—Eq. (4.17)—is derived under the condition that the nominal money supply M is unchanged, so the price level for the AD curve is a fixed-money price and thus a real term. If money supply M changes, the AD curve will shift up or down and the associated price change is a pure nominal phenomenon.

One criticism regarding deriving an AD curve from the IS/LM model is that the derived output for the AD curve is not an output demanded but an equilibrium output at the intersection of IS and LM. Thus, the derived AD curve is both an AD and AS curve because it includes not only the demand-side information but also the supply-side information (Moseley 2010). This argument looks reasonable because the points on AD are obtained through the equilibria in the IS/LM model, but the argument results from two misunderstandings which need rectifying. First, any point on any demand curve is an equilibrium point at a given supply. The reason is that we must use different levels of supply to reveal the consumer’s willingness to pay in order to construct a demand curve. This necessitates equality of supply and demand at each price the consumer is willing to pay. The way to construct an AD curve by shifting the LM curve is similar to the way of identifying a market demand curve by changing the level of supply. As a result, the obtained AD curve is comparable with the ordinary demand curve. No demand curve contains supply-side information, because a change of supply level or shifting of the LM curve is used only as a tool to reveal demand. Second, there is no active supply side in the IS/LM model. Because of the assumption of excess supply capacity in Keynesian models, the supply side is passively accommodating the change on the demand side. As such, there is no supply-side information in the IS/LM model nor in the derived AD curve.

The other criticism is that the AD curve derived from the IS/LM model is valid only when goods are in excess supply (Barro 1994). The reasoning is that, if the price falls below the market-clearing price, there will be a shortage of goods in the market and thus there will be no equilibrium point in the IS/LM model and thus no data to construct an AD curve. This reasoning is correct but it is an unnecessary concern. One reason is that there is always an excess supply capacity in the IS/LM model due to the Keynesian assumption of excess supply capacity. Thus, if one derives an AD curve from an IS/LM model, the supply capacity is not a concern. The other reason is that while supply capacity is an important tool in deriving a demand curve, a demand curve is not constrained by the capacity of supply. The current supply capacity may be limited but it may increase tomorrow or in the near future, and then we can add the new data to the demand curve. In the end, a demand curve indicates a consumer’s tastes and responses to price or quantity change, so it will not be affected by supply-side activities.

The arguments of Barro (1994) and Moseley (2010) have an ideological source from Patinkin (1965), who regarded the AD curve as an equilibrium curve because it is derived from the equilibrium points in the IS/LM model. This idea is actually a misunderstanding of the demand curve. This misunderstanding led to the wrongful claim of interdependency between the demand curve and the supply curve put forward by Rowan (1975), Corden (1978) and Field and Hart (1990). Following Patinkin (1965), Field and Hart (1990) argued that a point on a demand curve indicates an equilibrium between supply and demand so the supply-side is embedded in the demand curve. They further argued that the aggregate demand curve derived under Keynesian assumption implied a supply response of firms that is inconsistent with modern supply theory. Here, we expose the flaws in their arguments.

It is true that every point on a demand curve is an equilibrium between supply and demand, but the points on the demand curve indicate the responses of demand to different supply conditions, rather than the responses of supply (or firms) claimed by Field and Hart (1990). Here, they made a logical mistake. The shift of a supply curve is an instrument to test the response of demand, but Field and Hart (1990) mistook it for a response of the firm’s supply to demand. All the supply-side information needed for deriving a demand curve is the availability of supply quantity or price, so the response of supply or the behaviour of suppliers is totally irrelevant to a demand curve. If one considers the response of supply (or firms), the centre of concern is the supply curve (e.g. how to construct a supply curve), not the demand curve, so the charge Field and Hart (1990) laid on the AD curve derived from the IS/LM model is logically flawed. The irrelevance of the supply response in revealing a demand curve can be shown in Fig. 4.39.

Fig. 4.39

Using supply curves to reveal demand

Three types of supply curves are shown in Fig. 4.39 to reveal a demand curve. The left panel is a typical Keynesian supply curve—the horizontal supply curve shows that the price is fixed or that there is an excessive or unlimited supply at a given price. With this supply curve, the consumer faces a given price and decides how much to purchase, other things being equal. This is exactly what demand means. The middle panel shows the case of a fixed supply. With this supply curve, the consumer faces a given quantity and decides how much they should pay. This reveals the consumer’s willingness to pay, which is an alternative way to define demand. The right panel shows a normal upward-sloping supply curve. With this supply, the consumer faces neither fixed price nor fixed quantity, but the intersection between supply and demand reveals how much the consumer wants to buy and at what price. This can fit in either definition of demand. In all three cases, the shift of a supply curve indicates the response of the consumer: buying less when the price is higher (or paying more when the quantity is less). If the interpretation of Field and Hart (1990) is valid that a shift of supply curve indicates a response of firms, not only the Keynesian supply curve but also all other supply curves are at odds with modern supply theory. In the case of the Keynesian supply curve, the firm produces less when the price is higher. In the case of a fixed-supply curve, the firm requires a higher price when the consumer demand is weaker. In the case of an upward-sloping supply curve, the firm either supplies less when the price is higher or asks for higher price when the demand is weaker. Apparently, it is not the slope of the supply curve but the logic of Field and Hart (1990) that is problematic.

Others argued the interdependency of aggregate demand and aggregate supply based on the complexity of an aggregate model. Rowan (1975) argued that the price-induced change in aggregate supply will affect the aggregate demand through income, i.e. price affects aggregate supply, which in turn affects total output or income. A change in income will then affect aggregate demand. Corden (1978) argued that aggregate demand may be affected by autonomous or parametric shifts in the aggregate supply function. For example, a change in income distribution may affect the aggregate propensity to consume and may also affect investment through changed profit expectations.

It is obvious that an aggregate model for an economy involves many variables and feedback effects. However, in deriving any demand/supply curves, we must hold the condition of other things being equal. With this condition, we can exclude the impact on deriving the aggregate demand curve of the variables exampled by Corden (1978), and in the meantime, we can express the impact of any changes in these variables as a shift of the AD curve.

Although the reasoning of Rowan (1975) is irrefutable, namely a change in aggregate supply will cause a change in income and thus a change in aggregate demand, this statement does not lead us to the conclusion that a change in aggregate supply will affect the aggregate demand curve. Because of the condition of other things being equal when deriving a demand curve, the income level must be kept constant, so the income effect of a change in aggregate supply is excluded when one derives the AD curve. Here, the use of word ‘income’ may cause some confusion. When we say ‘keep income level constant in order to derive an aggregate demand curve’, we mean fixed-money income (just like the condition for deriving a Marshallian market demand curve). If one regards it as physical income or real output, one changes the value judgement standard from money value to output itself. This will contradict the meaning of price ($ per output) in the AD/AS model. As we will explain later (when we derive the AD curve from aggregating market demand curves) that, if the money supply and circulation velocity are unchanged, the money income of an economy is unchanged (a change in physical income affects the price level and thus leaves money income unchanged). As such, the aggregate demand curve will not be affected even if a change in aggregate supply affects the physical income and thus affects aggregate demand, because the changed aggregate demand provides another point on the AD curve. For example, an increase in physical income means a higher output level and a lower price level if the money supply and circulation velocity is fixed. This gives another point on the aggregate demand curve.

In short, it is the supply quantity or supply price that is important in revealing the demand of individuals, markets or the economy. The features of a supply response (e.g. the shape or slope of supply curves) are irrelevant, so a demand curve contains no information about a supply response. Although supply capacity is necessary for deriving a demand curve, it is not a component of a demand curve. In other words, sufficient supply capacity is implicitly assumed for a demand curve, so there is no need to restate it explicitly as a condition of a demand curve.

• 4. Examining alternatives to the AD curve—The DD curve and the HW curve

Based on the Keynes’ concept of deficiency of effective demand, the effective demand or aggregate demand may fall short of or exceed total income, so the Eq. (4.15) for the IS curve should be written as

$$ Y_{\text{D}} = c(Y - T) + G_{0} + C_{0} + I_{0} - b*r $$

(4.18)

where Y_D indicates effective demand which may differ from real output or physical income Y.

From the money demand function—the Eq. (4.16)—we can solve for the interest rate r:

$$ r = (kY + A_{0} - M/P)/h $$

(4.19)

Plugging Eq. (4.19) into Eq. (4.18), we have:

$$ Y_{\text{D}} = (c - bk/h)Y - cT + G_{0} + C_{0} + I_{0} - bA_{0} /h + bM/(Ph) $$

(4.20)

Equation (4.20) is called the HW curve, which is derived by Henry and Woodfield (1985). The HW curve is clearly downward sloping because Y_D and P are in an inverse relationship. Henry and Woodfield have also explained that the HW curve also shows that the income level Y positively affects demand Y_D. Everything looks plausible except that the HW curve may not fit into the definition of a demand curve.

To derive a demand curve, one must set the demand equal to the different levels of given supply, i.e. we must find the equilibrium points. As such, to qualify Eq. (4.20) as a demand curve, one must set Y_D = Y. This will give an AD curve shown in Eq. (4.17). Henry and Woodfield were also confused about income Y. They regarded it as the real income which affects aggregate demand. Based on our definition of different types of income, we know this income Y is actually the physical income or the amount of output which is not a determinant of demand.

Rowan (1975) went further than Henry and Woodfield (1985) to derive the DD curve. With an assumption that the planned supply was fully realized, Rowan derived Y = Y_S(P) based on a production function. This is essentially a positively sloping supply curve, so Y is positively related to price level P. Plugging this into Eq. (4.20), he had:

$$ Y_{\text{D}} = (c - bk/h)Y_{\text{S}} (P) - cT\, + \,G_{0} + C_{0} + I_{0} - bA_{0} /h + bM/(Ph) $$

(4.21)

Equation (4.21) is Rowan’s alternative aggregate demand curve, called DD curve (Rowan’s derivation used a nominal interest rate equation in more general form, so his equation is not exactly the same as Eq. 4.21, but the relationship between Y_D and P is the same). Rowan further argued that the impact of (c − bk/h)Y_S(P) might outweigh the impact of the bM/(hP), so his aggregate demand curve can be positively sloping. However, Rowan’s interpretation was a mistake because he changed the concept of real income when he plugged Y = Y_S(P) into the Eq. (4.20).

In Eq. 4.20, Y stands for physical income or real output of an economy, which is equilibrium outcome determined by both aggregate supply and aggregate demand. When he derived the equation Y = Y_S(P) from a production function, he implicitly assumed that Y is determined by supply side only, so what he derived is not an equilibrium outcome and thus is not real output of the economy. Rather, what he derived is simply an aggregate supply function Y_S = Y_S(P). This can easily be seen from their relationship with P. Aggregate supply Y_S is positively related to P, but the real output of an economy does not have this positive relationship, so it is invalid to plug Y_S = Y_S(P) into Eq. 4.20. By viewing Y as Y_S(P) through equation Y = Y_S(P), Rowan excluded any role of aggregate demand Y_D. This is similar to the situation that there is no role of aggregate supply in Keynes’s multiplier model. In other words, Rowan made Y_D an accommodating term to take any value required by Y_S, and thus, Y_D in Eq. (4.21) actually means Y_S. As a result, the DD curve was essentially an aggregate supply function, and the positive slope of the DD curve was of no surprise.

• 5. Obtaining an AD curve from the direct aggregation of the market demand

To be free from various scepticism and criticism about the AD curve, the best method is to obtain an AD curve by direct aggregation of market demands. Despite the difficulties shown previously, this section will demonstrate that, with some necessary restrictions, aggregation of market demands is not impossible.

We start with the practice of obtaining aggregate price indexes (e.g. CPI). These indexes are generally obtained through weighted average price. For example, the equilibrium outputs or demands for commodity 1 and 2 in a certain period is Q₁ and Q₂ (for simplicity, we temporarily assume only two commodities in the economy), the prices are P₁ and P₂, respectively. The value of commodities 1 and 2 and total outputs are V₁, V₂ and V, respectively. We have:

$$ V_{1} = P_{1} *Q_{1} ,\;V_{2} = P_{2} *Q_{2} ,\;V = V_{1} + V_{2} , $$

Using the value share of each commodity as the weighting, we can calculate the price and quantity of the aggregate output for the economy:

$$ P = P_{1} *V_{1} /V + P_{2} *V_{2} /V,\;Q = V/P = V^{2} /(P_{1} V_{1} + P_{2} V_{2} ) $$

Through this type of aggregation, we can obtain a point on the AD curve. If we can measure the demands for goods 1 and 2 at different sets prices, e.g. at price set (a) we have price P_1a and quantity demanded Q_1a for good 1, and price P_2a and quantity demanded Q_2a for good 2; at price set (b) we have P_1b, Q_1b, P_2b and Q_2b, using the same method as computing aggregate price indexes like CPI, one can calculate the aggregate price and demand at each set of prices and thus obtain an AD curve.

The key for this aggregation is to find the way to associate a point (e.g. a point determined by P_1b and Q_1b) on the demand curve for good 1 with a point (e.g. a point determined by P_2b and Q_2b) on the demand curve for good 2. In reality, this association is realized through the choices by households and firms in the economy. To aggregate market demands to an AD curve, we need to mimic the reality by applying restrictions on household demand decisions. This can be achieved in a number of ways.

• a. Strictly following the conditions for deriving market demand

Since a market demand curve is derived by keeping constant the income level and the prices of other goods, all points on the demand curve for good 1 are associated with the same points on demand curves for other goods, e.g. both the points (P_1a, Q_1a) and (P_1b, Q_1b) on demand curve for good 1 are associated with a point (P_2a, Q_2a) on the demand curve for good 2 and with a point (P_3a, Q_3a) on the demand curve for good 3. To obtain an AD curve, we can choose a market demand curve, associate each point on this curve to the same set of points on other market demand curves, and aggregate prices and quantities as being shown in CPI aggregation. The obtained AD curve is similar to a right shift of the chosen market demand curve and thus will be downward sloping. However, the position of the AD curve can vary, depending on which market demand curve is chosen as the base for aggregation.

• b. Applying multiple-stage budgeting

The two-stage budgeting method is popularly used in empirical studies on aggregate demand. In this approach, the researcher allocates some portion of the household budget to two different groups of commodities and thus obtains the quantity and prices for commodities in each group. For the details about composite goods and two-stage budgeting, readers are referred to Nicholson and Snyder (2017) and Blackorby et al. (1978). The method can be shown in a simple two-good case. If we know the market demand curves for two goods and the income spent on them, we have the following three equations:

$$ Q_{1} = f_{1} (P_{1} ), $$

$$ Q_{2} = f_{2} (P_{2} ). $$

$$ P_{1} *Q_{1} + P_{2} *Q_{2} = I $$

These three equations include four variables, P₁, Q₁, P₂ and Q₂. If the price of one good (e.g. P₁) is given, the values for other variables and thus the aggregate price and quantity can be obtained. By assigning different values to the price of the selected good, one can obtain different aggregation points and thus an AD curve.

Figure 4.40 shows the procedure to aggregate market demands to obtain the aggregate demand for an economy. Panel (a): For a given price of good 1, e.g. P_1a, we can calculate Q_1a, P_1b and Q_1b. Then, we can find the corresponding points on the market demand curve, e.g. A_1a and A_2a. Panel (b): Although we cannot add Q_1a and Q_2a directly, we can calculate the value of each good and construct the graphs of the demand value at each price, so we can obtain the new points A_1a′ and A_2a′ (A_1a′ is transformed from point A_1a, and A_2a′ is transformed from point A_2a).

Fig. 4.40

Deriving the AD curve using two-stage budgeting

Panel (c): Adding the value of goods 1 and 2, we have the value of aggregate good V_a, which is equal to total expenditure or total income (I). Moreover, using the calculated values as weighting, we can calculate the aggregate price P_a from P_1a and P_2a. Using V_a and P_a, we can produce a point B_a. Panel (d): Using Q_a = V_a/P_a to obtain the aggregate quantity Q_a, we can have a point C_a on the aggregate demand curve. Choosing another price P_1b, we can repeat the calculation and obtain another point B_b (with the same value V_a because of the fixed budget allocated) in panel (c) and another point C_b on the aggregate demand curve in panel (d). Choosing a third price P_1c and repeating this procedure, we can obtain more points and construct the aggregate demand curve.

At any sets of prices, the total value of spending on goods 1 and 2 is the same (equal to the income allocated). Since total spending is the same in each procedure, for a higher aggregate price P_a (compared with P_b), the corresponding aggregate quantity Q_a will be smaller (than Q_b). As such, the resulting AD curve will be downward sloping.

For the case of more than two goods, the aggregation can be achieved by repeated two-stage budgeting, i.e. multiple-stage budgeting. First, we aggregate in the way as shown in Fig. 4.40 the demand of any two markets to obtain the market demand for the first-stage aggregate good. Second, we aggregate the first-stage aggregate market demand and a third market demand to obtain the market demand for the second-stage aggregate good. Third, we aggregate the second-stage aggregate market demand and a fourth market demand to obtain the market demand for the third-stage aggregate good. Repeating this procedure, we can finally obtain the aggregate demand for the economy. The result of this multiple-stage budgeting approach is similar to that of using a Cobb–Douglas or LES utility function, which allocates spending to each commodity in a proportional or linear fashion.

• c. Using the concept of composite goods

The concept of composite goods is also popularly used in commodity consumption aggregation. This concept is based on the substitution effect between commodities. For close substitutes (e.g. different types of food), their prices tend to move in a similar fashion so one can consider that there is a fixed price ratio among these commodities. In considering commodity prices purely from the demand perspective (e.g. ignore supply condition or assume similar supply functions), if consumer preference does not change but the overall demand increases, the price of all commodities will increase in a similar fashion. As such, all commodities can be viewed as a composite good and a fixed price ratio between different types of commodities can be assumed. Using this assumption, we can demonstrate in Fig. 4.41 the procedure to aggregate market demands to obtain the aggregate demand for an economy.

Fig. 4.41

Deriving the AD curve using composite goods

Panel (a) of Fig. 4.41: for historically determined prices of goods 1 and 2, e.g. P_1a and P_2a, we can obtain from the market demand curves the quantity demanded Q_1a and Q_2a, so we have two points A_1a and A_2a. Panel (b): although we cannot add Q_1a and Q_2a directly, we can calculate the spending on each good V_1a = P_1aQ_1a and V_2a = P_2aQ_2a, so we can obtain the new points A_1a′ and A_2a′ (A_1a′ is transformed from point A_1a and A_2a′ is transformed from point A_2a). Panel (c): adding the value of goods 1 and 2, we have the value of aggregate good V_a = V_1a + V_2a. Using the calculated values V_1a and V_2a as weighting, we can calculate the aggregate price P_a = (P_1aV_1a + P_2aV_2a)/V_a. Using V_a and P_a, we can produce a point B_a in panel (c). Panel (d): the calculated V_a and P_a gives the aggregate quantity demanded Q_a = V_a/P_a, and thus the point C_a on the AS/AD curve.

The above procedure is the same as conducting a GDP or CPI aggregation, where the prices and quantities P_1a, P_2a, Q_1a, Q_2a are the realized results in history. For aggregation of market demand curves, we need more than one point.

Choosing a price P_1b which is less than P_1a, the constant price ratio for composite goods necessitates that P_2b is less than P_2a, so we have point A_1b and A_2b in panel (a). Assuming the demand for both goods 1 and 2 is elastic (we will discuss the other cases later), the spending on both goods will be greater than that when prices are P_1a and P_2a, so we have points A_1b′ and A_2b′ in panel (b), which lead to the downward-sloping curves in panel (b). Since P_1b and P_2b are less than P_1a and P_2a, respectively, the aggregated price P_b is less than P_a. Meanwhile, the aggregated spending V_b is greater than V_a thanks to the assumption of elastic demand, so we have in panel (c) a point B_b which is below and at the right of A_a. This produces a downward-sloping curve in panel (c). Consequently, we have point C_b and a downward-sloping aggregate demand curve in panel (d). If we change the preset value of the price ratio for goods 1 and 2 and repeat the above procedure, we can obtain another AD curve, which can be viewed as a shift of the previous AD curve.

However, at the top part of the linear demand curves for goods 1 and 2, the price elasticity of demand tends to become very small. If the demand for good 1 and/or for good 2 becomes inelastic, the spending may decrease when the prices decrease. This would lead to positively sloping curves in panels (b) and (c), and thus an upward-sloping aggregate demand curve when the aggregate price is small.

Next, we show that, with demand curves of constant price elasticity for goods 1 and 2, the aggregate demand curve is a curve similar to market demand curves, so the aggregate demand curve is downward sloping.

Assume a constant-elasticity demand curve for goods 1 and 2: Q₁ = a₁ * P ^−e1₁ , Q₂ = a₂ * P ^−e2₂ , (elasticity e₁ > 0 and e₂ > 0) and a constant price ratio: P₂ = λP₁, which λ > 0. We can calculate $V_{1} = P_{1} Q_{1} ,\;V_{2} = P_{2} Q_{2} ,\;V = V_{1} + V_{2} \;{\text{and}}\; P = (P_{1} V_{1} + P_{2} V_{2} )/V.$

$$ \begin{aligned} V & = P_{1} Q_{1} + P_{2} Q_{2} = P_{1} \,*\,a_{1} \,*\,P_{1}^{ - e1} + P_{2} \,*\,a_{2} \,*\,P_{2}^{ - e2} \\ & = P_{1} \,*\,a_{1} \,*\,P_{1}^{ - e1} + \lambda P_{1} \,*\,a_{2} \,*\,(\lambda P_{1} )^{ - e2} \\ & = a_{1} \,*\,P_{1}^{1 - e1} + a_{2} \,*\,\lambda^{1 - e2} P_{1}^{1 - e2} \\ & = P_{1}(a_{1} \,*\,P_{1}^{ - e1} + a_{2} \,*\,\lambda^{1 - e2} P_{1}^{ - e2})\\ \end{aligned} $$

$$ \begin{aligned} P & = (P_{1} V_{1} + P_{2} V_{2})/V = (P_{1}^{2} Q_{1} + P_{2}^{2} Q_{2})/V \\ & = P_{1}^{2} (a_{1} \,*\,P_{1}^{ - e1} + a_{2} \,*\,\lambda^{2 - e2} P_{1}^{ - e2})/V \\ & = P_{1}(a_{1} \,*\,P_{1}^{ - e1} +a_{2} \,*\,\lambda^{2 - e2} P_{1}^{ - e2})/(a_{1} \,*\,P_{1}^{ - e1} +a_{2} \,*\,\lambda^{1 - e2} P_{1}^{ - e2})\\ \end{aligned} $$

$$ Q = V/P = (a_{1} \,*\,P_{1}^{- e1} + a_{2} \,*\,\lambda^{1 - e2} P_{1}^{- e2} )^{2}/(a_{1} \,*\,P_{1}^{- e1} + a_{2} \,*\,\lambda^{2 - e2} P_{1}^{- e2} ) $$

This expression is hard to interpret generally, but we can examine two special cases: e₁ = e₂ or λ = 1. If λ = 1, P₁ = P₂ = P, Q = a₁ * P^-e1 +  a₂ * P^-e2, so the aggregate demand curve is the horizontal aggregation of market demand curve, so it must be downward sloping. If e₁ = e₂, P = P₁(a₁ + a₂ * λ^2-e2)/(a₁ + a₂ * λ^1-e2), Q = P^1-e1(a₁ + a₂ * λ^1-e2)²/(a₁ + a₂ * λ^2-e2). The aggregate price level is proportional to market prices and the aggregate demand curve is of the same elasticity as the market demand curves, so it must also be downward sloping. In a general case, it can be shown that the derivatives dQ/dP₁ < 0 and dP₁/dP > 0, so Q and P will be negatively related, i.e. the demand curve is downward sloping.

• d. Utilizing a general equilibrium model

The general solution to obtain an AD curve is to use a general equilibrium model. The model can mimic the behaviour of consumers and firms through mathematic functions and thus obtain the market prices and equilibrium quantities under different supply conditions. The results of a general equilibrium model can be illustrated by the PPF curves on the supply side and indifference curves (IC), as shown in Fig. 4.42.

Fig. 4.42

Obtaining the AD curve from a general equilibrium model

For a closed economy, the equilibrium outcome will be the point when PPF is tangent to an IC, e.g. point A. The tangent point A gives the quantity of goods demanded Q_x1 and Q_y1 and the price ratio P_x1/P_y1. Given that the exogenous money income is fixed (e.g. I = P_x1Q_x1 + P_y1Q_y1), we can obtain the monetary price P_x1 and P_y1. With given value for P_x1, Q_x1, P_y1 and Q_y1, we can obtain a point on an AD curve. If the supply-side situation changes to a situation represented by PPF₂, we can obtain values for P_x2, Q_x2, P_y2 and Q_y2, so we can have another point on the AD curve. With changing conditions on the supply side, we can obtain more points and draw the AD curve.

The flexibility of a general equilibrium model means that the resulting AD curve can mimic any situation in reality. However, since many variables in the model can affect the equilibrium outcome, the position and slope of the AD curve depend on these variables and thus can vary considerably.

Different ways of direct aggregation here have their limit. Although the theory of two-stage budgeting and the concept of composite goods are popularly used in empirical research, they are criticized for using unrealistic assumptions. Fixing the demand for other good so as to satisfy the condition of deriving a demand curve is also not realistic. Using a general equilibrium model to derive aggregate demand curve can produce the most realistic result, but this approach is both time-consuming and highly data demanding. Nevertheless, by applying these methods we can demonstrate theoretically that it is possible to derive an AD curve from aggregating different types of market demands. This refutes, once and for all, the claim that the AD curve is totally different from the market demand curve (we refuted this earlier) and the claim that it is invalid to aggregate the market demand to obtain the aggregate demand.

Appendix 2 (for Section 4.8.3.3): Aggregating Industrial Supply to Obtain a Short-run AS Curve

Based on historical observation on the prices of industrial outputs, we can set up a price ratio for industry 1 and industry 2 and demonstrate in Fig. 4.43 the way to obtain an AS curve by aggregating market supplies. Given the supply curves for these two and the price ratio, once we know one point on market supply of industry 1, e.g., quantity A_1a and price P_1a, we can calculate for industry 2 the corresponding price and quantity, P_2a and Q_2a. Using the prices and quantities P_1a, P_2a, Q_1a and Q_2a in panel (a), we can calculate the revenues V_1a = P_1a * Q_1a and V_2a = P_2a * Q_2a, thus we can obtain points A_1a′ and A_2a′ in panel (b). Aggregating the total revenue V_a = V_1a + V_2a and aggregate price P_a = (P_1aV_1a + P_2aV_2a)/V_a, we can obtain the point B_a in panel (c). Calculating aggregate quantities according to Q_a = V_a/P_a, we can obtain the point C_a in panel (d).

Fig. 4.43

Deriving the AS curve from short-run industrial supplies

Selecting another price P_1b for industry 1 and using the price ratio set up according to historical observations, we can obtain P_1b, P_2b, Q_1b, Q_2b and thus points A_1b and A_2b in panel (a). Since the industrial supply curve is normally upward sloping, if we set P_1b > P_1a, we have P_2b > P_2a, Q_2b > Q_2a and Q_1b > Q_1a. Calculating revenues for industries 1 and 2 according to V_1b = P_1b * Q_1b and V_2b = P_2b * Q_2b, we will have V_1b > V_1a and V_2b > V_2a. This produces two points A_1b′ and A_2b′ and thus two upward-sloping curves in panel (b). Aggregating point A_1b′ and A_2b′ according to V_b = V_1b + V_2b and P_b = (P_1bV_1b + P_2bV_2b)/V_b, we can obtain the point B_b and an upward-sloping curve in panel (c). Calculating aggregate quantity according to Q_b = V_b/P_b, we can obtain the point C_b and an upward-sloping AS curve in panel (d).

It can be proved that, given upward-sloping industrial supply curves, the resulting AS curve is upward sloping. For simplicity, we suppress the intercepts and thus write the industrial supply curves as Q₁ = a₁P₁ and Q₂ = a₂P₂, which a₁ > 0 and a₂ > 0. The price ratio for industries 1 and 2 can be written as P₂ = kP₁, with k > 0. Revenue for industries 1 and 2 can be calculated as V₁ = P₁Q₁ and V₂ = P₁Q₂, respectively. Total revenue of two industries (V) and the aggregate price (P) can be written as:

$$ V = P_{1} Q_{1} + P_{2} Q_{2} = P_{1} a_{1} P_{1} + P_{1} a_{2} P_{2} = a_{1} *P_{1}^{2} + a_{2} *\left( {kP_{1} } \right)^{2} = \left( {a_{1} + a_{2} k^{2} } \right)P_{1}^{2} . $$

$$ \begin{aligned} P = & {{\left( {P_{1} V_{1} + P_{2} V_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{1} V_{1} + P_{2} V_{2} } \right)} V}} \right. \kern-0pt} V} = {{\left( {P_{1}^{2} Q_{1} + P_{2}^{2} Q_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{1}^{2} Q_{1} + P_{2}^{2} Q_{2} } \right)} V}} \right. \kern-0pt} V} = {{\left( {P_{1}^{2} a_{1} P_{1} + P_{2}^{2} a_{2} P_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{1}^{2} a_{1} P_{1} + P_{2}^{2} a_{2} P_{2} } \right)} V}} \right. \kern-0pt} V} = {{\left( {P_{1}^{3} a_{1} + k^{3} P_{1}^{3} a_{2} } \right)} \mathord{\left/ {\vphantom {{\left( {P_{1}^{3} a_{1} + k^{3} P_{1}^{3} a_{2} } \right)} V}} \right. \kern-0pt} V} \\ = & {{P_{1} \left( {a_{1} + \, a_{2} k^{3} } \right)} \mathord{\left/ {\vphantom {{P_{1} \left( {a_{1} + \, a_{2} k^{3} } \right)} {\left( {a_{1} + a_{2} k^{2} } \right),}}} \right. \kern-0pt} {\left( {a_{1} + a_{2} k^{2} } \right),}} \\ \end{aligned} $$

Or,

$$ \begin{aligned} P_{1} = {{P\left( {a_{1} + a_{2} k^{2} } \right)} \mathord{\left/ {\vphantom {{P\left( {a_{1} + a_{2} k^{2} } \right)} {\left( {a_{1} + a_{2} k^{3} } \right)}}} \right. \kern-0pt} {\left( {a_{1} + a_{2} k^{3} } \right)}},\;V = {{P^{2} \left( {a_{1} + a_{2} k^{2} } \right)^{3} } \mathord{\left/ {\vphantom {{P^{2} \left( {a_{1} + a_{2} k^{2} } \right)^{3} } {\left( {a_{1} + a_{2} k^{3} } \right)^{2} }}} \right. \kern-0pt} {\left( {a_{1} + a_{2} k^{3} } \right)^{2} }}. \\ \end{aligned} $$

Consequently, So we have

$$ \begin{aligned} Q = V/P = \, {{P\left( {a_{1} + a_{2} k^{2} } \right)^{3} } \mathord{\left/ {\vphantom {{P\left( {a_{1} + a_{2} k^{2} } \right)^{3} } {\left( {a_{1} + \, a_{2} k^{3} } \right)^{2} }}} \right. \kern-0pt} {\left( {a_{1} + \, a_{2} k^{3} } \right)^{2} }}. \\ \end{aligned} $$

Since the parameters a₁, a₂ and k are all positive, aggregate quantity Q and aggregate price P are positively related, so the resulting AS curve must be upward sloping. If we change the value for parameter k, we can obtain another AS curve of a different but still positive slope.