Economics of the Firm Consumer Demand Analysis

Demand relationships are based off of the theory of consumer choice. We can characterize the average consumer by their utility function. “Utility” is a function of lemonade and hot dogs Consumers make choices on what to buy that satisfy two criteria: Their decision on what to buy generates maximum utility Their decision on what to buy generates is affordable These decisions can be represented by a demand curve

Example: Suppose that you have $10 to spend. Hot Dogs cost $4 apiece and glasses of lemonade cost $2 apiece. # Hot DogsMU (Hot Dogs) # LemonadeMU (Lemonade) This point satisfies both conditions and, hence, is one point of the demand curve

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend) # Hot DogsMU (Hot Dogs) # LemonadeMU (Lemonade) Your decision at the margin has been affected. You need to buy less hot dogs and more lemonade (Substitution effect) You can’t afford what you used to be able to afford – you need to buy less of something! (Income effect)

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend) # Hot DogsMU (Hot Dogs) # LemonadeMU (Lemonade) This point satisfies both conditions and, hence, is one point of the demand curve

Demand curves slope downwards – this reflects the negative relationship between price and quantity. Elasticity of Demand measures this effect quantitatively Quantity Price $ $6.00 1

Arc Elasticity vs. Price Elasticity At the heart of this issue is this: “How do you calculate a percentage change? Suppose that a variable changes from 100 to 125. What is the percentage increase? - OR - Note: This discrepancy wouldn’t be a big deal if these two points weren’t so far apart!

Consider the following demand curve: Quantity Price $ $ Arc Elasticity

Consider the following demand curve: Quantity Price $ $ Point Elasticity

If demand is linear, the slope is a constant, but the elasticity is not!! Quantity Price $ $ $

If demand is linear, the slope is a constant, but the elasticity is not!! Quantity Price $ $ $ Low prices = Low Elasticities High prices = High Elasticities Unit Elasticity

If you are interested in maximizing revenues, you are looking for the spot on the demand curve where elasticity equals 1. Revenues Price

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend. # Hot DogsMU (Hot Dogs) # LemonadeMU (Lemonade) Your decision at the margin is unaffected, but you have some income left over (this is a pure income effect)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend. # Hot DogsMU (Hot Dogs) # LemonadeMU (Lemonade) This point satisfies both conditions and, hence, is one point of the demand curve

For any fixed price, demand (typically) responds positively to increases in income. Income Elasticity measures this effect quantitatively Quantity Price $

Cross price elasticity refers to the impact on demand of another price changing Quantity Price $ Note: These numbers aren’t coming from the previous example!! A positive cross price elasticity refers to a substitute while a negative cross price elasticity refers to a compliment

Time Demand Factors t t+1t-1 Cross Sectional estimation holds the time period constant and estimates the variation in demand resulting from variation in the demand factors For example: can we predict demand for Pepsi in South Bend by looking at selected statistics for South bend

Estimating Cross Sectional Demand Curves Lets begin by estimating a basic demand curve – quantity demanded is a function of price. Next, we need to assume a functional form. For simplicity, lets start with a linear model

Price Quantity Next, Collect Data on Prices and Sales

Regression Results VariableCoefficientStandard Errort Stat Intercept Price (X) That is, we have estimated the following equation Regression Statistics R Squared.782 Standard Error Observations250

Values Price of X$2.50 Average Price of X $5 (3.004)(0.774)(10.02) Our forecast of demand is normally distributed with a mean of 23 and a standard deviation of 9.90.

If we want to calculate the elasticity of our estimated demand curve, we need to specify a specific point. $

Given our model of demand as a function of income, and prices, we could specify a variety of functional forms Linear Demand Curves Here, quantity demanded responds to dollar changes in price (i.e. a $1 increase in price lowers demand by 4 units.

Given our model of demand as a function of income, and prices, we could specify a variety of functional forms Semi Log Demand Curves Here, quantity demanded responds to percentage changes in price (i.e. a 1% increase in price lowers demand by 4 units.

Given our model of demand as a function of income, and prices, we could specify a variety of functional forms Semi Log Demand Curves Here, percentage change in quantity demanded responds to a dollar change in price (i.e. a $1 increase in price lowers demand by 4%.

Given our model of demand as a function of income, and prices, we could specify a variety of functional forms Log Demand Curves Here, percentage change in quantity demanded responds to a percentage change in price (i.e. a 1% increase in price lowers demand by 4%. Log Linear demands have constant elasticities!!

One Problem Suppose you observed the following data points. Could you estimate a demand curve? D

Estimating demand curves Market prices are the result of the interaction between demand and supply!! A problem with estimating demand curves is the simultaneity problem. D S

Estimating demand curves Case #1: Both supply and demand shifts!! D’’ S’ S S’’ D’ D D S’ S S’’ Case #2: All the points are due to supply shifts

An example… Supply Demand Equilibrium Suppose you get a random shock to demand The shock effects quantity demanded which (due to the equilibrium condition influences price! Therefore, price and the error term are correlated! A big problem !!

Suppose we solved for price and quantity by using the equilibrium condition

We could estimate the following equations The original parameters are related as follows: We can solve for the supply parameters, but not demand. Why?

S D D D By including a demand shifter (Income), we are able to identify demand shifts and, hence, trace out the supply curve!!

Time Demand Factors t t+1t-1 Time Series estimation holds the demand factors constant and estimates the variation in demand over time For example: can we predict demand for Pepsi in South Bend next year by looking at how demand varies across time

Time series estimation leaves the demand factors constant and looks at variations in demand over time. Essentially, we want to separate demand changes into various frequencies Trend: Long term movements in demand (i.e. demand for movie tickets grows by an average of 6% per year) Business Cycle: Movements in demand related to the state of the economy (i.e. demand for movie tickets grows by more than 6% during economic expansions and less than 6% during recessions) Seasonal: Movements in demand related to time of year. (i.e. demand for movie tickets is highest in the summer and around Christmas

Suppose that you work for a local power company. You have been asked to forecast energy demand for the upcoming year. You have data over the previous 4 years: Time PeriodQuantity (millions of kilowatt hours) 2003: : : : : : : : : : : : : : : :419

First, let’s plot the data…what do you see? This data seems to have a linear trend

A linear trend takes the following form: Forecasted value at time t (note: time periods are quarters and time zero is 2003:1) Time period: t = 0 is 2003:1 and periods are quarters Estimated value for time zero Estimated quarterly growth (in kilowatt hours)

Regression Results VariableCoefficientStandard Errort Stat Intercept Time Trend Regression Statistics R Squared.53 Standard Error 1.82 Observations16 Lets forecast electricity usage at the mean time period (t = 8)

Here’s a plot of our regression line with our error bands…again, note that the forecast error will be lowest at the mean time period T = 8

Sample We can use this linear trend model to predict as far out as we want, but note that the error involved gets worse!

Time PeriodActualPredictedError 2003: : : : : : : : : : : : : : : : One method of evaluating a forecast is to calculate the root mean squared error Number of Observations Sum of squared forecast errors

Lets take another look at the data…it seems that there is a regular pattern… Q2 We are systematically under predicting usage in the second quarter

Time PeriodActualPredictedRatioAdjusted 2003: (.87)= : (1.16) = : (.91) = : (1.04) = : (.87) = : (1.16) = : (.91) = : (1.04) = : (.87) = : (1.16) = : (.91) = : (1.04) = : (.87) = : (1.16) = : (.91) = : (1.04) = Average Ratios Q1 =.87 Q2 = 1.16 Q3 =.91 Q4 = 1.04 We can adjust for this seasonal component…

Now, we have a pretty good fit!!

Recall our prediction for period 76 ( Year 2022 Q4)

We could also account for seasonal variation by using dummy variables Note: we only need three quarter dummies. If the observation is from quarter 4, then

Regression Results VariableCoefficientStandard Errort Stat Intercept Time Trend D D D Regression Statistics R Squared.99 Standard Error.30 Observations16 Note the much better fit!!

Time PeriodActualRatio MethodDummy Variables 2003: : : : : : : : : : : : : : : : Ratio Method Dummy Variables

A plot confirms the similarity of the methods

Recall our prediction for period 76 ( Year 2022 Q4)

Recall, our trend line took the form… This parameter is measuring quarterly change in electricity demand in millions of kilowatt hours. Often times, its more realistic to assume that demand grows by a constant percentage rather that a constant quantity. For example, if we knew that electricity demand grew by g% per quarter, then our forecasting equation would take the form

If we wish to estimate this equation, we have a little work to do… Note: this growth rate is in decimal form If we convert our data to natural logs, we get the following linear relationship that can be estimated

Regression Results VariableCoefficientStandard Errort Stat Intercept Time Trend Regression Statistics R Squared.54 Standard Error.1197 Observations16 Lets forecast electricity usage at the mean time period (t = 8) BE CAREFUL….THESE NUMBERS ARE LOGS !!!

The natural log of forecasted demand is Therefore, to get the actual demand forecast, use the exponential function Likewise, with the error bands…a 95% confidence interval is +/- 2 SD

Again, here is a plot of our forecasts with the error bands T = 8

When plotted in logs, our period 76 ( year 2022 Q4) looks similar to the linear trend

Again, we need to convert back to levels for the forecast to be relevant!! Errors is growth rates compound quickly!!

QuarterMarket Share Consider a new forecasting problem. You are asked to forecast a company’s market share for the 13 th quarter. There doesn’t seem to be any discernable trend here…

Smoothing techniques are often used when data exhibits no trend or seasonal/cyclical component. They are used to filter out short term noise in the data. QuarterMarket Share MA(3)MA(5) A moving average of length N is equal to the average value over the previous N periods

The longer the moving average, the smoother the forecasts are…

QuarterMarket Share MA(3)MA(5) Calculating forecasts is straightforward… MA(3) MA(5) So, how do we choose N??

QuarterMarket Share MA(3)Squared Error MA(5)Squared Error Total = Total = 62.48

Exponential smoothing involves a forecast equation that takes the following form Forecast for time t+1 Actual value at time t Forecast for time t Smoothing parameter Note: when w = 1, your forecast is equal to the previous value. When w = 0, your forecast is a constant.

QuarterMarket Share W=.3W= For exponential smoothing, we need to choose a value for the weighting formula as well as an initial forecast Usually, the initial forecast is chosen to equal the sample average

As was mentioned earlier, the smaller w will produce a smoother forecast

Calculating forecasts is straightforward… W=.3 W=.5 So, how do we choose W?? QuarterMarket Share W=.3W=

QuarterMarket Share W =.3Squared Error W=.5Squared Error Total = 87.19Total = 101.5