Question

A few months ago Acme raised the price of the rice pudding from $1.19 to $1.29. As would be expected, sales fell a little, from an average of 535 per day to an average of 487 per day. After “doing the math”, the manager believes that the profit from selling the rice pudding actually increased, even though the number sold went down. He wants you to check his math, and recommend to him whether he should increase the price even further. The profit from selling the rice pudding is calculated by simply multiplying the number sold by the price per unit (this is the revenue) and then subtracting the cost of the rice pudding to the store (the cost). The cost of the rice pudding is 49 cents per unit.

2. The manager feels that if he raises the price by another 10 cents, the profit should increase by the same amount. Explain with a graph what this last statement means and why this amounts to expecting there to be a linear relationship between price and profit, and discuss whether or not you think it is reasonable to make this assumption. By thinking about what would happen if the store raises the price too high (no sales) or decreases it too much (what happens if the price drops to equal the unit cost?), make an argument that the actual function giving profit as a function of price should be concave down, and should therefore have a maximum.

(Profits increase at 1.29 and 1.39. Around 1.49 I found that profits began to dercrease, but I'm not sure what function I'd create along with the graph)

3. In order to find that maximum profit, proceed as follows. Instead of a linear relationship between profit and price, assume that there is a linear relationship between demand (the number of units sold per day, on average) and the price. Use the given data to find the linear model. Use this in turn to come up with a model for the profit as a function of price. Use calculus to find the price that gives the maximum profit, if your model turns out to have one. Don’t forget the usual check that you have indeed found a maximum

Answer 1

Let the first price of rice pudding be denoted by P₁, the next by P₂ and so on. Similarly, the quantities will be Q₁, Q₂ and so on. The Total revenue (TR₁) is given by P₁* Q_{1 .} Since per unit cost is $0.49, total cost is $0.49*Qi where i=1,2,3,..,n represents the number of times the variables are changing.

1. The formula for profit is given as, Profit = Total Revenue (Price * Quantity) – Total Cost Therefore, Profit₁ = ($1.19*535)-(535*0.49) = $374.5

                                   Profit₂ = ($1.29*487)-(487*0.49) = $389.6

Thus, the manager’s calculations are correct and profits have in fact increased after an increase in price even though there has been a fall in quantity.

2. Elasticity refers to the degree of responsiveness in quantity due to change in price. Elasticity of demand is expressed as: e_D = %change in Quantity demanded / % change in Price.

When the manager is expecting that a further price increase will bring about a same profit increase, he is assuming that the price elasticity of demand is unitary. This means that the % change in quantity demanded is equal to the % change in price.

When price changes from $1.19 to $1.29, the % change in price is ($1.29-$1.19/$1.19)*100 ~ 8.4%

Similarly,% change in quantity corresponding to those prices are: (535-487)/535 *100 ~ 8.9%

Therefore elasticity = 8.9%/8.4% = 1.05 ~ 1%.

Similarly, when Price increases from $1.29 to $1.39, % change in price is ~ 7.8%

When profit increases by the same amount, ($389.6- $374.5) = $15.1 then the new profit corresponding to price $1.39 is $389.6+ $15.1 = $404.70

Therefore if Quantity is Q then ($1.39Q – $0.49Q) = $ 404.7 or Q= 449.66~ 449.

Therefore % change in Quantity is (487-449)/487 * 100 ~ 7.8% which is equal to the % change in price. Therefore elasticity is 1.

Graphically, this is shown in Fig 1a and 1b where the price corresponding to unitary elastic price is graphed to get the profit function as the difference between TR and TC in fig 1c.

Since profits rose initially with increase in price, the relationship of price and profit was thought to be linear. The gap between TR and TC went on increasing with increase in price till the price was $1.39 and elasticity was 1.

However it is wrong to assume that with increase in price, profit will go on increasing and that price and profit have a linear relationship. In reality, if prices are increased further, profits start reducing. If one stops at $1.39 it might appear as if price and profit have a linear relationship. But going a little further, one can experience a decrease in profit levels.

When price is too high, there is no sale, and Q=0 (Fig 2a). This means that both TR and TC which are functions of Q will be 0 and as a result, profit (TR-TC) will be 0. Similarly, when price is so low that it equals to the per unit cost of production of $0.49, then profit will be (TR-TC= $0.49*Q- $0.49*Q) equal to zero. At this level, the firm just breaks even. (Fig 2b) However, in between the maximum price of say p’ and the minimum price of $0.49, the profit is at positive levels as was verified from part a). Thus the function HAS to be concave as shown in figure 2c, where the profit begins from 0, gradually increases to reach a maximum and eventually decreases again to level 0 and in fact negative if prices are increased further.

3. When there is a linear relationship between demand and price it can be expressed as

P=a-bQ where P is the price, Q is the quantity demanded (or the number of units sold per day on an average). Following the law of demand, price and demand have an inverse relationship. Therefore the demand curve is negatively sloped with slope= -b. Therefore Q= (a-P)/b or Q= a/b – P/b

Let cost be a function of Q as was stated above, having a form C=cQ where c is the per unit cost. Therefore cost = c(a/b – P/b)

Therfore

• TR= P*Q or TR= P(a/b – P/b) = aP/b – P²/b
• TC= c(a/b – P/b)
• Profit (P) = aP/b – P²/b - c(a/b – P/b)
• Or Profit (P) = aP/b – P²/b – ca/b + cP/b
• Therefore slope of profit function is dProfit/dP = a/b – 2P/b + c/b—(1)
• First order condition for profit maximization requires the slope of profit function = dProfit/dP = 0. Therefore, a/b – 2P/b + c/b = 0 or 2P/b= (a+c)/b or P= (a+c)/2 and Q= (a-c/2b) (By substituting the value of P in the demand function Q=a/b – P/b)
• If the profit function is concave then its curvature or d²Profit/dP² should be negative. This is also the Second order condition for profit maximization which requires that d²Profit/dP²<0 and validates the claim that the profit function is in fact concave.
• Therefore, Second order condition for profit maximization implies d²Profit/dP² = -2/b <0. (From equation 1)
• Thus profit is maximum at P= (a+c)/2 and corresponding quantity = (a-c)/2b