Experiment 981130 ec 11

The first section of this discussion gives theory and equations. The second section is the solution to the equations. These numbers can be compared to the data that can be found in both the graphs and the history of the Marketscape program.

THEORY AND EQUATIONS

This is the way that the theory can be used to predict the prices and other activities in the 981130 ec 11 experiment. First note the symmetry of the problem. There are four types of agents, two consumer/resource suppliers and two types of producers. The economy of the experiment was created by replicating the parameters in sets of four. That is, each type was one fourth of the subjects. Thus the model can be applied by considering only four agents.

For the four person economy, the theory gives 18 unknowns and 19 equations. On might think that there are more equations than unknowns but the theory itself gives an explanation of the seeming paradox. In addition separate theories can be used to determine the price of bonds and the overall price level.

Consumers type 1maximize utility subject to budget constraint

Max 100Z_{1} - (9X_{1} + 2X_{1} ^{2} ) - ( 32 Y_{1} + 5/2 Y_{1}^{2} )

+l
_{1} (P_{z} Z_{1} - P_{y} Y_{1} - X_{1})

from which one gets the following equations (which can be used to derive demand curves).

1. 100 +l
_{1} P_{z} = 0

2. -9 - 4X_{1} - l
_{1} = 0

3. -32 - 5 Y_{1} - l
_{1} P_{y} = 0

4. P_{z} Z_{1} - P_{y} Y_{1} - X_{1} = 0

Consumers type 2 maximize utility subject to budget constraint

Max 100Z_{2} - (9X_{2} + 2X_{2} ^{2} ) - ( 30 Y_{2} + 5/2 Y_{2}^{2} )

+l
_{2} (P_{z} Z_{2} - P_{y} Y_{2} - X_{2})

from which one gets the following equations (which can be used to derive demand curves).

5. 100 +l
_{2} P_{z} = 0

6. -9 - 4X_{2} - l
_{2} = 0

7. -30 - 5 Y_{2} - l
_{2} P_{y} = 0

8. P_{z} Z_{2} - P_{y} Y_{2} - X_{2} = 0

The profit maximization hypothesis means that Producers type 1 buy resources to equate value of marginal product and use them to produce/supply output as defined by the production function. That gives three equations per producer.

9. Z_{3} = 2 X_{3}^{1/2} Y_{3}^{1/3}

10. P_{z} X_{3}^{-1/2} Y_{3}^{1/3} = 1

11. P_{z} 2/3 X_{3}^{1/2} Y_{3}^{-2/3} = P_{y}

Producers also want to use the profits to purchase items that they can enjoy. In this experiment they can purchase X and Y with the franc profits and then sell the X and Y in their private markets for dollars. We will call these new variables X_{3}^{c} and Y_{3}^{c} so suggest their role as consumption units. Parameters are such that producers will buy one unit of Y and all remaining profit will be spent on X and this fact is used as a shortcut to an expression of the equations. The consumption activities of Producer type 1 are captured by the following equation which says consumption of X is the total of profits minus expenditures on the one unit of Y used in consumption.

12. X_{3}^{c} = P_{z} Z_{3}- X_{3} - P_{y}(Y_{3} + 1)

The profit maximization hypothesis means that Producers type 2 buy resources to equate value of marginal product and use them to produce/supply output as defined by the production function. That gives three equations per producer.

13. Z_{4} = 2.3 X_{4}^{1/3} Y_{4}^{1/2}

14. P_{z} 2.3/3 X_{4}^{-2/3} Y_{4}^{1/3} = 1

15. P_{z} 2.3/2 X_{4}^{1/3} Y_{4}^{-1/2} = P_{y}

Producer type 2 has the same incentives as the other producer to use profit for consumption. Thus an additional equation is implied by the theory.

16. X_{4}^{c} = P_{z} Z_{4} - X_{4} - P_{y}(Y_{4} + 1)

Symmetry means that we need only consider one of each type of consumer and producer. We need to add only that quantities supplied equals quantities demanded.

17. X_{1} + X_{2} = X_{3} + X_{4} + X_{3}^{c} + X_{4}^{c}

18. Y_{1} + Y_{2} = Y_{3} + Y_{4} + 2

Where the 2 represents the consumption of 1 unit each of Y by producers.

19. Z_{1} + Z_{2} = Z_{3} + Z_{4}

Now list the number of variables. We have assumed that the P_{x} = 1. This can be done by convention since prices are only ratios and one price is used as the unit in which to do the accounting. The price of a dollar is one dollar. They are the four X's, the four Y's, the four Z's, the two prices and two l
's. That gives sixteen unknowns. In addition there are two X^{c} 's (the two Y^{c}'s are each 1) giving a total of 18 unknowns. It might seem that we have nineteen equations (one equation too many) but equations (4), (8), (12), (17), (18) and (19) have a dependency. One of these equations can be removed giving us a total of eighteen independent equations to solve for the eighteen unknowns.

Now, the money supply was increased during the experiment. Initially each agent had 300 francs. With the increase in the money supply the francs were increased to roughly 600 francs each. There were 32 people in the experiment in total. However, not all participated all of the time.

To compute absolute price levels the quantity theory of money can be used. Use MV=PO, where O means GDP . Find the solution to the above problem and compute

O = Z_{1} + Z_{2} plus the consumption of X and Y by producers. We will neglect the consumption by the producers here so the computation can be interpreted as potential GDP. Now there were 32 people with a money supply of 300 francs each giving a total money supply of 9600 francs during the first part of the experiment. After the money supply increased the quantity of money was approximately 18000 francs. Choose a price index where the system seems to have equilibrated and compute V. During the experimenter intervention the money supply almost doubled.

Bonds should trade at a price of 1/2 the price of X, or a little less depending upon risk aversion.

SOLUTION

The solution to the equations above are as follow. Remember that the prices are determined only up to a proportion of the price of X. So the price of X is 1 and the other prices are in terms of the markup over the price of X.

l
_{1} = -53.5732

l
_{2} = -53.5732

P_{x} = 1

P_{z} = 1.8666

P_{y}= 1.2991

X_{1} = 11.1433

X_{2} = 11.1433

X_{3} = 11.1387

X_{4} = 4.7576

X_{3}^{c} = 2.4138

X_{4}^{c} = 3.9766

Y_{1} = 7.5196

Y_{2} = 7.9196

Y_{3} = 5.7160

Y_{4} = 7.7231

Z_{1} = 11.2033

Z_{2} = 11.4817

Z_{3} = 11.9347

Z_{4} = 10.7503

What should be the absolute level of prices? In a four person economy the money supply is 1200 francs (300 francs each) and if the money supply is doubled the money supply is 2400. Gross domestic product (neglecting inventory changes, etc.) should be the final consumption levels of 23 units of Z (rounded to the nearest integer) plus 2 units of Y consumed by producers plus 6 units of X consumed by producers. Thus the monetary value of GDP is [23 + (P_{y}/P_{z} ) 2 + (P_{x}/P_{z} )6] P_{z} = [23 + (.68) 2 + (.53) 6] P_{z} . If the real value of GDP is indicated as O, the price level as P, the money supply as M and velocity of money is V, the quantity equation becomes

MV = OP or, 1200 V = [23 + (.68) 2 + (.53) 6] P_{z} = 28 P_{z} during the low money supply periods

and 2400 V = 28 P_{z} during the high money supply periods.

The value of V has ranged somewhat less than 3 in previous experiments. So, if the value of V is chosen as 3 then the predicted price levels would be approximately: P_{x} = 136, P_{y} = 174, P_{z} = 257 during the high money supply periods near the end of the experiment. Of course increases in V yield proportional increases in prices as does a change in the money supply if V is constant.

THE DATA

Price ratios can be studied. As can be seen in the graphs or prices, the ratios of prices are similar to those predicted by the model. Go to the web page. Log in as a viewer and click on "graph of all markets". Remember that the money supply almost doubled at about 18 or 19 hours into the experiment. This event is reflected in the rapid upward movement of all prices. This change in parameters was implemented by the experimenter purchasing units in all markets.

If estimates the price of X near the end of the experiment are on the order of 115 then the price of Y, which the model predicts to be 1.3 times the price of X, should be about 150. The price of Z, which the model predicts to be 1.9 times the price of X should be about 218. While there is substantial variance in the data these prices are clearly in the range of the data. Prices of X are around 115, prices of Y are somewhere over 130 and prices of Z seem to be in the 210 - 220 range. The price of bonds should be one half of the price of X, which is 57 if the price of X is 115.

Given the models that lie behind the version of the quantity theory introduced above, the V in the experiment must have been less than 3. Somewhere between 2.5 and 3 might be a better model. Of course, no real control existed over the subjects, who came and left the experiment over the extended time over which it operated. So, no real measurements of the actual money supply are reflected in the model. In addition subjects acquired inventories, which should be part of GDP and are missing from these calculations. Thus the error of the model might be due to data errors as well as shortcomings of the model itself.