Deriving the short run aggregate supply curve

Individual income can grow as fast as productivity rises.
Alex Berenson

The law of diminishing returns gives us that for every unit of additional output produced, the addition to total cost increases more rapidly. Therefore, in perfect competition the supply curve is a positively sloped curve that as output increases is steeper. We ask the firms how much they are willing to produce at different prices and we derive the supply curve done in an empirical fashion. Perfect competition, monopoly, monopolistic competition, oligopoly comprise the economy and are all added up to define the short run aggregate supply curve (s.r.a.s.c.). In the long run, buildings and machinery increase but so do the wages therefore the long run aggregate supply curve is vertical at national income of full employment Y. An increase in money supply leads to a shift of both the aggregate demand curve (AD) and s.r.a.s.c resulting at the same level of output Y with higher prices. At liquidity trap the AD is vertical, we prove it as this essay unfolds, resulting in sluggish economic growth or if there is asymmetry in negative economic growth and negative inflation. Finally, we tackle economic problems of countries such as liquidity trap, inflationary pressures and try to solve them. The solutions are suggestions.

Marginal cost(M.C.) is the cost incurred in producing one additional unit of output. It is the addition to Total cost(T.C.) for producing 1 additional unit of output. The law of diminishing returns gives us that the marginal cost increases at a faster pace. Why?
When we add input to produce more goods, for example, when we add workers (labor) there comes a time that the output does not increase by the same proportion of the additional labor employed. Too many workers collide, their relative space decreases they are more disorganized. The output they produce is not proportional to the input of labor. For example, a factory building with the machines has a capacity of 100 workers if the workers are increased to 200 the output will not double. Workers are not the only variable cost. If the firms' demand for raw materials increases the price of raw materials will increase thus marginal cost progresses. Therefore, to produce one additional unit the cost (wages) is increasing at a faster pace. As output increases the cost is rising at a steeper rate: diseconomies of scale. The following diagram illustrates it:
Figure 1 As quantity produced increases the M.C. is steeper.

From the graph below we conclude the M.C. curve is the supply curve in the short run in perfect competition we explain below.
Figure 2 From figure 2 at output A, M.C. is more than the marginal revenue(M.R.) which in perfect competition is the price. M.C. is greater than the price. The firm loses money so it will produce less. At output B, greater than M.C. For example, the price of one unit of output is $5- the M.C.=$3-. The firm gains profit and it will produce more to gain more profit until M.C.=M.R. at C (see figure 2). If it produces more than C M.C.>M.R. the firm loses money. The firm will produce at C where Price=M.C. The firm produces where the price of the product=M.C. it produces that output. Therefore the M.C. curve is the supply curve in perfect competition(pc). At a given price how much it will produce: supply curve.

Technical information 1  
Consider the following example of a pizza firm, for simplicity values are rounded:
Output production
units of pizza
per year
Price per unit of pizza
3,000 3
4,000 4
5,000 6
6,000 9
7,000 13

The accountant of the firm can provide the selling price for different values of output and will be like the table above. Past data with other values than the current value can be used, adjusted for inflation, to give the values in the table. By an interpolation(given the values the equation is derived) calculator these values give the function:
P= (q2)/2.000.000 -5q/2.000 +6
P: price per unit of pizza
q: quantity in units of pizza

Now we take 2018 statistics, for example, the total production of units of pizza in 2018 of the economy is 100,000 and the price is $6- per unit. We take the 2018 prices to be price= 100-. Now if price increases by 50% the new price= 150-. This means that now the price of pizza is $9- (6x1.50= 9). From the table above we find the corresponding output which is 6,000 units of pizza.

Therefore at
priceoutputTotal production
x= Total production, at price $9-=
(6,000✕100,000)/5,000= 120,000- units of pizza

To repeat, what we do is if price increases by 50% we find the new price: $9- and the corresponding output from the equation:
P= (q2)/2.000.000 -5q/2.000 +6 from an inverse function calculator we find: q=500(√ 8P-23 +5). In the year 2018 the total output in the economy of pizza units was 100,000-
At price $6- quantity 5,000 the total output was 100,000
At price $9- quantity 6,000 the total output: (6.000✕100.000)/5.000= 120.000
assuming that the market share of pizzas of the firm in the example remains the same and also the supply curve of the pizza firm is the same shape with the other pizza firms which are both acceptable assumptions to make. We do the same with all other goods and services in GDP(national output or national income(Y)) we choose one representative firm and calculate the total production of the industry as it was with pizzas and add them all up.
Then we find:

150(6*✕120,000(no. of pizzas) +all other...)
*the price is $6 not $9 to be able to compare GDP outputs at the same price: Real GDP at price 150.

We plot a graph by finding the points:
(100, GDP2018)
(150, as calculated)
(200, calculated)

From the above, we use an interpolation calculator and find the equation and graph for perfect competition it will be:
P=aY n+bY n-1+...+g
aY n+bY n-1+...+g>0,   n integer >0,   dP/dY>0,   d2P/dY2>0
a1Ypcn+...+b1Ypc+g1>0,   n integer >0,   dPpc/dY>0,   d2Ppc/dY2>0, a1, b1, g1 are numbers to be found by interpolation
Figure 3 The above graph holds true for perfect competition.


In monopoly, there is no unique relation between market price and quantity supplied. There is no distinct supply curve under monopoly. Yet when demand curve does not pivot, and why should it pivot, there is a relation between increase in prices and increase in quantity supplied as the following graph shows:
Figure 4 The monopolist produces that output where M.C.=M.R. to maximize profit as it was explained. From the graph above you can see the output determined is at the intersection of the M.C. curve (black line-) and the M.R.1, M.R.2, M.R.3 curves. The monopolist sets the price at which consumers are willing to pay at the determined output the demand curves D1, D2, D3. Where the output meets the demand curve we get the price. From the above graph you can see that the change in quantity DQ increase is less than the change in price DP (increase). The price compared to quantity increases at a larger rate.

Please see figure 5, below.
The demand curve is the average revenue(A.R.) curve.
A.R.=Total revenue/quantity = price✕quantity/quantity = Pq/q = P
Let the equation of the demand curve be PD=mq+c where m: slope, c: intersection to the y-axis, Price-axis
Marginal Revenue= d(T.R.)/dq = d(PDq)/dq = d(mq2 + cq)/dq = 2mq+c
The slope of the demand curve is m and is negative. As proved the slope of the M.R. curve is 2m it is twice as steep always.
The A.R. curve meets the q axis:
mq=-c =>q=-c/m
The M.R. curve meets the q-axis:
0=2mq+c =>q=-c/2m
The horizontal distance from(0,0) is half for the marginal curve. When M.R. is zero it intersects the q-axis at q=-c/2m at that q the A.R.= mq+c= m(-c/2m) + c= c/2
When M.R.=0 A.R.= c/2

Now let:
mc: slope of the M.C. curve

x2=2mq2+c2 => Dx= 2mDq+ Dc

At the two points of intersection between M.R.s and M.C. we have: z1=mcq1+cc
z2=mcq2+cc => Dz= mcDq
M.R. curves and M.C. curve intersect at two points. The change in M.R. is the same as the change in M.C.: DM.R.=DM.C. please see figure 5.
Dz= Dx
∴mcDq= 2mDq+ Dc
Dc= mcDq- 2mDq -*

y1= mq1+ c1
y2= mq2+ c2
Dy= mDq+ Dc
From the above and from*: Dy= mDq+ mcDq- 2mDq
Dy=mcDq- mDq
Dy= (mc -m)Dq
DP= (mc -m)Dq
DP/Dq= mc -m
∴P= b2q+ g2
    b2= mc- m, mc>0 the slope of the marginal cost curve,   m<0 the slope of the demand curve which is negative so -m is positive therefore b2>0, g2 is a number
Figure 5 Why the demand of a monopolist is steep, |m| is high?
Because consumers cannot switch to another brand it is monopoly. A change in price brings a small change in quantity sold and thus produced. Therefore we relax on the assumptions that the demand curve shifts and not pivots(elasticity of demand changes dramatically) and also the M.C. is a straight line (please see figure 5).

From the above calculations we have P= b'2q+ g'2
To express q units of output produced into Dollars we multiply q by price2018: output produced in Dollars. We multiply every value of q by price2018 as we did earlier in perfect competition.
We find b'2 and g'2 by interpolation we know one point P=100, p2018q2018 output of the monopoly.
P= b'2pq+ g'2
If the above monopoly(1) is 30% of the output of all monopolies then:
0.3P= 0.3b'2pq+ 0.3g'2
∴0.3P= 0.3b'2✕0.3Ym+ 0.3g'2
We add all other monopolies so Pm= b2Ym+ g2
Therefore the point 2018 will be:
P=100, Ym(p(1)2018q(1)+p(2)2018q2+...)
p(1)2018 is the current price of 2018 of monopoly(1)
q(1) is the 2018 current output at price2018 of the monopoly(1)
Pm= b2Ym+ g2,     b2>0

Technical information 2  
Figure 5.1 As explained previously the perfect competitive firm will produce that output where M.R.=M.C. to maximise profits. The M.R. curve is the same as the A.R. curve which is the demand curve and it is parallel to the x-axis and to other demand curves. The points of intersection between M.R. curves and the M.C. curve describe the supply curve for example at PA it will produce qA but these points as it is illustrated is the M.C. curve: points A, B, C.
Figure 5.2 Please, see also figure 4. In monopoly the monopolistic firm will produce the output where M.R.=M.C. to maximise profits where M.C. curve intersects the M.R. curve. At that output we find the price we move upwards vertically until we meet the demand curve, D, of the monopolistic firm. There, is the price of the product. Therefore, at output qA the price is PA. At output qB the price is PB and at output qC the price is PC. The line which connects these points A, B, C,: (qA, PA) (qB, PB) (qC, PC) is the supply curve of the monopoly the philosophy is the same as in perfect competition. Of course we assume that the demand curve shifts parallel to the previous demand curve. If it pivots we connect the points as done above and we will find the supply curve.

Monopolistic competition

In monopolistic competition there are many firms with freedom of entry and exit (competition). Yet each firm has some power over price because each sells a product that is somehow different from the firms competitors (monopoly). The demand curve is rather elastic because there are substitute products from other firm competitors: As it was mentioned in monopoly the demand curve does not pivot because the elasticity of demand normally does not change it is a monopoly consumers cannot switch to other goods. In monopolistic competition, however, the demand curve of a firm pivots because consumers can consume other similar products as the graph below demonstrates.
Figure 6 When the demand curve pivots from D1 to D2 there is a small change in price DP.

We showed in monopolistic competition(mc) that variation in quantity changes the price.
Now let:
mc: slope of the M.C. curve
c the vertical distance between zero and the point of intersection of the A.R. or y curves to the y-axis
cc the vertical distance between zero and the point of intersection of the M.C. or z curve to the y-axis
y1= m1q1 + c
y2= m2q2 + c
z1=mcq1 + cc
z2=mcq2 + cc
But Dx=Dz
∴2m1q1 - 2m2q2= mcDq -*
But Dy= m1q1 - m2q2
From the above equation and from* Dy= 1/2mcDq
∴y= 1/2mcq + g3

Pmc= b3Ymc+ g3,     b3= 1/2mc, g3 is a number


In oligopoly few firms compete with each other. Each firm has enough power so that it cannot be a price taker(take price as given), but it is subject to enough inter-firm rivalry that it cannot consider the market demand curve as its own. This is probably the dominant market structure outside agriculture and basic industrial materials.

Figure 7

In the figure above the oligopolistic firm is assumed to be selling q1 units at a price of p1. It then considers altering its price and it makes two key assumptions. First, it assumes that if it cuts its price, all of its competitors will match its price cut. Its demand will then expand along the relatively steep curve below a, which indicates the effect of the firm's price cut when its share of the market is unchanged. Second, it assumes that if it raises its price, above p1, none of its competitors will raise theirs. Its demand for prices above p1 thus contracts along the relatively flat curve indicating the effects of raising prices with a declining market share. To summarize, above price p1 the competitors will not increase the price thus the firm loses a lot of demand: a small increase in price leads to a large decrease in demand the demand curve is flatter. At price below p1 the competitors will also reduce the price there will be no change in the firm's demand, competitors follow suit therefore the demand curve is steep a change in price brings a small change in output. Therefore changes in the (M.C.) do not alter the price: see figure 7 M.C.1, M.C.2.
Figure 8 As demand increases the demand curve shifts from D1 to D2 the quantity changes from q1 to q2 but the equilibrium price remains the same p1
In oligopoly price is constant.
Po= 100

To conclude, we add up all markets of the economy as displayed above to derive the short run aggregate supply curve (s.r.a.s.c.) of the economy. We assume that perfect competition= 10% of the economy, oligopoly= 50%, monopoly=10% and monopolistic competition= 30%. Therefore, Ypc= 0.1Y=> Ypcn= 0.1nYn and Ym= 0.1Y. We integrate the analysis done separately before and the just previous assumptions and we have:
Perfect competition
0.1P= 0.1a1✕0.1nY n+...+0.1✕0.1b1Y+0.1g1
0.1P= 0.1✕0.1b2Y+ 0.1g2
Monopolistic competition
0.3P= 0.3✕0.3b3Y+ 0.3g3
0.5P= 50

∴P=0.1a1✕0.1nY n+...+(0.1✕0.1b1+0.1✕0.1b2+0.3✕0.3b3)Y+0.1g1+0.1g2+0.3g3+50
For example in perfect competition we multiply everything by 0.1 because in the prior calculations and the graph the Price= 100 at GDP2018pc. For the whole economy the Price in perfect competition= 10.
P= aY n+...+bY+g
P= aY n+...+bY+g>0,       n integer >0,   dP/dY>0,   d2P/dY2>0,   it was shown that this polynomial is positive its slope is always positive and the rate of change of the slope is also always positive leading to a curve that becomes steeper and steeper.

You are right in thinking that in monopoly the demand curve might also pivot or in monopolistic competition the curve will also shift but overall the s.r.a.s.c is a positively sloped curve which becomes steeper as national output increases because in the short run firms have a specified capacity leading to a s.r.a.s.c. which has a vertical asymptote at a large national output, please see figure 3.

Below there is an example of a s.r.a.s.c.:
The given equations are not the real US GDP they are just an example which resemble reality more realistic equations can be obtained from empirical evidence that is at given prices how much the producers are willing to produce as explained before.
Perfect competition
0.1P= -6.0563403585579✕10-6Y4 + 0.00281504Y3 - 0.0826279Y2 + 0.97583Y + 6
0.1P= 0.03Y + 4
Monopolistic competition
0.3P= 0.045Y + 30
0.5P= 50

P=-6.0563403585579✕10-6Y4 + 0.00281504Y3 - 0.0826279Y2 + 1.05083Y + 90
The graph is as follows:
Figure 8.1

As output increases costs are rising at a faster pace. As output increases labor and raw materials become scarce so the wages, costs rise. Costs are also rising from diseconomies of scale and diminishing returns as previously explained. In all markets perfect competition, monopoly, monopolistic competition, oligopoly, the marginal cost curve leads to finding the output produced and the price. For example, if the M.C. curve moves upwards: the equilibrium output q is restrained while the price increases. Thus, as output increases the price increases at a faster pace giving us a short run aggregate supply curve which is upward sloping. At very high output the economy's potential is reached: full employment, full capacity the output remains constant while price escalates. The assumption in deriving the s.r.a.s.c. is that the factory buildings, heavy equipment remain the same. In the long run factory buildings, machinery increase. Thus the s.r.a.s.c. shifts to the right:
Figure 9 At equilibrium the economy is at its natural level of output (potential output) Y and the s.r.a.s.c. meets aggregate demand curve at point A. When buildings, machinery increase the s.r.a.s.c. shifts to the right s.r.a.s.c.2. At point B the actual output Y2 exceeds the potential output workers are scarce thus wages rise and s.r.a.s.c. shifts to the left to s.r.a.s.c.1. Therefore at point A after the shift we have more factories and machinery and higher wages. In reality this is what happens.

Suppose the economy is in equilibrium (Y, P) operating at full employment or potential output or natural level of output. The government increases the money supply. This leads to a shift in the aggregate demand curve from A.D.1 to A.D.2 see figure below. The new equilibrium national output is Y2 it exceeds potential output. Workers demand wage increases because Y2 exceeds full employment output Y. As wages increase the s.r.a.s.c.1 shifts to the s.r.a.s.c.2 achieving a new equilibrium with national output at its natural rate Y and higher prices P3, please see figure below.
Figure 10 It is important to note as it was illustrated that an increase in money supply leads always to higher prices, higher inflation in the long run (see the Cagan model).
When the equilibrium output Y2 exceeds the natural level of output in the short run the workers donot demand any wage increases due to existing contracts. In fact the marginal cost curve and the s.r.a.s.c1 describe points where there are no wage increases. In the long run labor ask for wage increases the s.r.a.s.c. shifts upwards to s.r.a.s.c.2. This proves that the long run aggregate supply curve is vertical at national output of full employment Y.

Liquidity trap

Figure 11 At liquidity trap the money demand curve (Md) is flat as money demand changes the rate of interest is the same r1. The money supply curve MS does not change with interest rate changes in fact the rate of interest does not have an effect on money supply it remains the same: 40. Suppose we have an increase in prices the real money supply decreases because real money supply= M/P= 40 where P is the price. If prices double then real money supply= M/2P= 20.
A change in prices does not change real money demand because the higher the prices the more money people will demand canceling each other out. Money demand= (M/P)d after doubling the price: (2M/2P)d= (M/p)d real money demand is the same.
From the previous graph the rate of interest is constant thus the LM curve is flat. Therefore national income does not change it remains the same Y1 as in the graph below.
Figure 12 Because the Y1 remains the same we have a vertical national income or aggregate demand curve. As price changes A.D. is the same:
Figure 13

In the short run a firm should produce if and only if total revenue is equal or greater than total variable cost. A firm to make a profit its total revenues must be greater than its total costs: T.R.>T.C.    Profit= Total revenues- Total costs
Total costs= Fixed costs + variable costs
Fixed costs: interest, rent, depreciation
Variable costs: labor, raw materials for the product
The fixed costs are there even if the output is zero the fixed costs exist.
In the short run the firm produces if the total revenues are equal or greater than the variable costs. Why? Even if: Total revenues- variable costs= $1- the firm subtracts this one dollar from the fixed costs. The fixed costs will be $1- less. The fixed costs always exist. Except when the firm closes down.
Therefore the firm to reduce its fixed costs produces where the total revenue is equal to or greater than the variable cost. At that point Total revenues= Total variable costs and at values where the total revenues are greater the firm produces and at that point onwards the marginal cost curve is the supply curve in perfect competition. Please bear in mind that
total revenues= price✕quantity produced
total variable cost= averange variable cost✕quantity produced
total revenues= total variable cost
Therefore price✕quantity produced= averange variable cost✕quantity produced
Therefore price= averange variable cost(A.V.C.)
Consequently, the firm produces where the price= averange variable cost or at a greater price. This applies to all the markets in the economy: perfect competition, monopoly, monopolistic competition, oligopoly.

Figure 14 As it was previously explained the supply curve is the red line-curve (please figure above). At price-output less than A the firms close down. The supply curve is a vertical line along the vertical-axis, price-axis at output 0 until point A. Then the supply curve is the curve from point A onwards.

Figure 15 Suppose wages, prices, costs, fall: A.V.C. falls (from A.V.C.1 to A.V.C.2) the M.C. also falls (from M.C.1 to M.C.2). When A.V.C. and M.C. fall output quantity increases, price falls. Supply curve shifts to the right, down (please, see the figure above). There is a vertical movement from A to B the supply curve moves vertically down from A to B the new supply curve is the red line curve. Please, note that at price less than B the supply curve moves from B to C. There is a horizontal movement from B to C.

Figure 16 At the above graph there is an asymmetry: point A the s.r.a.s.c. does not meet the A.D. curve. There is excess supply. When there is excess supply the prices fall. When prices fall below A the supply curve moves from point A to point B. What that means is that supply moves to zero, firms not having at least their average variable costs covered stop production they are closing down there is a movement: a arrow (please, see figure below). When firms are closing down unemployment increases demand for raw materials decrease. Wages and prices decrease therefore A.V.C. and M.C. fall there is a movement: b arrow. The result is a diagonal movement of the s.r.a.s.c. from point A to point C. At point C prices fell and output fell. This is what happened during the Great Depression in U.S.A. and to a lesser effect in Germany and Japan in 2009.
Figure 17

The problem with liquidity trap is that individuals do not deposit their money in the banks. The European Central Bank (ECB) injects money into the economy individuals take the money out of the economy by holding money as cash, money demand is flat please figure 11. Why is this happening? Europeans do not feel safe by depositing their money in the banks because the banks may default. What the ECB, Bank of England and Bank of Japan should do is to guarantee all bank deposits: in case a bank goes bankrupt the central bank will pay for the deposits and be legally bound to keep their word. Central banks have an unlimited supply of money. Secondly, even if all bank deposits are guaranteed individuals are indifferent of depositing their money because the rate of interest is very low. They do not forego any interest so they keep money as cash. Therefore, the interest rates should be increased this can be achieved by increasing the discount rate (the interest rate central banks charge the commercial banks for lending the commercial banks) and by increasing the funds rate (the interest rate banks charge each other). When banks need money they will sell bonds that they hold they will not borrow from the central bank or other banks because the rate of interest is higher. By selling bonds the price of bonds will go down the rate of interest will go up. There is the paradox how interest rates can increase (and they should be increased) and the economy and inflation grow? At a higher interest rate the opportunity cost of holding money increases individuals will deposit their cash injecting thus money into the economy. This will lead to banks using and/or lending money because they will have more cash from depositors and because the lending rate will increase. Banks will trade off between risk of loan and higher interest. I suggest the interest rate on deposits be driven to 3% even if selling bonds is required. As US interest rate rose there is an upward pressure on the interest rates in the rest of the world. Do not fight these upward pressures let the interest rates rise. Thus evading the liquidity trap. The above hold true for the Eurozone, United Kingdom and Japan. Also with E.U. citizens leaving the U.K. there has to be a way of increasing the labor force. I think this could be achieved if a tax is imposed on people who live with each other in the same house and only one is working and they do not have child/ren below 6 years of age. The tax should be an additional tax of 10% on the income of the person who is working. If both people work this tax does not apply. The money received could be given to married couples who both work for example in the form of increased marriage allowance (allowance: a sum to be deducted from gross income in the calculation of taxable income). This is an incentive for people to work and be married. It will increase the British labour force participation rate. British housewives will displace E.U. citizens in U.K. The classic measure in eliminating liquidity trap is a generous fiscal policy please see figure 12 IS curve shifts to the right. It proved to work in USA but not in Japan which ran generous fiscal policies. Also, if the government wants to reduce the expenses to achieve budget surplus the generous fiscal policy measure cannot be used.

In conclusion, the s.r.a.s.c. is a diagonal curve starting from low price and low national output (Y) and as Y increases it becomes steeper, at large Y there is a vertical asymptote because in the short run firms have a maximum capacity and because of diminishing returns and diseconomies of scale. The s.r.a.s.c is derived empirically and perfect competition, monopoly, monopolistic competition, oligopoly are taken into account. In the long run the s.r.a.s.c. shifts to the right because more buildings and machinery are used but the s.r.a.s.c. shifts back to the left because wages increase therefore the long run aggregate supply curve is vertical. An increase in money supply shifts the aggregate demand curve(AD) to the right but as wages increase the s.r.a.s.c. shifts to the left resulting in the same Y as before (Y) but with higher prices. At liquidity trap the AD is vertical and if it is low the s.r.a.s.c. does not intersect it leading to a reduction in both prices and Y. Now, if s.r.a.s.c. shifts to the right and long run supply curve shifts to the right because natural level of output increases then prices fall and Y is increased, please see figure below. This can be achieved if the labor is increased if the number of workers increases. The US government can increase labor if it taxes couples living in the same house and only one is working and they do not have child/ren below 6 years of age by decreasing the tax deduction from their adjusted gross income. The US government can increase the tax deduction of couples who both work and are married with a church wedding. This measure will increase the US labor force participation rate increasing the natural level of output Y decreasing the prices, please see figure below. How can productivity increase? The US government can introduce a measure that when an employer dismisses an employee the salary of the employee is deducted from the net profit before tax of the employer, the maximum duration of this is for a year and can apply if no other person is employed in a similar job of the dismissed employee. When a new employee takes the similar job this measure stops. This measure cannot be used if the period of the dismissal and employment is less than 3 months. This measure will reduce the number of unproductive employees and compel the employees to work more productively in order not to be dismissed.
Figure 18

Macroeconomics, N. Gregory Mankiw
Macroeconomics, Rudiger Dornbusch, Stanley Fischer, Richard Startz
Macroeconomic Theory and Policy, William H. Branson
Macroeconomics, Dernburg, McDougall
An introduction to Positive Economics, Richard G. Lipsey
Economics, David Begg, Stanley Fischer, Rudiger Dornbusch
Price theory and applications, Jack Hirshleifer

The author is a graduate from City.

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