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Unformatted text preview: nd Gum. Assume the general production functions that follow: Gum = f(LGum) Chips = g(LChips) LGum + LChips = LTotal 20 where, LGum = the amount of labour used in the production of Gum. LChips = the amount of labour used in the production of Chips. LTotal = the fixed total amount of labour in the economy. We can generalize the equation of the PPF as: LTotal = L(Gum,Chips) and we know that any movement along the PPF does not change the total amount of labour, LTotal. Thus, LTotal = 0. The total change in labour can be broken down into the change due to labour movements in the Gum sector and changes due to labour movements in the Chips sector. (L / Gum) Gum + (L / Chips) Chips = 0 Observe that (L / Gum) refers to the change in labour as a result of a change in the output level of Gum. This is essentially the marginal cost of Gum (technically, not exactly correct). Similarly, (L / Chips) can be interpreted as the marginal cost of Chips. We re-write in terms of marginal costs and get: MCGum Gum + MCChips Chips = 0 MCGum Gum = - MCChips Chips MCGum / MCChips = -Chips Gum Recall that we defined MRT above (just before the squiggly line) as Chips !!!!! Gum Thus, MCGum / MCChips = MRT. Q.E.D. Now let's turn our attention to three popular economic functions Cobb-Douglas, Liontief, and CES (Constant Elasticity of Substitution). As we know, these functions can be applied to both utility functions (theory of consumption) and production functions (theory of production). 21 COBB-DOUGLAS FUNCTIONS The general form of a Cobb-Douglas utility function is defined as: U(x,y) = XY with positive constants, > 0 expenditure share of good X > 0 expenditure share of good Y > 0 scale constant Cobb-Douglas utility functions have the following marginal utilities: MUX = U X MUY = U Y Don't take my word for it...let's derive the Cobb-Douglas marginal utilities. To do this we must take the partial derivatives of the utility function with respect to X and Y, respectively. MUX = U X MUX = Y(X-1) M...
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- Spring '10