General Equilibrium - Production Economy PDF

14 General Equilibrium - Production Economy These notes extend the general equilibrium model introduced in the last notes to a model with production. This final model for the class combines many of the concepts we have seen so far. Concepts Covered The Production Economy The Product...

14 General Equilibrium - Production Economy These notes extend the general equilibrium model introduced in the last notes to a model with production. This final model for the class combines many of the concepts we have seen so far. Concepts Covered The Production Economy The Production Possibility Frontier The Rate of Product Transformation Profit Maximization with 2 Goods Equilibrium 14.1 The Production Economy Setup For this economy, we will keep the basic structure of the exchange economy but with some important differences. We will assume 1. 2 Goods: {x, y} 2. A single firm produces both goods using different production technologies for each and one input (labor) x = F (Lx ) y = F (Ly ) 3. The amount of labor available to the firm is fixed at some quantity L̄ 4. Firms sell their products to consumers at prices px and py 5. Consumers all have the same utility function, own equal shares the firms, and receive all profits that result from production (this assumption implies that total income to consumers is equal to the firm’s revenue). It’s worth thinking about the different implications of these assumptions compared to the exchange economy we discussed before. The biggest difference is that the supply of each good is no longer fixed. Instead, the firm will choose how much of each good to produce based on the prices of each good. Since we haven’t yet worked on a firm that can produce more than one product, we will first take a closer look at the firm’s problem. 1 14.2 The Production Possibility Frontier Definition To represent the tradeoff between producing the two goods, we will use a concept called the Production Possibility Frontier (PPF). Essentially what the PPF represents is the set of possible combinations of x and y that can feasibly be produced by the firm. Assuming the firm uses all labor available, we can write L̄ = Lx + Ly Our goal is to get a function that relates x and y. In other words, we need to eliminate Lx and Ly from the equations above using the production functions for x and y. Example This idea is best illustrated with an example. Let’s assume the firm has technologies given by x = L1/2 x y = L1/2 y We can use the production functions to write Lx = x2 Ly = y 2 And plug these into the labor supply equation L̄ = x2 + y 2 Solve for y
1/2 y = L̄ − x2 This function is the PPF. Below is a plot of the PPF with L̄ = 100. 12 10 8 y 6 4 2 0 0 2 4 6 8 10 12 x 2  Any point in the bottom left (inside the curve) can be produced by the firm given the amount of labor in the economy. If they use all of the labor on good y, so x = 0, then clearly we would have y = 1001/2 = 10 x=0 And the opposite if all labor was used producing x. Notice that as we produce more of a good, the opportunity cost of producing that good increases. In other words, to produce the first unit of x, we have to give up very little y, but to go from 9 units of x to 10, we have to give up a lot more. This result is directly related to the functions we chose for our production functions. Let’s look at the marginal product of labor for each good to see why. ∂x M P Lx = = 1/2L−1/2 x ∂Lx ∂y M P Ly = = 1/2L−1/2 y ∂Ly Both marginal products are decreasing as we increase the amount of labor on that good. The first unit of labor for each good is very efficient but each subsequent worker is less and less productive. This property is what gives the PPF its concave shape. 14.3 The Rate of Product Transformation To formalize the idea above, we can solve directly for the slope of the PPF. We will call the negative of the slope of the PPF the Rate of Product Transformation (RPT). If we have already solved for y as a function of x, we could just directly take the derivative, but there is a slightly more intuitive way to do it. In particular, let’s look at the equation L̄ = Lx + Ly = x2 + y 2 we can think of this as representing the total cost the firm has to pay for production (at least proportional to the cost - it would have to be multiplied by a wage, but as long as workers all receive the same wage we won’t need to worry about it for now). Because of this interpretation, we will label the denote the right hand side of the function above as C(x, y). Since the total amount of labor is always fixed at L̄, the firm’s cost is also fixed. In this way, the firm’s PPF is similar to an indifference curve for the consumer’s problem. In that case, we found all combinations of x and y that gave the same utility to the consumer. Here, the PPF is showing all the combinations of x and y that have the same cost to the firm. Similarly, in the same way that the MRS related the marginal utilities of x and y, the RPT will be the ratio of the marginal costs of the two goods ∂C(x,y) M Cx ∂x RP T = = ∂C(x,y) M Cy ∂y In the example above, since our cost (written in terms of x and y) is given by x2 + y 2 , we have 2x x RP T = = 2y y 3 The proof that this is equal to the (negative) slope of the PPF is similar to the proof that the MRS is the slope of an indifference curve, but we can see that it appears to match the graph. When x = 0 and y = 10 the slope of the PPF is 0 and the RPT is 0 as the cost of x relative to y is very small. As y goes to 0 and x goes to 10 the slope goes to negative infinity as the cost of x relative to y gets very large. As we will see below when we connect the PPF to a profit maximization problem for the firm, it will be important that the PPF is concave, which means the RPT is getting larger as we increase the ratio of x to y. Note that this is the opposite of what we wanted for an indifference curve, where we wanted it to be convex for the utility maximization problem. Does it make sense for a PPF to be concave? There is an intuitive explanation for why it would be. Imagine that workers are specialized - some workers are better at producing x and others are better at producing y. Then when we want to produce a small amount of x, we can take only the workers who are very good at producing x and not so good at producing y. But as we increase production of x, we will start having to employ workers who aren’t as good at producing x and force them to start producing the good even though they aren’t very efficient. As a result, it becomes harder and harder to produce x the more x the firm produces. 14.4 Profit Maximization with 2 Goods As in our earlier discussion of firms, we will assume that firms aim to maximize profits. However, there are a couple major differences between the problem here and the problem in the earlier profit maximization section. The first is that firms are now maximizing profits over 2 goods. This means that the firm’s revenue is still price times quantity but now summed up across goods R(x, y) = px x + py y The second difference is that the firm is constrained by the amount of labor in the economy. They cannot produce more than the production possibility frontier. We will always assume that the firm uses all of the labor available to it (labor is like a fixed cost in this model) so their cost will be the same no matter which point along the PPF they choose. If the firm uses all of this labor and wages are taken as given, the firm always has the same cost regardless of which good they actually use the labor to produce (and regardless of what the wage is, which is why we don’t have to worry about it here). Since cost is always the same along the PPF, maximizing profit is equivalent to maximizing revenue, with the constraint that we have to be on the PPF. We will always try to write the constraint in the form L̄ = C(x, y) Where again we use C to capture the idea described above that the function represents the firm’s cost of production. So our full problem is max px x + py y x,y subject to L̄ = C(x, y) 4 Which we can set up as a Lagrangian
L = px x + py y + λ L̄ − C(x, y) Taking our first order conditions with respect to x and y, ∂L ∂C(x, y) ∂C(x, y) = px − λ = 0 =⇒ px = λ ∂x ∂x ∂x ∂L ∂C(x, y) ∂C(x, y) = px − λ = 0 =⇒ py = λ ∂y ∂y ∂y Taking the ratio of these two conditions gives us ∂C(x,y) px ∂x = ∂C(x,y) = RP T py ∂y Which tells us that the price ratio should be equal to the ratio of marginal costs, or the RPT. 14.5 Equilibrium We are finally ready to bring it all together. To recap, above we got the result that firms maximize profits we must have px = RP T py Since the consumer problem is standard, we know that for a consumer to be maximizing utility we must have px = M RS py As in the exchange economy, equilibrium will imply optimization and market clearing. In this case, that means that firms must be maximizing profits and consumers must be maximizing utility, so both of the above equations must hold at once. In other words, in equilibrium, we should have px = RP T = M RS py This condition guarantees that firms are maximizing profits and consumers are maximizing utility. In general, this condition will give us a relationship between x and y that must hold in equilibrium. The market clearing conditions can be a bit trickier here than in the exchange economy because now the firms decisions change the supply of both x and y as well as the consumer’s income (recall from above that in this model the consumer’s income is equal to the firm’s revenue). Rather than write out the market clearing conditions explicitly, we will instead use the production possibility frontier to find a feasible point that also satisfies the condition above. It turns out that this will be sufficient for market clearing and we will show that this is true in the example below. 5 Example Assume consumers have utility function U (x, y) = x1/2 y 1/2 And firms have production functions x = L1/2 x y = L1/2 y Assume there are 200 workers available for the firm to use in production (L̄ = 200). Let’s find the equilibrium. On the consumer side we have y M RS = x On the firm side, we first need to find the PPF. Inverting the production functions gives us Lx = x2 Ly = y 2 Setting these equal to the total labor supply gives 200 = x2 + y 2 This is the PPF. Now we can find the RPT x RP T = y Using the equilibrium condition x y M RS = RP T =⇒ = =⇒ x2 = y 2 y x Plug this into the PPF equation 200 = x2 + y 2 = y 2 + y 2 = 2y 2 =⇒ y 2 = 100 =⇒ y = 10 And therefore x = 10 and ppxy = M RS = RP T = 1. Once again we can only solve for the ratio of prices, but we can always set one of the prices to 1. so we could say px = 1, py = 1 are equilibrium prices. Now let’s make sure this result actually clears the market. In other words, does the consumer actually demand 10 units of x and 10 units of y when the price of each good is equal to 1? We know that the Marshallian demand given the utility function is 1 I 1I x∗m = ∗ ym = 2 px 2 py The consumer’s income is equal to the firm’s revenue so I = (1)(10) + (1)(10) = 20 Plugging this result into Marshallian demands 1 20 1 20 x∗m = = 10 ∗ ym = = 10 2 1 2 1 Therefore demand for each good is equal to supply and the market clears. 6  The graph below shows what this equilibrium looks like (note that the line represents both the consumer’s budget constraint and the firm’s revenue). Note that at the equilibrium point, the firm’s revenue line is tangent to the PPF (which is true whenever RP T = px /py ) and the consumer’s indifference curve is tangent to the budget constraint (M RS = px /py ) 20 15 y 10 5 0 0 5 10 15 20 x 7