ECON-3020 Lecture Notes - Lecture 2: Production Function, Marginal Product, Autarky
Topic 2: Production and Income
❖ Value added and production
➢ Notation
■ Y=output, what is produced
■ K=capital
■ L=labor
■ A=technology, how much you can get per K or L
➢ Production function: Y=F(A,K,L)
■ We will no longer write A as an argument for the production function since
A is assumed constant
❖ The production function
➢ Y=F(K,L)
➢ How much ice cream can we make if we use L workers and K ice cream
machines?
❖ Cobb-Douglas production function
➢ Y=F(K,L)=AKαL1-α
➢ Constant returns to scale: twice as much of every input yields twice as much
output
❖ Firm’s problem
➢ profit=revenue-labor cost-rental cost
➢ profit=PY-WL-RK or profit=PAF(K,L)-WL-RK where P is the price, W is the
worker, and R is the price of a machine
➢ Assume that firms take prices as given (competitive market), so firms maximize
profits by choosing L and K.
■ Choose optimal level of capital by maximizing profit with respect to K,
treating L fixed
● Differentiate profit function with respect to K and set it equal to
zero
●
●
where
is the real rental rate of capital
■ Choose optimum level of labor by maximizing profit with respect to L,
treating K fixed
● Same differentiation for labor, with respect to L
● MPL=
where
● Capital demand by firm=MPK=r
● Labor demand by firm=MPL=w
● Firm sets marginal product of labor equal to the price of labor
➢ Plugging in the Cobb-Douglas function
■ Capital demand by firm MPK=
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Document Summary
A=technology, how much you can get per k or l. We will no longer write a as an argument for the production function since. Constant returns to scale: twice as much of every input yields twice as much output. Firm"s problem worker, and r is the price of a machine. Profit=py-wl-rk or profit=paf(k,l)-wl-rk where p is the price, w is the. Assume that firms take prices as given (competitive market), so firms maximize. Choose optimal level of capital by maximizing profit with respect to k, profits by choosing l and k. treating l fixed. Choose optimum level of labor by maximizing profit with respect to l, zero. Differentiate profit function with respect to k and set it equal to. (cid:1839)(cid:1837)=(cid:4666)(cid:3012),(cid:3013)(cid:4667) (cid:3012) =(cid:1870)where (cid:1870)=is the real rental rate of capital (cid:3012) =(cid:1875)where (cid:1875)=(cid:1871) (cid:1872) (cid:1857) (cid:1870)(cid:1857)(cid:1853)(cid:1864) (cid:1875)(cid:1853)(cid:1859)(cid:1857) Firm sets marginal product of labor equal to the price of labor. Same differentiation for labor, with respect to l treating k fixed.
